Politecnico di Torino
First School of Engineering
M. Sc. Course in Mathematical Modelling in Engineering
Reduced-Basis Approximation and
A Posteriori Error Estimation
for Saddle-Point Problems
Author:
Lorenzo Zanon
Main Supervisor:
Prof. Claudio Canuto
Politecnico di Torino
Supervisor:
Prof. Karen Veroy-Grepl
AICES - RWTH Aachen
Co-Supervisor:
Prof. Alfio Quarteroni
Politecnico di Milano, EPFL Lausanne
This work has been carried out at the Aachen Institute for Advanced Study in
Computational Engineering Science (AICES, RWTH University Aachen) to which I am
indebted for providing hard- and software, and technical support. I would like to thank
my advisor at AICES, Prof. Karen Veroy-Grepl, for the precious and constant help.
i
Abstract
The Stokes problem is a saddle point problem. A stable discretization enables applying classical results from the Brezzi and Babuška Theories in order to compute a
posteriori upper bounds for the approximation error. The error can be the norm of the
difference between the exact solution and the Galerkin Finite Elements discretization
or - in the case of this work - between the FE and the Reduced Basis approximation.
The RB approximation allows a quick computation of the problem solution accordingly
to the variation of geometric parameters. A correct implementation of the RB method
requires a verification of the stability: in particular the role of the supremizer operator
is analyzed. A rigorous error bound is required for enhancing the application of the RB
method and the motivation for a research of a Brezzi-based error bound relies on the
possibility of efficiently computing lower or upper bounds for the involved constants.
The classical Babuška approach for variational problems provides the error bound for
the velocity and the pressure jointly, whereas the Brezzi approach deals with the two
quantities alone. This does not necessarily mean a sharper and improved bound. Numerical tests show that for reasonable (O(1)) values of the ratio of the dual norms of the
momentum and continuity equations residuals, the Brezzi error is sharper (by a factor
10) than the Babuška error only for the velocity. In a parametrized domain, e.g. a
channel with a rectangular obstacle, the behaviour of the Brezzi error bound gets worse
as the parameters assume values which are far from the reference ones.
ii
Contents
1 Introduction
1
2 Variational saddle point problems and their discretization
2.1 Preliminar ideas . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A bilinear form: an example of a priori bound . . . . . . . . .
2.3 An abstract saddle point problem . . . . . . . . . . . . . . . .
2.4 Derivation of a saddle point problem . . . . . . . . . . . . . .
2.5 The Stokes problem on a parametrized domain . . . . . . . .
3 The
3.1
3.2
3.3
3.4
3.5
3.6
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RB Approximation and the Stokes problem as application
Truth FE Approximation . . . . . . . . . . . . . . . . . . . . . . .
RB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Building stable RB spaces . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The supremizer . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Options for the RB spaces . . . . . . . . . . . . . . . . . . .
Comparison among the enrichment options . . . . . . . . . . . . .
A priori convergence for the RB Approximation . . . . . . . . . . .
Appendix: computation of the dual norms of the residuals . . . . .
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4 A posteriori error estimation
4.1 Babuška and Brezzi Theories . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Babuška’s combined bound . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Brezzi’s separated bounds . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 General ideas for the proof of the Closed Range Theorem for linear
tinuous operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Comparison between Babuška and Brezzi bounds . . . . . . . . . . . . . . .
4.3 Applications and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The error equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 The channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Appendix: the Penalty Approach . . . . . . . . . . . . . . . . . . . . . . . .
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Conclusion
87
Bibliography
88
iii
Notation and Abbreviations
N, R, Rn
a
<(a), =(a)
U/M
kvkV
kBk
L(X, Y )
X∗
hx0 , xiX ∗ ×X
(u, w)V
u◦v
Tt
ker A
Im(A)
C(Ω)
natural numbers, real numbers, n-dimensional Euclidean space
complex conjugate of a ∈ C
real and imaginary parts of a ∈ C
quotient space of U over the closed subspace M
norm of v in the normed space V
norm of the continuous operator B
space of bounded linear operators between Banach spaces X and Y
dual space of the Banach space X
duality product of x0 ∈ X ∗ and x ∈ X
inner product of u and w in the Hilbert space V
Euclidean product between vectors in Rn
adjoint operator T t ∈ L(Y ∗ , X ∗ ) of T ∈ L(X, Y )
kernel of the operator A
range of the operator A
continuous functions in Ω
Wpk (Ω)
Sobolev space of functions with k-th order weak derivatives in Lp (Ω)
H k (Ω)
Hilbert space W2k (Ω)
H0k (Ω)
space of the functions in H k (Ω) which vanish on the boundary of Ω
H −k (Ω)
dual space of H0k (Ω)
∇J, HJ
sup ϕ(v)
gradient and Hessian of the scalar function J
supremum of ϕ(v): if v is a factor in the denominator, then v = 0 is excluded
v∈V
(the same for the inf)
span{vi ∈ V, i ∈ I}
space generated by the linear combinations over the field R of vi ∈ V, i ∈ I
RB
FE
PDE
ODE
b.c.
Reduced Basis
Finite Element
Partial Differential Equations
Ordinary Differential Equations
boundary conditions
The Einstein notation is adopted: sum over repeated indices.
Other notations are defined when used for the first time.
iv
Chapter 1
Introduction
The subject of this work is the application of the RB Method to a saddle point problem
(in particular the Stokes problem) and the research of a suitable a posteriori upper
bound for the error committed through the RB approximation, in the perspective of an
optimization of the method itself.
Chapter 2 of this thesis deals with the description of the stability conditions for a
saddle point problem and their application in a Galerkin FE approximation framework.
The Stokes problem is then presented as a saddle point problem. In addition, the
considered problem is parametrized: the domain of interest is a rectangle with an obstacle
of varying dimensions (in the perspective of an e.g. microfluidics application). The goal is
not a precise solution for a particular parameter, but an overview including the possibility
of comparing many solutions at different parameters. Thus, the FE approximation is
considered as the truth solution and another suitable discretization tool is examined.
In Chapter 3 the RB Method is introduced. It was applied at first to linear and
non-linear structural analysis problems, before expanding to many classes of PDE (and
ODE) problems, different spaces of approximation (Taylor, Lagrange, Hermite) and
consequently different applications. It consists in computing the solution of the problem
as a linear combination of truth solutions computed at some sample parameters: the loss
of accuracy is compensated by the gain of computational time. While the pre-processing
offline stage depends on N , the (usually big) dimension of the FE space, the online
phase only depends on N , the low dimension of the RB space. When each component
of the problem is affine in the parameters, then we can easily separate the parameter
-dependent and -independent terms of the problem and easily deduce the offline-online
procedure.
The motivation for this work is therefore a contribution to exploring how to apply
the RB method in an efficient and accurate way, taking the Stokes problem as a case
study. First of all, there are stability issues. A RB solution is good if the stability
requirements in the exact spaces are also fulfilled in the approximation: that is why
Babuška and Brezzi dissertations on the stability of variational problems and saddle
point problems must be invoked. The conditions (continuity, coercivity, inf-sup constant)
are tested on the RB spaces and an investigation is carried out in order to determine
1
whether these conditions are actually necessary. Specifically, the role of the supremizer
operator in this context will be highlighted.
The nature of the initial applications of the RB Method, but above all the lack
of a good a posteriori error estimation and of proper stability considerations were the
reasons for the RB Method to have limitations in its first stage. Since the RB spaces
depend on truth solutions at some sample parameters, an inexpensive a posteriori error
bound is crucial to properly choose the next sample in the basis, e.g. through a Greedy
Algorithm, and know when the basis is sufficiently rich. That means, at each enrichment
step, a good estimation (depending on the approximated solution itself) of the error
between the exact solution and the RB approximation should be computed over many
test parameters. The parameter which maximizes the error is then added to the basis.
Chapter 4 therefore deals with the research of a suitable a posteriori error bound:
the estimation arising from the Babuška theory is widely recognized as trustworthy for
all non-coercive problems, but it is not efficient. The possible adavntages of applying
Brezzi’s rather than Babuška’s theory are investigated instead.
Wrapping everything up, these topics are going to be faced:
• theory of saddle point problems;
• description of the offline-online RB procedure in an affine parameter dependence
framework;
• stability considerations for the RB spaces, following from the Brezzi theory for
saddle point problems;
• a posteriori error estimation, in particular comparing the role that Babuška and
Brezzi constants play.
These subjects will not be discussed:
• non-affine parameter dependence;
• an optimal sampling strategy: random choice of the parameters entering the basis
is always assumed;
• an N -only-dependent a posteriori error estimation: there are widely known techniques (e.g. Successive Constraint Method) which can be applied to get N independent bounds for the Brezzi constants, but additional work would be needed.
It can nevertheless be interesting to provide an example where the last two points
are taken into account, following the work in [1], as we will see at the end of Chapter 4.
A penalty approach for the Stokes problem is exploited in order to build an a posteriori
error bound which does not involve the use of the inf-sup constant. Although the SCM is
a very efficient method for computing inexpensive upper and lower bounds of continuity
and coercivity constants, a lower bound for the inf-sup constant requires a greater effort,
see for instance [14]. Circumventing this computation can therefore be a great advantage.
2
This introductory part is concluded with some original concepts of Babuška and
Brezzi and a short introduction of the Stokes problem.
Relevant steps forward in the theory of the approximation of variational problems
have been made with the articles [5] and [6], which came out respectively in 1971 and
1974. The first one by Babuška deals with variational problems applied to bilinear forms
in Hilbert spaces:
u ∈ H1 : B(u, v) = f (v) , ∀v ∈ H2 .
(1.1)
Brezzi instead considers saddle point problems, in terms of the bilinear forms a(·, ·)
and b(·, ·):
a(u, v) + b(v, ψ) = hf, viV ∗ ×V , ∀v ∈ V ,
= hg, ϕiW ∗ ×W , ∀ϕ ∈ W .
b(u, ϕ)
The first issue is finding a solution in the exact - or continuous - context. Wellposedness of the problems is guaranteed under some conditions involving the bilinear
forms. A continuous dependence over the initial data is provided, as a consequence of
the Generalized Lax-Milgram Theorem.
The following step is the discretization of the problems. A solution is looked for in
closed subspaces of the initial Hilbert spaces: we remark that transferring the stability
conditions from the exact case to the discretized one is not always trivial. In particular,
a Galerkin or Petrov-Galerkin FE approximation is chosen and an error with respect
to the exact solution must be evaluated. Babuška gives, for the bilinear form problem,
an upper bound for the norm of the difference between the exact solution and the FE
approximation in terms of the best approximation error and on other problem-related
constants:
0
ku − uh kH1 ≤ σh
inf ku − vh kH1 , u ∈ H1 ;
vh ∈H1h
the same procedure can be applied to the saddle point problem:
00
ku − uh kV + kψ − ψh kW ≤ σh inf ku − vh kV + inf kψ − ϕh kW ,
vh ∈Vh
ϕh ∈Wh
u ∈ V, ψ ∈ W ,
with σh0 > 0 and σh00 > 0. These are both a priori error bounds, which means that the
actual approximated solution does not appear on the right-hand side of the estimation.
In this work the theory of saddle point problems is applied for the particular case
of the Stokes problem, which describes a steady-state, low velocity incompressible flow.
The Stokes problem is obviously a restricted case of the Navier-Stokes Equations for an
incompressible flow. We can give a quick glimpse to the state of the art for the resolution
of NS equations:
• if the domain Ω has dimension d = 2, there exists a unique solution over the whole
considered time-interval (0, T ];
• if d = 3:
3
– there exists a weak solution over (0, T ], but uniqueness has not been proved
yet;
– a solution which is more regular - in time and space - than the weak one is
unique, but existence has not been proved yet;
– there exists a unique strong solution of the problem with small initial data
(in an interval (0, τ ] ⊂ (0, T ], with low Reynlods number, . . . )
4
Chapter 2
Variational saddle point problems
and their discretization
2.1
Preliminar ideas
Variational problems and saddle point problems and their discretization are going to be
analyzed in this Chapter.
An example of bilinear form variational problem like equation (1.1) is the Laplace
equation with fully homogeneous Dirichlet b.c.:
∂ ∂u
−
= f in Ω ⊆ Rd ,
∂xj ∂xj
u = 0 on δΩ .
The weak formulation introduces the bilinear form on the left-hand side of the first
equation:
Z
∂u ∂v
B(u, v) =
dx ,
Ω ∂xj ∂xj
for all v ∈ D(Ω), the space of functions with a compact support in Ω and continuous
derivatives of every order in Ω (test functions). The search for a (Galerkin-like) approximated solution means looking for a solution into a closed subspace of the Hilbert space
of the solutions. In particular, given the continuity dependence on the initial data of the
unique solution of the well-posed problem (1.1):
kukH1 ≤
kf kH2∗
,
L
L>0,
we can restrict the problem to the closed subspaces M1 ⊂ H1 , M2 ⊂ H2 . The constant
L, witness of the stability of the problem, is replaced by a subspaces-dependent constant
L(M1 , M2 ), witness of the stability of the approximation. The solution in the subspace
5
depends on the distance between the solution u ∈ H1 and the subspace itself M1 : there
exist δ > 0 and ω ∈ M1 such that
ku − ωkH1 ≤ δ .
Thus, there exists a unique solution û ∈ M1 , with the estimation:
L1
ku − ûkH1 ≤ 1 +
δ,
L(M1 , M2 )
where L1 > 0 is the continuity constant for the bilinear form B(·, ·). In the FE approximation case, we can give a more precise a priori estimation, depending on the Hilbert
space (and dual space) to which the solution u (and the right-hand side of the equation
◦
f ) belong. In particular, the function spaces of interest are Sobolev spaces: u ∈W2α (Ω)
means that u belongs to the closure in D(Ω) of W2α (Ω), the Sobolev space of functions in
L2 (Ω) with derivatives up to the order α ∈ R in L2 (Ω). With the multi-index notation
Dk u :=
∂ |k| u
,
∂x1 . . . ∂xknn
k ∈ Nn ,
k1
n ≥ 1,
we can introduce the norm (α ∈ N):
kuk2W α (Ω) =
X
kDk uk2L2 (Ω) .
2
|k|≤α
If α ∈ R, then α = bαc + σ, σ ∈ (0, 1):
kuk2W α (Ω) = kuk2
bαc
W2 (Ω)
2
+
X
kDk uk2W σ (Ω) ,
2
|k|=α
where
|u(t) − u(τ )|2
dtdτ .
n+2σ
Ω |t − τ |
An a priori estimation for uh ∈ Mh , FE closed subspace, is then:
kuk2W σ (Ω)
2
Z
=
ku − uh kW β (Ω) ≤ Khp kf kW2α (Ω)
K>0,
2
for some α and β > − 23 , and p = p(α, β). For instance, if n = 2, the error can be
computed with respect to the norm in the continuous functions space:
0
ku − uh kC(Ω) ≤ Khp kf kW α0 (Ω) ,
2
with
− 21
<
α0
< −1,
p0
=
α0
+ 1 − ε, ε > 0 arbitrary.
That was the analysis conducted in [5]: we will examine instead a procedure for an a
priori error in terms of the best approximation error, both for the bilinear form and for
the abstract saddle point problem. This terminology can be rather confusing, since, as
we have seen in the previous Chapter, a saddle point problem is just a more elaborated
variational problem. A justification will be therefore given and an example introduced:
the Stokes problem, which will be the main object of inquiry in the next Chapters.
6
2.2
A bilinear form: an example of a priori bound
Let us take two Hilbert spaces U and V . We can define a form B(·, ·) : U × V → R
which is bilinear (linear with respect to both variables), and continuous. That means,
there exists a constant M > 0 such that
B(u, v) ≤ M kukU kvkV ,
and actually M = kBk. The norm of a bilinear operator is in fact defined as
kBk = inf {C : B(u, v) ≤ CkukU kvkV } =
=
sup
u∈U,v∈V
=
B(u, v)
=
kukU kvkV
sup
B(u, v) ,
kukU ,kvkV =1
where the last identity is justified by the substitution u = kukU ū with ū a unit-norm
element of U .
We consider now the variational problem, find u ∈ U such that
B(u, v) = hf, vi
∀v ∈ V ,
(2.1)
with f ∈ V ∗ , dual space of V . The bilinear form generates two corresponding continuous
operators:
B : U → V ∗ , hBu, viV ∗ ×V = B(u, v) , ∀u ∈ U, ∀v ∈ V ,
B t : V → U ∗ , hB t v, uiU ∗ ×U = B(u, v) , ∀u ∈ U, ∀v ∈ V .
If B is bounded from below and B t is injective, then there exists a constant γ > 0 such
that (inf-sup condition):
inf sup
u∈U v∈V
B(u, v)
B(u, v)
= inf sup
=: γ > 0 .
v∈V u∈U kukkvk
kukU kvkV
(2.2)
We will see in a more detailed way the implications of the nature of the operators B and
B t on γ when we will analyze the Brezzi theory in Chapter 4.
The Generalized Lax-Milgram Theorem [5] states that the variational problem (2.1) is
well posed if and only if Equation (2.2) holds, then the solution u ∈ U is bounded as
follows:
hf, vi
B(u, v)
= sup
,
v∈V kvkV
v∈V kvkV
B(u, v)
B(u, v)
= sup
≥ inf sup
=γ,
u∈U
kvk
kuk
kvk
V
U
V kukU
v∈V
v∈V
kf kV ∗
kukU ≤
.
γ
kf kV ∗ = sup
kf kV ∗
kukU
7
(2.3)
In order to actually solve the problem, we abandon the exact formulation and consider the discretization of problem (2.1), in terms of a Galerkin FE approximation. In
particular, we look for a solution uh ∈ Uh for
B(uh , vh ) = hf, vh i ,
∀vh ∈ Vh .
We suppose that the problem is well posed: the inf-sup condition in the discrete spaces
holds, with a constant γh > 0. The form B is obviously continuous in the subspaces too,
by a constant kBh k. We can therefore easily a priori evaluate the approximation error
in terms of the error of best approximation:
kBh k
ku − uh kU ≤ 1 +
inf ku − wh kU .
(2.4)
wh ∈Uh
γh
We can prove that by mean of the Galerkin orthogonality property: since Uh ⊂ U , then
both equations
∃u ∈ U : B(u, vh ) = hf, vh i , ∀vh ∈ Vh
and
∃uh ∈ Uh :
B(uh , vh ) = hf, vh i ,
∀vh ∈ Vh ,
hold. Then, by the equations above and the inf-sup and continuity discrete conditions:
γh kuh − wh kU ≤ sup
vh ∈Vh
B(uh − wh , vh )
B(u − wh , vh )
= sup
≤ kBh kku − wh kU ,
kvh kV
kvh kV
vh ∈Vh
and by the triangle inequality
ku − uh kU
≤ ku − wh kU + kwh − uh kU
kBh k
≤ ku − wh kU +
ku − wh kU =
γh
kBh k
1+
ku − wh kU .
γh
We can take the inf over wh ∈ Vh , since there are no limitations on the choice, and get
the bound (2.4). Following a different procedure, it is interesting to see how to provide
a sharper bound, getting rid of the ”1” in (2.4), as in [9] and [10]. In order to do that
we need the following
Lemma 1 [9] Let X ⊂ U be a subset of the Hilbert space U and P a non-trivial (neither the null-operator nor the identity), idempotent, not necessarily orthogonal linear
projection onto X, that is
P :U →X,
P2 = P ,
P 6= 0, I .
Then
kI − P k = kP k .
8
(2.5)
A proof is given at the end of the Section.
We can define the mapping Ph : U → Uh , Ph u = uh , which is linear and idempotent,
so that the identity:
kPh k = kI − Ph k ,
(2.6)
holds, by Lemma 1. We observe that, by the discrete analogue of equation (2.3), Galerkin
orthogonality and continuity of the form B:
kPh ukU = kuh kU ≤
1
B(uh , vh )
1
B(u, vh )
kBk
1
=
≤
kf kV ∗ =
sup
sup
kukU .
γh
γh vh ∈Vh kvh kV
γh vh ∈Vh kvh kV
γh
From the relation (2.6) we get:
ku − uh kU
= ku − uh − wh + wh kU = k(I − Ph )(u − wh )kU
≤ kI − Ph kku − wh kU = kPh kku − wh kU
= sup
u∈U
kPh ukU
ku − wh kU .
kukU
Since wh is arbitrary in Uh , and combining the two last results, we get the bound we
were looking for:
kBk
inf ku − wh kU .
ku − uh kU ≤
γh wh ∈Uh
Proof of Lemma 1:
Showing that the equality (2.5) is true means proving both inequalities:
kP k ≤ kI − P k ,
kP k ≥ kI − P k .
Since we can reverse the roles of P and I − P because I − P is a projection too, it will be
enough to prove one of the two inequalities, the first for instance. Furthermore kP k ≥ 1
because P is a projection. In fact, let us suppose kP k < 1. Then
kP k = sup
x∈U
kP xkU
<1,
kxkU
kP xkU
<1.
kxkU
Since this holds for every x ∈ U , we can replace x with P x, getting
kP 2 xkU
<1,
kP xkU
and
kP xkU < kP xkU ,
9
which is absurd. Now, if kP k = 1, since I − P is a projection too, kI − P k ≥ 1 and we
would be done. We consider then kP k > 1:
kP k= sup
x∈U
kP xkU
>1.
kxkU
Then, for each a ∈ R such that 1 < a < kP k there exists u ∈ U such that:
a≤
kP ukU
,
kukU
kP ukU ≥ akukU > 0 .
We take
v := u −
(u, P u)
Pu ,
kP uk2U
which is orthogonal to P u:
(v, P u) = (u, P u) −
(u, P u)
(P u, P u) = 0 .
kP uk2U
We can use the Pythagoras formula:
kP uk2U kuk2U − |(u, P u)|2
|(u, P u)| 2
2
2
2
kP
uk
=
.
kvkU = kukU −
U
kP uk2U
kP uk2U
Since u 6= 0, then kP ukU ≥ akukU > kukU > 0, and u 6= P u, v 6= 0. It is also easy to
see that (I − P )v = (I − P )u 6= 0, we can therefore write
k(I − P )vk2U
k(I − P )uk2U kP uk2U
kP uk2U
=
≥
.
kvk2U
kP uk2U kuk2U − |(u, P u)|2
kuk2U
The last inequality must be verified. We observe that it is equivalent to:
k(I − P )uk2U kuk2U − kP uk2U kuk2U + |(u, P u)|2 ≥ 0 ,
kuk2U | ((I − P )u, (I − P )u) | − kP uk2U kuk2U + |(u, P u)|2 ≥ 0 ,
kuk2U kuk2U + kP uk2U − (P u, u) − (u, P u) − kP uk2U kuk2U + |(u, P u)|2 ≥ 0 .
By definition of inner product: (a, b) = (b, a), then we have:
kuk2U (kuk2U − 2<(u, P u)) + |(u, P u)|2 ≥ 0 ,
kuk4U − 2kuk2U <(u, P u) + <2 (u, P u) + =2 (u, P u) ≥ 0 ,
(kukU − <(u, P u))2 + =2 (u, P u) ≥ 0 .
10
which is definitely true. So:
kP uk2U
k(I − P )vk2U
≥
≥ a2 .
kvk2U
kuk2U
Since the sup of the first term in the inequality above is the norm of the operator I − P ,
we conclude that:
kI − P k ≥ a ,
and, as 1 < a < kP k and a is arbitrary,
kI − P k ≥ kP k . 2.3
An abstract saddle point problem
The main object of interest in this work are saddle point problems. They result in
a composition of bilinear forms, allowing us to exploit the previous arguments. As a
matter of fact it is possible to show that the variational problem defined above is also a
special case of saddle point problem. We define again two Hilbert spaces, V and Q, and
two bilinear forms:
• a continuous (by a constant kak∗ ) and not necessarily symmetric form
a:V ×V →R,
• a continuous (by a constant kbk) form
b:V ×Q→R.
So, given f ∈ V ∗ , g ∈ Q∗ , an abstract saddle point problem is: find u ∈ V, p ∈ Q such
that:
a(u, v) + b(v, p) = hf, viV ∗ ×V , ∀v ∈ V ,
(2.7)
b(u, q)
= hg, qiV ∗ ×V , ∀q ∈ Q .
As we have pointed out in the previous Section, the form b originates two operators
B and B t which are one the adjoint of the other:
b(v, q) = hBv, qiQ∗ ×Q = hB t q, viV ∗ ×V ,
∀v ∈ V, q ∈ Q .
(2.8)
As for variational problems, we deal at first with the continuous/exact form of the
problem, and then with its discretization. Brezzi theory in [6] explains under which
conditions the problem (2.7) admits a unique solution:
1. the form a is coercive over S := ker B, which is to say there exists α > 0 such that
a(v0 , v0 ) ≥ αkv0 k2V ,
11
∀v0 ∈ S ;
(2.9)
2. the form b satisfies an inf-sup condition, that is, there exists β > 0 such that
inf sup
q∈Q v∈V
b(v, q)
:= β .
kvkV kqkQ/ ker B t
(2.10)
The norm in the quotient space is defined as:
kqkQ/ ker B t :=
inf
q0 ∈ker B t
kq + q0 kQ .
(2.11)
Brezzi points out how the bilinear-form variational problem can be considered as a special
case of the saddle point problem. To this end, we take the variational problem related
to the form a, defined over V0 × V0 , where V0 is a closed subspace of V . We look for
u ∈ V0 such that:
a(u, v) = hf, viV ∗ ×V , ∀v ∈ V0 ,
where a(·, ·) is a V0 -elliptic (coercive over V0 ) form. This is equivalent to finding (u, ψ) ∈
V × W (where W = {w ∈ V ∗ : hw, viV ∗ ×V = 0 ∀v ∈ V0 }: the polar space of V0 ) such
that:
a(u, v) + hψ, viV ∗ ×V = hf, viV ∗ ×V , ∀v ∈ V ,
hϕ, uiV ∗ ×V
=0,
∀ϕ ∈ W .
In this special saddle point problem, b(v, ϕ) = hϕ, viV ∗ ×V : B ≡ B t ≡ I, so the stability
condition is fulfilled:
inf sup
ϕ∈W v∈V
= inf
ϕ∈W
b(v, ϕ)
hB t ϕ, viV ∗ ×V
= inf sup
=
ϕ∈W v∈V kvkV kϕkV ∗ / ker B t
kvkV kϕkV ∗ / ker B t
kB t ϕkV ∗
kϕkV ∗
= inf
≥1.
ϕ∈W
kϕkV ∗ / ker B t
kϕkV ∗ / ker B t
Furthermore, S ≡ V0 , so the coercivity condition for a is fulfilled as well, since a was
supposed to be V0 -elliptic.
We focus again on the issue of the well-posedness of problem (2.7). If conditions 1 and
2 above hold, then for each f ∈ V ∗ , g ∈ Im(B), there exists unique u ∈ V, p ∈ Q/ ker B t
solution of (2.7) with the following continuity estimate:
1
1
kak∗
kukV
≤
kf kV ∗ +
1+
kgkQ∗ ,
α
β
α
(2.12)
1
kak∗
kak∗
kak∗
∗ +
∗ .
t
kpk
≤
1
+
kf
k
1
+
kgk
V
Q
Q\ker
B
β
α
β2
α
Before looking at the FE discretized form of the problem, let us make some remarks
about condition (2.9). The coercivity of the form a is actually a sufficient condition for
12
the well-posedness of problem (2.7). Actually, a necessary and less restrictive condition
is the existence of a constant α0 such that:
a(u0 , v0 )
≥ α0 ,
inf
sup
u
∈ker
B
ku
0
0 kV kv0 kV
v0 ∈ker B
(2.13)
a(u
,
v
)
0
0
sup
≥ α0 .
inf
v0 ∈ker B u0 ∈ker B ku0 kV kv0 kV
These conditions are not actually a consequence of an inf-sup condition over the whole
space V . In fact, the assumption that for instance there exists α∗ > 0 such that
inf sup
u∈V v∈V
a(u, v)
≥ α∗ ,
kukV kvkV
is equivalent to asserting:
∃v ∈ V :
a(u, v)
≥ α∗ .
kukV kvkV
∀u ∈ V
(2.14)
This does not imply that
∃v0 ∈ ker B :
∀u0 ∈ ker B
a(u0 , v0 )
≥ α∗ ,
ku0 kV kv0 kV
since v ∈ V in condition (2.14) does not necessarily belong to ker B.
We can now go on with the discretization issue. Choosing a good FE approximation
is crucial: in the RB context, the exact solution will be identified with the FE truth
solution. For this reason, the conditions for a stable discretization are here analyzed.
In order to discretize our problem, we take Galerkin finite-dimension subspaces Vh ⊂
V, Qh ⊂ Q and we reformulate the abstract saddle point problem. We must find uh ∈
Vh , ph ∈ Qh such that
a(uh , vh ) + b(vh , ph ) = hf, vh iV ∗ ×V , ∀vh ∈ Vh ,
(2.15)
b(uh , qh )
= hg, qh iQ∗ ×Q , ∀qh ∈ Qh .
The continuous conditions of existence and uniqueness of a solution cannot unfortunately
be considered automatically valid in the discrete case, so we examine them carefully.
First of all we observe that the continuous inf-sup condition for the form b is satisfied
if and only if (see Chapter 4) the operator B has a closed range. Now, the discrete
analogous of B, Bh : Vh → Q∗h is obviously defined as
hBh vh , qh iQ∗ ×Q = b(vh , qh ) ,
∀qh ∈ Qh .
Bh has always a closed range in Q∗h , since it is a mapping between finite-dimensional
spaces. So, there exists βh > 0, in general h-dependent, such that:
inf
sup
qh ∈Qh / ker Bht vh ∈Vh
b(vh , qh )
=: βh .
kvh kV kqh kQ
13
This is not enough for guaranteeing the solvability of the problem (2.15). It is also
necessary that gh := g|Qh (the restriction of g on Qh ) lies in the range of Bh , which
is not a direct consequence of g ∈ Im(B). A useful characterization comes from the
following Lemma [8].
Lemma 2 For each g ∈ Q∗ :
Zh (g) := {vh ∈ Vh :
b(vh , qh ) = hg, qh iQ∗ ×Q
∀qh ∈ Qh } .
We remark that Sh := ker Bh = Zh (0). The following conditions are equivalent:
1. ∀g ∈ Im(B), Zh (g) 6= ∅, which means gh ∈ Im(Bh );
2. ∀v ∈ V , ∃vh = Πh v ∈ Vh such that
b(v − Πh v, qh ) = 0 ,
∀qh ∈ Qh ;
3. ker Bht = ker B t ∩ Qh .
A third issue is the coercivity of the form a in the discrete space. We have seen that the
coercivity of a over ker B is a sufficient condition for well-posedness in the continuous case
(2.7). If ker Bh ⊂ ker B, then this condition is automatically valid for the discretization:
there exists αh > 0 such that
a(vh , vh ) ≥ αh kvh k2V ,
∀vh ∈ Sh .
(2.16)
In particular, coercivity over the whole space V is guaranteed for the case-study we will
analyze, the Stokes problem. Both ker B and ker Bh are subspaces of V , then condition
(2.16) is h-uniformly satisfied: there exists α such that αh ≥ α > 0.
In general, coercivity over S might not be transferred over Sh , if Sh 6⊂ S. Theoretically, we could circumvent this problem by introducing a Hilbert super-space of V such
that V ,→ H (continuous inclusion). Now we can use a norm k · kH in order to write a
condition over the whole space V , which would imply V -coercivity over ker B as well:
a(v, v) ≥ αkvk2H
∀v ∈ V .
This is of course valid over Vh too:
a(vh , vh ) ≥ αkvh k2H ,
∀vh ∈ Vh .
Since Vh is a finite-dimensional space, then all the norms over Vh are equivalent, up to
a suitable constant S(h):
kvh kV ≤ S(h)kvh kH ,
α
and αh :=
.
S(h)2
Finally, for every g such that Zh (g) 6= ∅, if condition (2.16) holds and letting βh be the
discrete inf-sup condition, problem (2.15) has a unique solution uh ∈ Vh , ph ∈ Qh / ker Bht ,
with the upper bound with respect to to the initial data:
14
kuh kV
kph kQ/ ker B t
h
≤
1
1
kf kVh∗ +
αh
βh
≤
1
βh
kak∗
1+
αh
kak∗
1+
kgkQ∗h ,
αh
kf kVh∗
kak∗
+ 2
βh
kak∗
kgkQ∗h .
1+
αh
(2.17)
It is important to have an h-independent (uniform) estimation for the discretized solution. First of all, it is easy to see that
kf kVh∗ ≤ kf kV ∗ ,
kgkQ∗h ≤ kgkQ∗ .
Therefore the approximation is h-uniformly stable:
kuh kV + kph kQ/ ker B t ≤ C (kf kV ∗ + kgkq∗ ) ,
h
with C 6= C(h) ,
if αh and βh have h-independent lower bounds:
αh ≥ α0 > 0,
βh ≥ β > 0 .
In order to get a uniform inf-sup discrete condition, we can for instance build a Fortinoperator Πh (operator at point 2 of Lemma 2):
Proposition 1 If there exists a Fortin operator for the couple Vh , Qh then ker Bht ⊂
ker B t . If the continuous inf-sup condition holds (by a constant β), then
inf
qh ∈Qh / ker Bht
sup
vh ∈Vh
b(vh , qh )
β
≥
,
kvh kV kqh kQ
νh
with νh := kΠh kL(V,V ) .
If νh has an h-independent upper bound, then the uniform discrete inf-sup condition
holds. A proof of the Proposition can be found in [8].
There are troubles arising when βh does not have a uniform bound from below. There
will be asymptotic problems in the estimate of ph , since from (2.17)
1
ph = O
.
βh2
Furthermore, if inf βh = 0, we can call pseudo-spurious (pressure) components, those
h
qh ∈ Qh \ ker Bht for which
sup
vh ∈Vh
b(vh , qh )
' βh kqh kQ .
kvh kV
This means that, in spite of not belonging to ker Bht , these components minimize the
output of b. There are also spurious (pressure) components. As a matter of fact, p and
15
ph are defined up to a term belonging to ker B t and to ker Bht respectively (Lemma 2
characterizes the condition ker Bht ⊂ ker B t ). For the Stokes problem this means that
both the exact and the FE approximated pressure are defined up to a constant. When
ker Bht 6⊂ ker B t then the pressure components in ker Bht \ ker B t are called spurious and
must be eliminated in order to get an unbiased approximation of p.
After having chosen a suitable FE approximation, fulfilling all the conditions above,
we can provide an estimation of the error made with the FE approximation of the saddle
point problem. It is again based on the error of best approximation:
≤ cuu Eh (u) + cup Eh (p) ,
ku − uh kV
(2.18)
kp − ph kQ/ ker B t ≤ cpu Eh (u) + cpp Eh (p) ,
where
Eh (v) = inf kv − vh kV ,
Eh (q) = inf kq − qh kV ;
vh ∈Vh
cuu =
kak∗
kbk
1+
1+
,
αh
βh
cpu =
kak∗
βh
1+
kak∗
αh
qh ∈Qh
cup =
kbk
,
αh
kbk
kbk
kak∗
1+
, cpp = 1 +
1+
.
βh
βh
αh
A proof of (and conditions for sharpening) bound (2.18) can be found again in [8].
2.4
Derivation of a saddle point problem
The terminology saddle point still needs a justification: a sketch of the proof of the
equivalence between problem (2.7) and an actual saddle point problem is given in this
Section.
First of all, we can take the variational problem (2.1) and show that it is equivalent
to a minimization problem. We rewrite (2.1) replacing the bilinear form B with the
form a, which we now assume being coercive and symmetric. Symmetry is a stronger
assumption with respect to the previous Section, but for the Stokes problem a actually
is symmetric and coercive by a constant α. Therefore we claim that the problem, find
u ∈ V such that
a(u, v) = hf, viV ∗ ×V , ∀v ∈ V ,
(2.19)
is equivalent to the minimization of the functional J : V → R defined as
1
J(v) = a(v, v) − hf, viV ∗ ×V .
2
16
For a δv ∈ V , we can recognize in J(v + δv ) the terms of a Taylor expansion:
J(v + δv ) =
=
1
2 a(v
1
2
+ δv , v + δv ) − hf, v + δv iV ∗ ×V =
[a(v, v) + a(v, δv ) + a(δv , v) + a(δv , δv )] − hf, viV ∗ ×V − hf, δv iV ∗ ×V =
= J(v) + [a(v, δv ) − hf, δv iV ∗ ×V ] + 21 a(δv , δv ) =
= J(v) + h∇J(v), δv iV ∗ ×V + 12 hHJ(v)δv , δv iV ∗ ×V .
As δv is arbitrary, replacing v with u, by (2.19):
h∇J(u), δv iV ∗ ×V = a(u, δv ) − hf, δv iV ∗ ×V = 0,
∀δv ∈ V .
So
∇J(u) = 0 in V ∗ ,
and u ∈ V is a solution of (2.19) if and only if it is a stationary point for J. Since a is
also coercive, then
1
1
J(u + δv ) = J(u) + a(δv , δv ) ≥ J(u) + αkδv k2V > J(u) ,
2
2
if δv 6= 0 ,
and u is a minimum point for J.
We now consider problem (2.7) with g ≡ 0: this actually corresponds to the Stokes
problem, as we will see. In terms of minimizing J, we look for u ∈ S such that:
J(u) = min J(v) or J(u) = min J(v) .
v∈S
v∈V
Bv=0
Classically, in order to solve this problem, we can define the Lagrangian form
L(v, q) = J(v) + hBv, qiQ∗ ×Q = J(v) + b(v, q) ,
and we look for a saddle point (u, p) ∈ V × Q of L(u, v) such that (see Figure 2.1)
L(u, q) ≤ L(u, p) ≤ L(v, p),
∀q ∈ Q, ∀v ∈ V .
The element p ∈ Q is called the Lagrangian multiplier associated to u. It is also possible
to show that (u, p) is such that:
min sup L(v, q) = L(u, p) = max inf L(v, q) .
v∈V q∈Q
q∈Q v∈V
The obvious conditions for (u, p) to be a saddle point are:
∂L
(u, p) = 0 in V ∗ ,
∂v
and
17
∂L
(u, p) = 0 in Q∗ .
∂q
Figure 2.1: Geometrical interpretation of the saddle point: saddle point for the function
z = x2 − y 2 , [19]
.
Taking again a Taylor expansion, for δv ∈ V :
L(u + δv , p) = J(u + δv ) + b(u + δv , p) =
1
= J(u) + a(u, δv ) − hf, δv iV ∗ ×V + a(δv , δv ) + b(u, p) + b(δv , p) =
2
= L(u, p) + h
∂L
∂2L
(u, p), δv iV ∗ ×V + h 2 (u, p), δv iV ∗ ×V ,
∂v
∂v
with
h
∂L
(u, p), δv iV ∗ ×V = a(u, δv ) − hf, δv iV ∗ ×V + b(δv , p) = 0, ∀δv ∈ V ,
∂v
(2.20)
∂2L
1
h 2 (u, p), δv iV ∗ ×V = a(δv , δv ) ≥ 0,
∂v
2
We see that (u, p) is actually a V -minimum for L(v, q).
18
∀δv ∈ V .
Then, for any δq ∈ Q:
L(u, p + δq ) = J(u) + b(u, p + δq ) =
= J(u) + b(u, p) + b(u, δq ) =
= L(u, p) + h
with
h
∂L
(u, p), δq iQ∗ ×Q ,
∂q
∂L
(u, p), δq iQ∗ ×Q = b(u, δq ) = 0 ,
∂q
∀δq ∈ Q .
(2.21)
Since δv and δq are arbitrary, the first equation of (2.20) and equation (2.21) correspond
to (2.7), g ≡ 0.
2.5
The Stokes problem on a parametrized domain
We show in this Section how the Stokes problem is derived from the Navier-Stokes
equations and why it is a saddle point problem.
The Navier-Stokes equations for an incompressible flow are:
∂
1 ∂ ∂ui
∂ p̂
∂ui
+ uj
ui −
+
= fˆi ,
in Ω × (0, T ] ,
∂t
∂x
Re
∂x
∂x
∂x
j
j
j
i
∂uj
= 0,
in Ω × (0, T ] ,
∂xj
ui
= gi ,
on Γ × (0, T ] ,
ui (0)
= u0i , on Ω ,
where:
• the first equation is the momentum equation for the velocity u and the pressure
p̂, the second is the continuity equation;
• the right-hand side of the continuity equation set to 0 corresponds to the incomprimibility constraint;
• the third and fourth equations are the boundary and initial conditions;
• the domain Ω is connected and ⊂ Rd : i, j = 1, . . . , d. For us d = 2;
• the boundary Γ := δΩ must be regular enough (e.g. Lipschitzian);
• the density ρ can be neglected, since we are dealing with an incompressible case:
ρ = ρ0 = 1;
19
• the Reynolds number is is a function of a reference velocity, a reference length and
the kinematic viscosity:
Uref Lref
Re =
.
ν
The functional frame is:
d • u ∈ L2 0, T ; H 1 (Ω)
;
• p̂ ∈ L2 0, T ; L2 (Ω) ;
d • f̂ ∈ L2 0, T ; H −1 (Ω)
;
d 1
2
,
• g ∈ L 0, T ; H 2 (Γ)
1
where H 2 (Γ) is the space of the traces on the boundary of functions in H 1 (Ω).
Assuming now g ≡ 0, time-discretization (e.g. through schemes such as Euler
implicit/explicit, Crank-Nicholson or Adam-Bashfort) generates the generalized Stokes
problem:
∂ ∂ui
∂p
σui −
+
= fi , in Ω ,
∂x
∂x
∂x
j
j
i
∂uj
∂xj
= 0,
in Ω ,
with:
• p = Re p̂ at the new time step and p ∈ L2 (Ω);
2
• u is assumed at the new time step and u ∈ H01 (Ω) ;
• f contains f̂ and all the quantities referred to the current time step;
cRe
> 0 with c and ∆t dependent on the temporal scheme adopted.
∆t
We are actually interested in the steady-state case. Furthermore, the velocity is assumed to be low: the convective term can be neglected:
∂
ui = o( other terms ), i = 1, 2, in Ω .
uj
∂xj
• σ=
Thus, the equations of the Stokes problem can be derived directly from the original
equations and correspond to the generalized problem imposing σ = 0:
∂ ∂ui
∂p
−
= fi , in Ω ,
∂xi
∂xj ∂xj
(2.22)
∂u
j
= 0,
in Ω ,
∂xj
with
20
• p = Re p̂ and p ∈ L2 (Ω);
2
• u ∈ H01 (Ω) ;
• f = −Re f̂ .
We consider now the Stokes problem on a parametrized domain, changing the
notation with respect to (2.22), because of issues with the boundary conitions of the
velocity (now uin (µ)), which will be clear later on:
in
∂ ∂ui (µ) − ∂p(µ) = 0 ,
∂ x̂j ∂ x̂j
∂ x̂i
(2.23)
in Ω̂ ≡ Ω̂(µ) .
in
∂ui (µ)
= 0,
∂ x̂i
The uin
i (µ), i = 1, 2, are the velocity components (in the x̂1 and x̂2 directions) and p is
obviously the pressure. The domain we are considering for our case study is a rectangle
with a parametrized obstacle, as we can see in Figure 2.2. A possible application is a
microchannel with an obstacle with variable dimensions to optimize the mixing.
Figure 2.2: The channel: the parametrized domain Ω̂(µ) with the obstacle (µ1 , µ2 ) and
the boundary conditions: homogeneous Dirichlet on Γ̂0 (µ), inhomogeneous Dirichlet on
Γ̂in (µ) and natural Neumann outflow conditions on Γ̂N .
The space of the parameters µ is
D = [0.1, 0.5]2 ⊂ R2 .
The boundary conditions are mixed (Neumann and Dirichlet) and imposed as follows,
21
(Γ̂· ≡ Γ̂· (µ)):
uin
i (µ)
= 0,
on Γ̂0 ,
uin (µ)
= uinlet ,
on Γ̂in ,
∂u(µ)in
i
− p(µ)ni = 0,
∂n
on Γ̂N ,
∂·i
∂·i
nj is the normal derivative: nj are the components of the vector normal
=
∂n
∂·j
to the Neumann boundary. Physically, the velocity is imposed on the inlet boundary, noslip conditions are set on the above and below boundary and natural Neumann outflow
conditions on the outlet boundary. The boundary is then partitioned like
where
Γ̂ = Γ̂D ∪ Γ̂N ,
with the homogeneous and inhomogeneous Dirichlet conditions:
Γ̂D = Γ̂0 ∪ Γ̂in .
The velocity space is
n
o
V̂I = v : vi ∈ H 1 (Ω̂) : v = uinlet on Γ̂in , vi ≡ 0 on Γ̂0 ,
which is slightly different to the one for the velocity test functions:
n
o
V̂ = v : vi ∈ H 1 (Ω̂) : vi ≡ 0 on Γ̂D .
(2.24)
Both pressures and their test functions belong to
n
o
Q̂ = q ∈ L2 (Ω̂) .
(2.25)
We can now write the weak formulation of the problem: we transform the differential
formulation of problem (2.23) into an integral formulation. The first equation is multiplied by a test function of V̂ and the second by a test function of Q̂. Integrating over
the domain, we have the set of equations:
Z
Z
∂ ∂uin
∂p(µ)
i (µ)
vi −
vi = 0, ∀v ∈ V̂ ,
Ω̂ ∂ x̂j ∂ x̂j
Ω̂ ∂ x̂i
(2.26)
Z
in (µ)
∂u
i
q = 0, ∀q ∈ Q̂ .
∂ x̂i
Ω̂
As the first equation in (2.26) is concerned, by integration by parts and by the Divergence
Theorem, we have:
Z
Z
Z
∂uin
∂uin
∂ ∂uin
i (µ)
i (µ)
i (µ) ∂vi
vi =
vi nj −
,
∂ x̂j
∂ x̂j
∂ x̂j ∂ x̂j
Ω̂ ∂ x̂j
Γ̂
Ω̂
Z
−
Ω̂
∂p(µ)
vi = −
∂ x̂i
Z
Z
p(µ)vi ni +
Γ̂
22
p(µ)
Ω̂
∂vi
.
∂ x̂i
Since the test functions vi vanish over Γ̂D , we can write for the boundary integrals:
Z
Z in
∂uin
∂ui (µ)
i (µ)
vi nj =
nj vi ,
∂ x̂j
∂ x̂j
Γ̂
Γ̂N
Z
Z
p(µ)vi ni = −
−
(p(µ)ni )vi .
Γ̂N
Γ̂
Summing these two last equations:
in
Z
∂ui (µ)
vi
nj − p(µ)ni = 0 .
∂ x̂j
Γ̂N
The first equation of (2.26) becomes:
Z
Ω̂
∂u(µ)in
i ∂vi
−
∂ x̂j ∂ x̂j
Z
p(µ)
Ω̂
∂vi
=0,
∂ x̂i
∀vi ∈ V̂ .
We now introduce a known function uL ∈ V̂I such that
uLi
=0,
on Γ̂0 ,
uL
= uinlet ,
on Γ̂in ,
uin (µ) = u(µ) + uL ,
in Ω̂ ,
where
u(µ) = 0 on Γ̂D .
We assign a parabolic prophile for the inlet boundary Γ̂in . In particular, uL is parameterindependent and has the following expression:
4x̂2 (1 − x̂2 )(1 − x̂1 ), on [0, 1] × [0, 1] ,
uL =
0,
on Ω̂(µ) \ [0, 1] × [0, 1] .
It is therefore possible to replace uin (µ) in (2.26) and write the problem in terms of
u(µ) ∈ V̂ with all-homogeneous Dirichlet boundary conditions.
The new weak formulation of problem (2.23) is, find ui (µ) ∈ V̂ , p(µ) ∈ Q̂ such that
Z
Z
Z
∂ui (µ) ∂vi
∂vi
∂uLi ∂vi
−
p(µ)
=−
, ∀vi ∈ V̂ ,
Ω̂ ∂ x̂j ∂ x̂j
∂ x̂i
Ω̂
Ω̂ ∂ x̂j ∂ x̂j
(2.27)
Z
Z
∂u
(µ)
∂u
i
Li
q
=
q,
∀q ∈ Q̂ .
−
∂
x̂
∂
x̂
i
i
Ω̂
Ω̂
23
Each term in the system (2.27) can be seen as a parameter-dependent bilinear form:
Z
∂vi ∂wi
,
â(·, ·; µ) : V̂ × V̂ → R,
â(v, w; µ) =
∂
Ω̂ x̂j ∂ x̂j
Z
b̂(·, ·; µ) : V̂ × Q̂ → R,
=−
b̂(v, q; µ)
q
Ω̂
∂vi
,
∂ x̂i
or linear form
fˆ(·; µ) : V̂ → R,
fˆ(v; µ) = −
Z
Ω̂
Z
ĝ(·; µ) : Q̂ → R,
ĝ(q; µ)
=
Ω̂
∂uLi ∂vi
,
∂ x̂j ∂ x̂j
∂uLi
q.
∂ x̂i
It is necessary to compute the solution of the parametrized problem for many values
of the parameter, for instance for a real-time quest for the optimal mixing. In order to
do that efficiently, we must separate the parameter-dependent and -independent parts
of the forms, before applying the RB Method. Here, the approximated solution is a
linear combination of solutions at particular parameters. First of all a reference domain
Ω must be chosen, corresponding to one particular value of the parameters. In order to
easily map the reference domain into the parametrized one - and the other way round
-, the domains are split into R = 12 non-overlapping regions. These have a triangular
shape so that the mapping between the domains is the sum of affine mappings between
the regions. The regions are chosen accordingly to the design parameters µ ∈ D, as we
can see in Figure 2.3.
Figure 2.3: The channel: the mesh for the FE approximation and the 12 regions in which
the domain has been subdivided, accordingly to the design parameters.
We take as reference domain
Ω := Ω̂(µref ),
µref := [0.3, 0.3] ,
and Ω is partitioned in regions:
Ω=
R
[
r=1
24
Ωr .
The mapping from the parametrized regions Ω̂r to the reference ones Ωr is an affine
transformation Gr ≡ Gr (µ):
x = Gr x̂ + g r ,
x ∈ Ωr , x̂ ∈ Ω̂r ,
Gr ∈ R2×2 , g r ∈ R2×1 ,
r ∈ {1, . . . , R} .
The derivatives and the integrals in problem (2.27) must be transformed through a
variable changing:
∂
∂
= Grji
,
∂ x̂i
∂xj
dΩ̂r = (det Gr )−1 dΩr .
We can finally write the (bi-)linear forms on the reference domain, introducing the spaces
V and Q, obtained from definitions (2.24) and (2.25) erasing the hats 1 :
V = v : vi ∈ H 1 (Ω) : vi ≡ 0 on ΓD ,
Q = q ∈ L2 (Ω) ;
a(·, ·; µ) : V × V → R, a(v, w; µ) =
R
X
Z
Grlj Grkj (det Gr )−1
Ωr
r=1
b(·, ·; µ) : V × Q → R,
b(v, q; µ)
= −
R
X
Grli (det Gr )−1
r=1
f (·; µ) : V → R,
f (v; µ)
= −
R
X
Z
Ωr
Grkj Grlj (det Gr )−1
g(·; µ) : Q → R,
g(q; µ)
=
Grji (det Gr )−1
r=1
∂vi
,
∂xl
q
r=1
R
X
∂vi ∂wi
,
∂xk ∂xl
Z
q
Ωr
Z
Ωr
∂uLi ∂vi
,
∂xk ∂xl
∂uLi
.
∂xj
We can lump the parameter-dependent terms and get an affine decomposition for
every form:
a(v, w; µ) =
Qa
X
Θqa (µ)aq (v, w),
b(v, q; µ) =
q=1
f (v; µ) =
Qf
X
Θqf (µ)f q (v),
Qb
X
Θqb (µ)bq (v, q) ,
q=1
g(q; µ) =
q=1
Qg
X
(2.28)
Θqg (µ)g q (q) ,
q=1
1
The sum over the indices q and r is always explicitly formulated and it is an exception to the Einstein
notation.
25
so that:
Z
∂vi ∂wi
q=q(r,k,l) (v, w) =
,
a
Ωr ∂xk ∂xl
Θq=q(r,k,l)
(µ) = Grlj Grkj | det Gr |−1 ,
a
(r, k, l) ∈ {1, . . . , R} × {1, 2} × {1, 2},
1 ≤ q ≤ 4R ;
Z
∂vi
q=q(r,i,l)
,
b
(v, q) = −
q
Ωr ∂xl
q=q(r,i,l)
Θb
(µ) = Grli | det Gr |−1 ,
(r, i, l) ∈ {1, . . . , R} × {1, 2} × {1, 2},
1 ≤ q ≤ 4R ;
Z
∂uLi ∂vi
q=q(r,k,l) (v) = −
,
f
∂xk ∂xl
r
Ω
q=q(r,k,l)
Θf
(µ) = Grkj Grlj | det Gr |−1 ,
(r, k, l) ∈ {1, . . . , R} × {1, 2} × {1, 2},
1 ≤ q ≤ 4R ;
Z
∂uLi
q=q(r,j,i) (q) =
,
g
q
∂xj
r
Ω
Θq=q(r,j,i)
(µ) = Grji | det Gr |−1 ,
g
(r, j, i) ∈ {1, . . . , R} × {1, 2} × {1, 2},
1 ≤ q ≤ 4R .
In order to enhance computational efficiency, from each sequence of Θ coefficients,
actual parameter-dependency is highlighted. The final coefficients will be, e.g. for a:
a
Θpar-dep1
, . . . , Θpar-depQ
a
a
or rather simply
a
Θ1a , . . . , ΘQ
a
with
1 ≤ q ≤ Qa ≤ 4R .
The same procedure is applied to the other forms, where:
1 ≤ q ≤ Qb ≤ 4R,
1 ≤ q ≤ Qf ≤ 4R,
1 ≤ q ≤ Qg ≤ 4R .
Finally, writing the linear forms f (µ) and g(µ) as duality relations:
26
f (v; µ) = hf (µ), viV ∗ ×V ,
g(q; µ)
= hg(µ), qiQ∗ ×Q ,
system (2.27) can be re-written as:
a(u(µ), v; µ) + b(v, p(µ); µ) = hf (µ), viV ∗ ×V , ∀v ∈ V ,
= hg(µ), qiQ∗ ×Q ,
b(u(µ), q; µ)
(2.29)
∀q ∈ Q ,
with the unknown velocity u(µ) ∈ V and pressure p(µ) ∈ Q. System (2.29) lies in the
framework of the abstract saddle point problems (2.7). In the next Chapter we will
shortly introduce a suitable FE discretization for the Stokes problem and then finally
apply the RB approximation.
27
Chapter 3
The RB Approximation and the
Stokes problem as application
We have seen in the previous Chapter which requirements must be satisfied by a saddle
point problem and the space we choose for its approximation, so that stability is guaranteed. The issue of stability is therefore crucial for both the truth (Galerkin-FE) and the
RB Approximation, which are both introduced in this Chapter for the Stokes problem.
The literature about a good Galerkin approximation is vast, and there are many examples of stable approximation spaces, for instance the Taylor-Hood or the MINI element,
see [8]. On the other side, even though the RB technique allows us to save up a great
amount of time, for non-coercive problems in general, and saddle point problems in
particular, the RB method does not inherit the stability of the FE discretization.
Among the stability requirements, continuity of the linear and bilinear forms and
coercivity of the a form, at least in the Stokes case, are also fulfilled over the RB spaces.
A uniformly bounded from below inf-sup constant is a sufficient requirement for stability
in the FE. That is not automatically inherited in the RB: we will see what happens when
the RB inf-sup condition is satisfied and when it is not.
After establishing stability for the RB approximation, we must define a suitable a
posteriori error bound with respect to the FE solution. That is a goal for the next Chapter: we start here by exctracting from the problem the relevant data for the computation
of the Brezzi error bound (inf-sup, continuity and coercivity constants, computed as solution of eigenvalue problems) and for a comparison between the Brezzi and the Babuška
bounds (the ratio of the dual norm of the momentum and continuity residuals).
3.1
Truth FE Approximation
Before describing a suitable FE approximation space for the Stokes problem (2.29), we
recall that each element consists in a triple [17]:
(E, P (E), {li∈I }) ,
where
28
• E is a compact, non-empty, convex subset of the domain Ω;
• P (E) a finite-dimensional (dim = I) functions space over E;
• {li }i∈I a family of linear forms
li : P (E) → R ,
which are unisolvent over P (E). This means
p ∈ P (E),
li (p) = 0,
∀i ∈ I
⇒
p≡0.
We will use different elements for velocity and pressure, according to the classic TaylorHood element. The Es are triangles, both for velocity and pressure, such that
[
Ω=
Ei , Ei ∩ Ej = ∅, ∀i 6= j .
i
Velocities are approximated by second-order polynomials with a Lagrangian basis:
VN = span{ξi , i = 1, · · · , Nu :
ξi |E ∈ P2 |E } ⊂ H 1 (Ω) .
There are six degrees of freedom over each triangle (the vertices and the middle-points
of the edges). Each degree of freedom identifies a basis-function.
Pressures are continuous in the approximation space and approximated by first-order
polynomials:
QN = span{ψi , i = 1, · · · , Np :
ψi |E ∈ P1 |E } ⊂ L2 (Ω) .
The vertices of each triangle correspond to the degrees of freedom.
Consistently with the definition of a finite element, we point out that, for each E, the
spaces P (E) are VN |E and QN |E respectively, while the linear forms {li }i∈I correspond
to the degrees of freedom, such that:
li (p ∈ P (E)) = p(xi ) ,
where xi are the vertices and, for the velocities, the middle-points in the triangles as
well. It is easy to see that such forms are unisolvent. The FE approximation applied
to (2.29) generates a system of dimension N = (2Nu + Np ). This N is usually very
high: for our problem it has been taken Nu = 4977 and Np = 1287 so that N = 11241
(correspondent to the rather coarse mesh we can see in Figure 2.3).
We now examine quickly how the exact formulation is modified in the FE approximation. First of all, the inner products and norms, in the exact spaces (respectively
V = H01 (Ω) and Q = L2 (Ω)) are:
Z
p
∂vi ∂wi
(v, w)V =
, kvkV = (v, v)V ,
Ω ∂xj ∂xj
Z
(q, ψ)Q =
kqkQ =
qψ,
Ω
29
q
(q, q)Q .
They are computed, in the discretized spaces, through the matrices
Z
∂ξl ∂ξk
Ω ∂xj ∂xj
, l, k = 1, . . . , Nu ;
V Nu Nu =
Z
∂ξl ∂ξk
Ω ∂xj ∂xj
Z
Q Np Np =
l, k = 1, . . . , Np .
ψl ψk ,
Ω
The norms correspond then to matrices and vectors products:
kvk2VN = (v, v)VN =
2N
Xu
vi VNu Nu vj ,
i,j=1
kqk2QN
= (q, q)QN =
Np
X
qi QNp Np qj ,
i,j=1
where vi and qj are now the components of v and q in the FE decomposition (the first
and second component of v are not distinguished here to make the notation easier). The
whole system (2.29) can be re-written in terms of matrices and vectors, instead of bilinear
and linear forms. We will not go further into this topic, since the FE approximation
(with a sufficiently big number of degrees of freedom) is for us the truth approximation.
Truth approximation and exact solution are identified 1 .
Some considerations are nevertheless necessary as far as the linear and bilinear forms
are concerned, in the FE approximation. First of all, the discrete formulation of system
(2.29) actually is:
u1 (µ)
t
A(µ) B (µ)
f (µ)
u2 (µ)
=
,
(3.1)
B(µ)
g(µ)
p(µ)
with
A(µ) ∈ R2Nu ×2Nu , B(µ) ∈ RNp ×2Nu , f (µ) ∈ R2Nu , g(µ) ∈ RNp ,
ui (µ) ∈ RNu , p(µ) ∈ RNp .
For the Stokes problem, the bilinear forms are continuous, with positive continuity constants, both in the exact and in the FE formulation. We can recall the definition given
1
So, even if in this whole Chapter the FE approximation is concerned, the N subscripts are omitted.
30
in the previous Chapter:
a(v, w; µ) ≤ ka(µ)k∗ kvkV kwkV , ∀v, w ∈ V ,
(3.2)
b(v, q; µ) ≤ kb(µ)kkvkV kqkQ ,
∀v ∈ V, q ∈ Q .
In order to compute more efficiently the continuity constant for a, we can derive an
estimation from above. That will be sufficient for our purposes of error estimation,
which relies on the already anticipated equations (2.12): here ka(µ)k∗ appears only in
the numerator field, so an upper bound for it can be tolerated. For the Stokes problem,
a is symmetric: it therefore defines an inner product
(v, w)a(µ) := a(v, w; µ) ,
and it is possible to apply the Cauchy-Schwarz inequality: from the first equation of
(3.2):
a(w, v; µ)
ka(µ)k∗ = sup sup
≤
w∈V v∈V kwkV kvkV
1
1
a 2 (w, w; µ)a 2 (v, v; µ)
≤ sup sup
=
kwkV kvkV
w∈V v∈V
1
=
a 2 (w, w; µ)
sup
kwkV
w∈V
= sup
w∈V
!2
=
a(w, w; µ)
,
kwk2V
and from now on we will use
ka(µ)k := sup
w∈V
a(w, w; µ)
,
kwk2V
instead of ka(µ)k∗ .
The form a is coercive over the whole V :
a(v, v; µ) ≥ α(µ)kvk2V , ∀v ∈ V .
The inf-sup constant for the form b is satisfied as well, by the good properties of the
Taylor-Hood approximation:
β(µ) = inf sup
q∈Q v∈V
b(v, q; µ)
> 0,
kqkQ\ker B(µ)t kvkV
∀v ∈ V, q ∈ Q .
(3.3)
For the Stokes problem, we can actually get easily rid of ker B(µ)t in the expression
above: we can show that it is a trivial space. For instance, without loss of generality, let
us take µ = µref . Then:
ker B t = {q ∈ Q :
b(v, q) = 0
31
∀v ∈ V } .
This means, if q ∈ QN (this immediate proof is not valid for q ∈ L2 (Ω)):
Z
Z
Z
∂(qvi )
∂q
∂vi
=−
+
vi =
b(v, q) = − q
Ω ∂xi
Ω ∂xi
Ω ∂xi
Z
Z
∂q
vi = 0, ∀v ∈ V .
= − qvi ni ds +
∂x
| Γ {z
} | Ω {z i }
1
2
From term 2 in the last equation, it follows that q must be piecewise constant. As v does
not vanish over the whole boundary (there is a Neumann condition for the outlet flow),
q ≡ 0 (from term 1). This means that for every µ ∈ D, ker B(µ)t = {q ∈ Q : q ≡ 0}.
Finally, we can show a FE solution for the Stokes problem over the channel: µ =
(0.1, 0.4) is taken as parameter. The solution can be plotted on the reference domain,
so that it is easy to compare solutions corresponding to different parameters, or mapped
onto the parametrized domain. We can see the results in Figures 3.1 and 3.2.
32
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
(a) Velocity field on the reference domain Ω.
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
(b) Velocity field on the mapped domain Ω̂(µ).
Figure 3.1: Solution of the Stokes problem on the channel domain (reference or mapped
- µ = (0.1, 0.4) - domain) with the specified boundary conditions. The Taylor-Hood
element with 4977 degrees of freedom for each velocity component has been used.
33
0
−5
−10
−15
−20
−25
−30
−35
−40
−45
(a) Pressure patch on the reference domain Ω.
0
−5
−10
−15
−20
−25
−30
−35
−40
−45
(b) Pressure patch on the mapped domain Ω̂(µ).
Figure 3.2: Solution of the Stokes problem on the channel domain (reference or mapped
- µ = (0.1, 0.4) - domain) with the specified boundary conditions. The Taylor-Hood
element with 1287 degrees of freedom for the pressure has been used.
34
3.2
RB Approximation
We can start with a quick introduction to the RB Method [7]. In many problems,
the need for an efficient input-output computation makes the RB method worth being
applied. We can specify that:
• the input consists in the geometric configuration, physical or effective properties
of the model, boundary conditions and initial values, loads and sources;
• the output of interest is a particular information deduced by the solution of the
problem: a maximum temperature, a pressure drop, . . .
• the field variable connects the input and the output through a system of PDE.
The method is suited for two particular cases: real-time and many-query problems.
Real time problems are faced in three steps:
1. assess: the current state must be evaluated, in terms of parameters estimation or
inverse analysis, e.g. a tumor location;
2. predict: given the results of the first stage, (many) possible subsequent states must
be foreseen, e.g. the potential lethality of the tumor;
3. act: decisions, according to some requirements, must be taken in order to influence
the evolution, e.g. decide upon the kind of intervention.
Two examples which belong to the many-query problems category can be:
• real-time problems with a robust assess phase: an extensive parameters exploration
has to be performed so that all possible future scenarios of the model can be
determined;
• multi-scale and -physics models: the behavior at a big scale depends on many
spatial or temporal realizations at a smaller scale.
In these situations it can be necessary to evaluate many O(104 ) input-output relations.
The machinery of the FE method is not adequate as it could be for single- (or few-)
query problems, where an established problem must be solved without a considerable
variation of parameters. The RB method is therefore an alternative way which is built
upon the FE (truth) approximation: the solution is computed as a linear combination
of FE solutions at some chosen values of the parameters.
Besides the challenges, the typologies of problems we are considering give some opportunities for the RB approach:
• since we are looking at one problem with varying parameters, the space of the
solutions can be reduced to a low-dimensional manifold (the parameters space
dimension). The FE space is therefore unnecessarily big and expensive;
35
• the goal is computing many input-output combinations: the RB machinery can be
onerous and heavy in the pre-processing (offline) phase, as long as the marginal
cost of each online evaluation is kept small.
We can now analytically describe the RB method for the Stokes problem (2.29). Solutions at chosen parameters are pre-calculated and the RB solution at the desired
parameter lies in the vectorial space generated by them. We define the set of sample
parameters:
SM := {µ1 , · · · , µM } .
A new set of basis functions for velocities and pressures is introduced:
VN = span{ζ n } ,
QM = span{ϕn } ,
(3.4)
where ζ n , ϕn , n = 1, . . . are the basis functions for the spaces of the RB solutions.
They will be defined soon.
VN and QM have to form a stable pair. Continuity and coercivity of a and the
continuity of b are satisfied, as VN ⊂ V, QM ⊂ Q. Unlike them, the inf-sup condition
over the RB spaces:
b(v, q; µ)
>0,
βN M (µ) := inf sup
q∈QM v∈VN kqkQ kvkV
is not satisfied for every choice of VN and QM . More precisely, it is not always possible
to show that βN M (µ) has a lower bound:
βN M (µ) ≥ β(µ) > 0 ,
(3.5)
which, with β(µ) as defined in (2.10), would guarantee that the FE stability has been
inherited by the RB approximation. This issue will be faced in the next Section.
Let us now suppose that VN and QM form a stable pair, and that their dimensions
are respectively N and M , both Np < Nu . We can describe here the procedure
to compute the RB solutions. All µ-independent quantities can be formed and stored
within an offline stage that depends on the large finite element dimension N and that
is performed only once. For any given parameter µ ∈ D, we then calculate the RB
approximation in an online stage that depends only on the significantly smaller dimension
of the RB approximation spaces VN and QM and that is therefore very efficient. There
are therefore two phases:
• the expensive offline phase: we have to solve the FE problem at most M times,
and perform the procedure to build VN and QM . The FE approximation system
we must solve in order to compute the pressure and velocity corresponding to the
k-th parameter in SM is:
u1 (µk )
t
A(µk ) B (µk )
f (µk )
u2 (µk )
=
.
g(µk )
B(µk )
p(µk )
36
Each solution is a linear combination of the FE test functions:
ui (µk ) =
Nu
X
uji (µk )ξj ,
i = 1, 2;
p(µk ) =
j=1
Np
X
pj (µk )ψj ,
(3.6)
j=1
from which we derive the ξ n and ϕn of VN and QM respectively, as we will see in
the next Section. Once we have built VN and QM , we can express the RB solution
at a parameter µ as a linear combination as well:
uN (µ) =
N
X
uN j (µ)ζ j ,
j=1
pM (µ) =
M
X
pM j (µ)ϕj .
(3.7)
j=1
The system which allows us to compute the RB coefficients uN j (µ) and pM j (µ) is
also required.
Let us take the matrix A(µ). Recalling the affine decomposition for the FE quantities, equations (2.28), we see that:
A(µ) =
Qa
X
Θqa (µ)Aq ,
q=1
and each matrix Aq is computed through the form aq :
Aqnm = aq (ξn , ξm ),
n, m = 1, . . . , 2Nu ;
ξn , ξm ∈ V ,
taking into account that obviously the velocity test functions are the same for the
first and the second component (in the FE context, while in the RB context both
the velocity and the pressure have only one component, since the velocity basis is
made up of velocity vectors already). Changing our basis, we see that
!
N
N
N X
N
X
X
X
aq (vN , uN ) = aq
vi ζ i ,
uj ζ j =
vi aq (ζ i , ζ j )wj .
i=1
i=1
i=1 j=1
We can define, for the RB linear system, the matrices :
2N
Xu
q
q
n
m
ζin Aqij ζjm ,
n, m = 1, . . . , N ;
AN N nm = a (ζ , ζ ) =
i,j=1
Np 2Nu
n = 1, . . . , N
X
X
q
m q n
q
n
m
B
=
b
(ζ
,
ϕ
)
=
ϕ
B
ζ
,
m
= 1, . . . , M ;
i
ij j
M N mn
q = 1, . . . , Qa ;
q = 1, . . . , Qb ;
i=1 j=1
(3.8)
37
and the vectors:
2N
Xu
q
q (ζ n ) =
f
=
f
ζin fiq ,
Nn
i=1
n = 1, . . . , N ;
q = 1, . . . , Qf ;
(3.9)
Np
X
q
q
q (ϕm ) =
g
=
g
ϕm
m = 1, . . . , M ;
Mm
i gi
q = 1, . . . , Qg .
i=1
All the quantities which required FE computations have been stored. The computational cost for solving M FE linear systems is then O(M N 3 ), keeping in mind
that FE matrices and vectors are sparse.
• the inexpensive online phase consists in assembling the matrices and vectors at the
desired parameter µ:
Qf
Qa
X
X
q
q
q
Θ
(µ)A
,
A
(µ)
=
f
(µ)
=
Θqf (µ)fN
,
NN
N
a
NN
q=1
q=1
Qb
X
q
Θqb (µ)BM
N ,
BM N (µ) =
q=1
Qg
X
q
,
Θqg (µ)gM
gM (µ) =
q=1
and solving the parameter-dependent, low-order problem:
t
AN N (µ) BM
N (µ)
uN (µ)
fN (µ)
=
BM N (µ)
pM (µ)
,
gM (µ)
with
AN N (µ) ∈ RN ×N , BM N (µ) ∈ RM ×N , fN (µ) ∈ RN , gM (µ) ∈ RM ,
uN (µ) ∈ RN , pM (µ) ∈ RM ,
and the solution is given by (3.7). The computational cost for solving the RB
linear system is O (M + N )3 . The RB matrices are usually full: it is even more
important to optimize M , and consequently N , and keep them as small as possible.
3.3
Building stable RB spaces
So far, we learnt that the RB spaces are originated by FE solutions computed at parameters in a sample set SM . It will be shown that this is not enough for reaching stability.
A new tool which allows us to enrich the velocity RB space VN with new functions, so
that (VN , QM ) forms a stable pair, is the supremizer operator. The supremizer is now
introduced, then its employment for the actual computation of the RB spaces is faced.
38
3.3.1
The supremizer
The supremizer operator [1, 2, 7] is a mapping
Tµ : Q → V ,
such that
(Tµ q, v)V = b(v, q; µ),
∀v ∈ V, q ∈ Q .
(3.10)
Proposition 2 [7] If definition (3.10) holds, then:
Tµ q = arg sup
v∈V
b(v, q; µ)
.
kvkV
Proof :
By definition of B we can give a Riesz representation of Tµ q:
(Tµ q, v)V = hB t (µ)q, viV ∗ ×V .
Then:
kTµ qkV = kB t (µ)qkV ∗ = sup
v∈V
hB t (µ)q, viV ∗ ×V
b(v, q; µ)
= sup
.
kvkV
kvkV
v∈V
By definition of supremizer:
b(Tµ q, q; µ)
= kTµ qkV ,
kTµ qkV
then
sup
v∈V
b(Tµ q, q; µ)
b(v, q; µ)
= kTµ qkV =
,
kvkV
kTµ qkV
and finally
Tµ q = arg sup
v∈V
b(v, q; µ)
.
kvkV
The supremizer is in the first place useful for computing the inf-sup constant. Using
Proposition 2:
b(Tµ q, q; µ)
b(v, q; µ)
β(µ) = inf sup
= inf
=
q∈Q v∈V kvkV kqkQ
q∈Q kTµ qkV kqkQ
by definition of supremizer
(Tµ q, Tµ q)V
kTµ qkQ
= inf
.
q∈Q kTµ qkV kqkQ
q∈Q kqkQ
= inf
Definition (3.10) in terms of FE matrices corresponds to:
vt V Tµ q = vt B t (µ)q ,
Tµ q = V −1 B t (µ)q .
39
It is therefore possible to express the inf-sup constant as a Rayleigh quotient and compute
it through a linear system and a generalized eigenvalue problem:
q t BV −1 B t q
.
q∈Q
q t Qq
β 2 (µ) = inf
(3.11)
The same procedure can be applied to the RB inf-sup constant (3.5): we note that, if
v ∈ VN and q ∈ QM , then:
N
N
N
N
X
X
X
X
2
i 2
i
j
kvkV = k
vi ζ kV =
vi ζ ,
=
vj ζ
vi (ζi , ζj )V vj = kvk2VN ,
i=1
i=1
i=j
V
i,j=1
and analogously
kqk2Q = kqk2QM .
Under the same assumptions:
N
M
N X
M
X
X
X
i
j
t
b(v, q; µ) = b(
vi ζ ,
qj ϕ ) =
vi b(ζ i , ϕj )qj = vN BM
N qM .
i=1
j=1
i=1 j=1
Finally, defining the supremizer in the RB context as:
t
TµM N qM , vN V = vN BM
N qM ,
N
we see that
2
βN
M (µ) = inf
q∈QM
−1 t
t B
qM
M N VN BM N qM
.
t Q q
qM
M M
(3.12)
The same procedure, with obvious modifications in the RB context, can be used to
evaluate the coercivity constant:
α(µ) = inf
v∈X
a(v, v; µ)
vt A(µ)v
=
inf
,
v∈X vt Xv
kvk2X
(3.13)
the continuity constant:
ka(µ)k = sup
v∈X
a(v, v; µ)
vt A(µ)v
= sup
,
2
t
kvkX
v∈X v Xv
(3.14)
and also the Babuška constant (which will be introduced, for the Stokes problem, in the
next Chapter).
40
3.3.2
Options for the RB spaces
The solutions (3.6) are used for constructing the RB spaces VN and QM . Te building
up procedure for VN can be improved with the use of the supremizer operator itself, in
order to deal with the stability issues. We start with a set SM of M randomly chosen
sample parameters µi ∈ D: the RB pressure space QM is then:
QM = span p(µi ), i = 1, . . . , M ,
with dimQM = M ≤ M , due to possible linear dependence.
We can now examine some options for the RB velocity space VN :
• option 0: we do not enrich VN :
VN0 = span {u(µi ),
i = 1, . . . , M } ,
and condition (3.5) cannot be proved;
• option 1 (originally suggested in [2]): keeping the sampling procedure random, we
can define, for each term in the affine decomposition of b, a particular supremizer:
T k q, v
:= bk (v, q), ∀q ∈ Q, v ∈ V, k = 1, . . . , Qb ;
V
so that
T k q = arg sup
v∈V
bk (v, q)
,
kvkV
The new RB velocity space is:
n
VN1 = span u(µi ), T k p(µi ),
∀q ∈ Q, k = 1, . . . , Qb .
i = 1, . . . , M ;
k = 1, . . . , Qb
We can prove that condition (3.5) is satisfied:
0 < β(µ) = inf sup
q∈Q v∈V
b(v, q; µ)
=
kvkV kqkQ
Qb
X
= inf sup
k=1
= inf
Θkb bk (T k q, q)
k=1
q∈Q
kT k qkV kqkQ
Qb
X
≤ inf
q∈QM
41
=
kvkV kqkQ
q∈Q v∈V
Qb
X
Θkb bk (v, q)
≤
Θkb bk (T k q, q)
k=1
kT k qkV kqkQ
.
(3.15)
o
.
by definition of VN1 , if q ∈ QM , then T k q ∈ VN , so that, using again (3.15):
Qb
X
0 < β(µ) ≤ inf
sup
Θkb bk (v, q)
k=1
kvkV kqkQ
q∈QM v∈VN
= βN M (µ) ,
and stability for option 1 is proved;
• option 2: Enrichment for the velocity space is provided by the parameter-dependent
supremizer applied to each pressure function:
VN2 = span {u(µi ), Tµi p(µi ),
i = 1, . . . , M } ,
as for option 0, we cannot prove that option 2 satisfies the RB inf-sup condition,
but computations show a remarkably different behaviour with respect to the 0alternative.
A stable or unstable approximation could be reflected on the behavior of the error.
Let (u(µ), p(µ)) ∈ V × Q and (uN (µ), pM (µ)) ∈ VN × QM be the FE and RB solutions
of the Stokes problem for a particular value of the parameter µ. The errors are defined
as:
u
eN (µ) ≡ u(µ) − uN (µ) ,
(3.16)
p
eN (µ) ≡ p(µ) − pM (µ) .
The residuals of the equations are considered too: they are both indicators for the
precision of the approximation and important quantities for the error bounds, treated in
the next Chapter. We call residual of an equation the quantity obtained by embedding
an approximated solution into the exact problem. In our case, the RB solution of the
Stokes problem is embedded into the truth equations:
u
rN (v; µ) = f (v; µ) − a(uN (µ), v; µ) − b(v, pM (µ); µ) ,
(3.17)
p
rN (q; µ) = g(q; µ) − b(uN (µ), q; µ) .
u (µ) := r u (·; µ), r p (µ) := r p (·; µ), dual operators in V ∗ and Q∗ . We need
We define rN
N
N
N
to compute their norms (as defined, for instance, in equation (2.3) in Chapter 2; the
computational procedure is described in Section 3.6):
u
krN
(µ)kV ∗ ,
p
krN
(µ)kQ∗ .
An unstable couple of function spaces will lead to bad results for the pressure, as
pointed out at the end of Section 2.3. So, while the continuity equation can be well
satisfied anyway, that could not occur for the momentum equation. That is why we
consider this new quantity, the ratio of the dual norms of the residuals:
42
u (µ)k ∗
krN
V
.
p
krN (µ)kQ∗
ρ(µ) :=
(3.18)
It is interesting to introduce one last enrichment option, whose aim is keeping the
ρ(µ) ratio as low as possible:
• option 3: at the first step
QN = span {p(µ1 ),
µ1 ∈ SM } ,
VN = span {u(µ1 ),
µ1 ∈ SM } .
At the following step, we compute the parameter which maximizes the residuals
ratio:
µ01 = arg max ρ(µ) ,
µ∈Sopt3
where Sopt3 is a sample space (e.g. 100 random parameters) such that Sopt3 ∩SM =
∅; and we enrich the velocity space like this:
n
o
0
VN = span u(µ1 ), T µ1 p(µ01 ) .
Thus, the final velocity space is:
n
0
VN3 = span u(µi ), T µj p(µ0j ),
i = 1, . . . , M,
o
j = 1, . . . , M − 1 ,
while the pressure space is the same as for the other options.
It is important to remark that, although the pressure and velocity FE solutions are
computed at the same random parameters as in the other options (in the set SM ),
this option has a greedy component, due to the additional terms in the velocity
space. They are calculated through a worst-case procedure, over another pool of
parameters. The results of applying this hybrid option can therefore be surprising
and lead the way to new sample strategies.
The actual basis functions ζ i and ϕi which appear in definitions (3.4) are orthonormal
functions. They are obtained through the Gram-Schmidt orthonormalization procedure.
Let τn be the n-th computed pressure - p(µn ) - or velocity - u(µj ) (or T k u(µj ) or
T µj u(µj )) - and η n the correspondent basis function. Then (assuming the inner product
and norms in V or Q respectively)
τ1
if n = 1, η 1 =
,
kτ1 k
otherwise
n−1
X
t = τn −
(τn , τm )τm ,
m=1
ηn
t
.
=
ktk
If kη n k < ε, with ε a sufficiently small prescribed constant, then η n is linearly dependent
on the other basis functions and M = n ≤ M is the final dimension of the RB pressure
space (and a suitable multiple of M is the dimension of the velocity space).
43
3.4
Comparison among the enrichment options
We can show now some results from testing options 0, 1, 2, 3. Taking M = 50, the
dimensions of the RB spaces, before reaching linear dependence, are the following:
option
option
option
option
0
1
2
3
M = dim(QM )
49
23
35
42
dim(VN )
49
23 ∗ (Qb + 1)
35 ∗ 2 = 70
42 ∗ 2 − 1 = 83
We can reach richer spaces for both the pressure and the velocity in the case of
option 3. That was expected, since a greedy approach allows at each step the choice of
parameters far from those already picked up for the previously computed basis functions.
The quantities taken into account for the comparison are: the RB inf-sup constant,
the norm of the errors and the residuals, and the ratio ρ(µ). Actually we are considering
the normalized errors:
kepN (µ)kQ
keuN (µ)kV
,
.
(3.19)
ku(µ)kV
kp(µ)kQ
The values of interest are computed at the enrichment steps M ∗ = 5, 10, 15, . . . , as far
as it is allowed from the dimension of QM for each option.
The values of the RB inf-sup constant (computed according to the procedure described at the end of Section 3.3.1) allow us to predict the stability of the approximation:
o0
comparing βN
M (option 0) with β, we see that, as it was expected, the RB constant is
o0 is usually < 0.02, while β ≥ 0.08).
greater than 0, but lower than the FE constant (βN
M
The worst case is always considered: in Figure 3.3 we plot, at each enrichment step M ∗ ,
o0 evaluated at:
the minimum βN
M
o0
0
µ = arg 0min βN
M (µ ) ,
µ ∈Stest
where Stest is a sample of 60 randomly generated test-parameters, such that Stest ∩SM =
∅. The FE inf-sup constant is of course evaluated at the same parameter µ.
Comparing the same constants referred to options 1, 2, we see that in both cases
o1 reaches the best values (it
βN M > β, both at values ' 0.08, Figures 3.4. Although βN
M
is the only enrichment option for which stability is analytically proved), we claim that
we can expect a good approximation from both options 1 and 2.
Now we look at the errors. As a matter of fact, the normalized error is on average
one order of magnitude lower for options 1, 2 than for option 0 for the velocity, and at
least 2 orders of magnitude lower for the pressure, Figures 3.5. For both options 1, 2 the
decrease is monotonic and very similar, at least as far as linear dependence is reached
for option 1. On the contrary, at option 0 peaks of instability are witnessed. Option
1 reaches a normalized velocity error of approximately 10−3 and 10−4 for the pressure.
Option 2, at M ∗ = 35, almost reaches 10−4 for the velocity and 10−4.5 for the pressure.
A Greedy algorithm for the choice of the parameters, instead of a random procedure,
44
RB− and FE− inf−sup constant
0.12
0.1
βNM and βFE
0.08
βNM for option0
0.06
βFE
0.04
0.02
0
5
10
15
20
25
M MAX
30
35
40
45
o0 (µ) and β(µ) where µ = arg min β o0 (µ0 ) at each enrichment step
Figure 3.3: βN
NM
M
0
µ ∈Stest
M ∗.
RB− and FE− inf−sup constant
0.083
option1
option2
0.0811
βFE
0.082
βNM for option2
0.081
0.0809
0.081
βNM
βNM and βFE
RB−inf−sup constant
0.0812
βNM for option1
0.08
0.0808
0.0807
0.0806
0.079
0.0805
0.078
5
10
15
20
M MAX
25
30
o1
o2
(a) βN
M (µ), βN M (µ), β(µ) .
0.0804
5
35
10
15
20
M MAX
25
30
35
o1
o2
(b) βN
M (µ), βN M (µ): zoom of the picture on the
left.
Figure 3.4: Comparison between the RB and FE inf-sup constants for the options 1 and
o1,o2 0
2. The parameter considered at each enrichment step M ∗ is µ = arg 0min βN
M (µ ),
µ ∈Stest
which is the same for both options.
45
Normalized Pressure Error
5
Normalized Velocity Error
0
10
10
option0
option0
option2
option2
ep
eu
option1
option1
−1
10
−2
10
0
10
−3
10
−5
−4
10
5
10
15
(a)
20
25
M MAX
30
35
40
10
45
ku(µ) − uN (µ)kV
.
ku(µ)kV
5
10
15
(b)
20
25
M MAX
30
35
40
45
kp(µ) − pM (µ)kQ
.
kp(µ)kQ
Figure 3.5: Normalized errors between RB and FE solution of the Stokes problem, for options 0, 1, 2. The parameter considered at each enrichment step M ∗ is
µ = arg 0max keuN (µ0 )kV or, respectively, µ = arg 0max kepN (µ0 )kQ .
µ ∈Stest
µ ∈Stest
should let the RB spaces for option 1 reach a higher dimension and better approximation
results.
We can now consider the behaviour of the residuals ratio, ρ(µ). We consider options
1, 2 only, as we have witnessed that options 0 results are not relevant for an error bound
investigation. We can see in Figure 3.6 that ρ(µ) takes values in the range [1, 4] (with
the isolated peak of 10 for option 2 at M ∗ = 35). The importance of this result will be
cleared in the next Chapter.
As we can see from the residual norms plots in Figures 3.7, option 3 gives very
good results, comparable to option 2. Furthermore, it reaches linear dependence only
at M = 42, thanks to the partly greedy procedure which characterizes it. Again, we
can observe a monotonic decrease for options 1, 2 (and 3, almost always) and even more
peaks of instability for option 0 than we remarked in the errors plots.
46
2
Ratio of the Momentum and Continuity Residuals
10
option1
ρ
option2
1
10
0
10
5
10
15
Figure 3.6: For options 1 and 2: ρ(µ) =
µ = arg 0max ρ(µ0 ) .
20
M MAX
25
30
35
u (µ)k ∗
krN
V
, where at each enrichment step M ∗ ,
p
krN (µ)kQ∗
µ ∈Stest
47
Momentum Equation Residual
6
10
option0
option1
4
10
option2
option3
2
ε
u
10
0
10
−2
10
−4
10
5
10
15
20
25
M MAX
30
35
40
45
u
(µ)kV ∗ .
(a) krN
Continuity Equation Residual
0
10
option0
option1
−1
10
option2
option3
−2
εp
10
−3
10
−4
10
−5
10
5
10
15
20
25
M MAX
30
35
40
45
p
(b) krN
(µ)kQ∗ .
Figure 3.7: Dual norm of the residuals for the RB approximation of the Stokes problem,
for options 0, 1, 2, 3. The parameter considered at each enrichment step M ∗ is µ =
p
u
arg 0max krN
(µ0 )kV ∗ or, respectively, µ = arg 0max krN
(µ0 )kQ∗ .
µ ∈Stest
µ ∈Stest
48
3.5
A priori convergence for the RB Approximation
The error between the RB approximation and the FE one (but also between the RB
approximation and the exact solution) for a saddle point problem can have a descending
trend as M increases. This kind of trend is not necessarily monotonic, as we can witness
for the stable options analyzed above. The possible rate of decrease depends on many
factors: the choice of the parameters, the dimension of the parameters space, the RB
spaces, the chosen norm for the error. For a simple problem and a special choice for
the (1D-)parameters, it is indeed possible to prove that exponential decay is achieved:
a sketch of the proof in [11] is shown in this Section. On the contrary, we cannot still
formulate a general analytical result, although an exponential trend is witnessed also in
other cases where the choice of the parameters is accurate enough [12].
Let us consider the variational problem:
a(u(µ), v; µ) = f (v),
u, v ∈ V , µ ∈ D0 ,
(3.20)
with, as usual, V a Hilbert space, a(·, ·; µ) a bilinear form, f (·) a linear form in V ∗
and D0 = [0, µmax ] a set of one-dimensional parameters. We identify as usual the
FE approximation and the exact problem. It is also assumed that a has a simple affine
decomposition:
a(w, v; µ) = a0 (w, v) + µa1 (w, v),
∀v, w ∈ V,
µ ∈ D0 ,
where:
• a0 is symmetric, continuous and coercive over V (by a constant αa0 ), thus defining
an inner product and norm:
1
|k · |k := a0 (·, ·) 2 ;
• a1 is symmeytric, continuous (by a constant ka1 k) and semi definite positive.
Lax-Milgram conditions are fulfilled, so that problem (3.20) has a unique solution, and
we can establish a constant γ1 such that:
0≤
a1 (v, v)
ka1 k
≤
≤ γ1 .
a0 (v, v)
αa0
The Galerkin RB approximation space is
WN = span {u(µk ),
k = 1, . . . , N )} ,
with µk ∈ SN , a set of N sample parameters. They are log-equidistributed in D0 , such
that:
!
k
X
µk = exp − log γ +
δ̃lN − γ −1 , k = 1, . . . , N ,
l=1
49
where γ is an upper bound for γ1 and δ̃lN has the following property:
N
X
δ̃lN = log (γµmax + 1) ,
l=1
and, defining δN = log (γµmax + 1) and c∗ a real positive constant:
δ̃kN
≤ c∗ ,
δN
k = 1, . . . , N .
If all these conditions hold, we can write the following
Proposition 3 [11] For N ≥ Ncrit := c∗ e log (γµmax + 1):
1
|ku(µ) − uN (µ)|k ≤ (1 + µmax γ1 ) 2 |ku(0)|ke−N/Ncrit ,
∀µ ∈ D0 .
It can be interesting, just here, to distinguish the FE approximation from the exact
solution, so that we can compare the latter and the RB approximation. The result above
must therefore be rewritten in terms of N , the dimension of the FE approximation space
V N:
1
N
−N/Ncrit
2
|kuN (µ) − uN
, ∀µ ∈ D0 .
N (µ)|k ≤ (1 + µmax γ1 ) |ku (0)|ke
By Galerkin approximation properties, we can prove that:
1
|kuN (µ)|k ≤ (1 + µmax γ1 ) 2 ku(µ)k,
∀µ ∈ D0 .
Furthermore, we assume that the FE approximation is good enough, so that for each
ε > 0 there exists a Nε such that
|ku(µ) − uNε (µ)|k ≤ ε .
Thus,
Nε
Nε
Nε
ε
|ku(µ) − uN
N (µ)|k ≤ ku(µ) − u (µ)k + ku (µ) − uN (µ)k ≤
1
≤ ε + (1 + µmax γ1 ) 2 |kuNε (0)|ke−N/Ncrit ≤
≤ ε + (1 + µmax γ1 ) |ku(0)|ke−N/Ncrit ,
∀µ ∈ D0 .
We wonder if this result is confirmed by the RB normalized approximation errors (3.19)
for the channel problem (2.29). We take a single test parameter now: µ = (0.4, 0.1).
In Figures 3.8 we can see that, even if piecewise exponential decrease can be witnessed
for options 0 (velocity, and neglecting the peak of instability at M ∗ = 25), 1 and 2, we
cannot generalize the result above, at least with a random sampling strategy. The error
convergence in these contexts is still an object of study.
50
Normalized Velocity Error
−1
10
−2
10
option0
option1
option2
−3
eu
10
−4
10
−5
10
−6
10
5
10
15
20
(a)
25
M MAX
30
35
40
45
ku(µ) − uN (µ)kV
.
ku(µ)kV
Normalized Pressure Error
1
10
0
10
option0
option1
option2
−1
10
−2
ep
10
−3
10
−4
10
−5
10
−6
10
−7
10
5
10
15
20
(b)
25
M MAX
30
35
40
45
kp(µ) − pN (µ)kV
.
kp(µ)kV
Figure 3.8: Normalized errors between RB and FE solution of the Stokes problem, for
options 0, 1, 2 at µ = (0.4, 0.1) .
51
3.6
Appendix: computation of the dual norms of the residuals
We present the procedure for actually calculating the dual norm of the velocity residual
(or residual for the momentum equation), the first equation of (3.17). It is easy to
modify it for the pressure residual (or residual for the continuity equation).
First, we write down the affine decomposition for each term in the Stokes problem:
u
rN
(v; µ)
=
Qf
X
Qa
Qb
N
M
X
X
X
X
Θqf (µ)f q (v)−
Θqa (µ)aq (
Θqb (µ)bq (v,
uN i (µ)ζ i , v)−
pM i (µ)ϕi ) .
q=1
q=1
q=1
i=1
i=1
If we group together the parameter -dependent and -independent terms, we get:
u
rN
(v; µ)
=
R
X
Φi (µ)Fiu (v) .
i=1
By mean of the Riesz representation theorem, each Fiu := Fiu (·) ∈ V ∗ can be identified
with a correspondent element in Γui ∈ V :
Fiu (v) = hFiu , viV ∗ ×V ,
hFiu , viV ∗ ×V = (Γui , v)V ,
(3.21)
such that
kFiu kV ∗ = kΓui kV .
Thus, we compute the Γui by solving (in the FE approximation) the linear systems (3.21).
Since
!
R
R
R
X
X
X
u
u
u
hrN (µ), viV ∗ ×V =
Φi (µ)hFi , viV ∗ ×V =
Φi (µ)(Γi , v)V =
Φi (µ)Γi , v
,
i=1
i=1
i=1
V
we can compute the norm:
u
krN
(µ)kV ∗ = k
R
X
Φi (µ)Γui kV =
R X
R
X
i=1
i=1 j=1
52
Φi (µ)Φj (µ) Γui , Γuj
V
.
Chapter 4
A posteriori error estimation
We look for an a posteriori upper bound for the error in the RB approximation of the
parametrized saddle point problem (2.29). First of all we analyze Babuška and Brezzi
strategies which provide an a priori upper bound for the exact solution. This one can
be turned into an a posteriori bound for the RB approximation, if applied to the error
equations. We therefore obtain two a posteriori error bounds, one from the Babuška
strategy and one from the Brezzi one. Finding a suitable a posteriori bound is especially
important if the sampling strategy for the parameters choice is not random, like in this
work case, but, for instace, a Greedy algorithm. A sharp and efficient error bound would
allow us to accelerate convergence and therefore optimize M . We would like to know
which error bound strategy works better for the Stokes problem, whether Babuška or
Brezzi. So, before comparing the two bounds for the channel problem, we will test them
on a sequence of increasingly complicated problems and analyze the numerical results
(the constants involved, the ratio between the two bounds), in order to find a trend
which could hide an analytic relation.
4.1
Babuška and Brezzi Theories
We apply the Babuška Theory, already examined in Section 2.2 to the saddle point
problem for getting an a priori error bound for the truth solution. Then, Brezzi Theory
for saddle point problems is analyzed in more detail than Section 2.3, in order to get
another a priori bound for the solution: in this case separated bounds for the pressure
and the velocity are considered.
4.1.1
Babuška’s combined bound
Considering the abstract saddle point problem (2.7), we can define the combined operator
B((u, p), (v, q)) = a(u, v) + b(v, p) + b(u, q),
∀(v, q) ∈ V × Q ,
B((u, p), (v, q)) = hf, viV ∗ ×V + hg, qiQ∗ ×Q ,
∀(v, q) ∈ V × Q .
such that
53
Problem (2.7) is well posed if and only if B(·, ·) : (V × Q) × (V × Q) → R satisfies the
inf-sup condition:
B((u, p), (v, q))
=
(u,p)∈V ×Q (v,q)∈V ×Q k(u, p)kV ×Q k(v, q)kV ×Q
B((u, p), (v, q))
inf
sup
=: γ > 0 ,
(v,q)∈V ×Q (u,p)∈V ×Q k(u, p)kV ×Q k(v, q)kV ×Q
inf
sup
(4.1)
where we can define the l2-norm in the product space (B and B t are the operators
associated with the form b as in (2.8)):
q
k(v, q)kV ×Q−l2 = k(v, q)kV ×Q := kvk2V + kpk2Q/ ker B t .
The pressure is always defined up to an element in ker B t , so its norm must be taken
like in (2.11). Condition (4.1) means that problem (2.7) has a unique solution with the
bound
1
k(u, p)kV ×Q ≤ k(f, g)kV ∗ ×Q∗ ,
(4.2)
γ
with the l2-norm for the dual product space as well.
We point out that, in the FE approximation for the Stokes problem, the Babuška
inf-sup constant (4.1) can be computed through a generalized eigenvalue problem. The
procedure is equivalent to the Brezzi inf-sup one, at expression (3.11) (and its derivation).
The FE matrix B corresponding to the form B((·, ·), (·, ·)) is the whole system matrix in
(3.1), and and the FE matrix X corresponding to the l2-norm in the product space is
the composition of the V and Q norms:
A Bt
V
, X =
.
B=
B
Q
Thus, it is possible to compute γ as
γ2 =
(v, q)t BX −1 B t (v, q)
.
(v, q)t X (v, q)
(v,q)∈V ×Q
inf
(4.3)
This system is usually huge: for the - actually rather coarse - mesh we are using for
the Stokes problem we have approximately 5000 degrees of freedom for the velocity
and 2500 for the pressure. Since the velocity has two components, the whole system
matrix has 12500 × 12500 elements: it is a sparse matrix, but the matrix appearing at
the numerator of expression (4.3) is probably full and will not be actually computed for
solving the eigenvalue problem. That is why we search in the Brezzi approach an equally
good bound, which allows us to circumvent this onerous computation.
54
4.1.2
Brezzi’s separated bounds
In problem (2.7) both a (which does not have to necessarily be symmetric) and b are
continuous,
a(u, v) ≤ kakkukV kvkV ,
b(v, q) ≤ kbkkvkV kqkQ .
The bilinear form b originates the operators B : V → Q∗ and B t : Q → V ∗ , as in (2.8).
The bilinear form a defines a linear continuous operator too, A : V → V ∗ ,
hAu, viV ∗ ×V = a(u, v),
∀v, u ∈ V .
We can now write problem (2.7) as dual relations:
Au + B t p = f, in V ∗ ,
Bu = g,
in
Q∗
(4.4)
.
Defining under which conditions there exists a unique solution (u, p) ∈ V × Q of (2.7)
or (4.4), we will obtain the bound (2.12). The procedure is now examined step by step.
Proposition 4 [6] If g ∈ Im(B) and a is a continuous form, coercive over ker B,
a(v0 , v0 ) ≥ αkv0 k2V ,
∀v0 ∈ ker B ,
then there exists a unique u ∈ V solution of
a(u, v0 ) = hf, v0 iV ∗ ×V ,
∀v0 ∈ ker B ,
(4.5)
and
Bu = g .
(4.6)
Proof :
If g ∈ Im(B) then there exist ug ∈ V such that Bug = g. We now write u = u0 + ug
where u0 ∈ ker B is such that
a(u0 , v0 ) = hf, v0 iV ∗ ×V − a(ug , v0 ),
∀v0 ∈ ker B .
Now:
• ker B is a subspace of the Hilbert space V ;
• a is a bilinear, continuous and coercive form over ker B;
• the right-hand side of (4.7) is a continuous form over ker B.
55
(4.7)
Therefore, all the Lax-Milgram hypotheses are satisfied and there exists a unique solution
u0 ∈ ker B for the variational problem (4.7).
We still have to check that u = u0 + ug is unique, that is, it does not depend on the
choice of ug . Indeed, if u1 , u2 are two solutions of (4.5) and (4.6), then
B(u1 − u2 ) = B((u01 + ug1 ) − (u02 + ug2 )) = (0 + g) − (0 + g) = 0 ,
so u1 − u2 ∈ ker B. From (4.5) we can see that
a(u1 − u2 , v0 ) = 0,
∀v0 ∈ ker B ,
and from coercivity:
0 = a(u1 − u2 , u1 − u2 ) ≥ αku1 − u2 k2V ≥ 0 ,
so u1 − u2 = 0 and u1 = u2 : the solution is unique. It is possible to give an upper bound for u0 ∈ ker B observing that, by coercivity
and (4.7):
α0 ku0 k2V ≤ a(u0 , u0 ) ≤ kf kV ∗ ku0 kV + kakkug kV ku0 kV ,
and therefore
1
(kf kV ∗ + kakkug kV ) .
(4.8)
α
We must now deal with the pressure solution p ∈ Q: for that purpose, we need some
preliminar considerations. We assume first of all that Im(B) is closed in Q∗ (each
convergent sequence in Im(B) ⊂ Q∗ converges to a point in Im(B) ⊂ Q∗ ). In finitedimensional spaces, this is always true. We can apply the Banach closed range Theorem
[16], which is referred to any closed operator. The Stokes case does not require such a
generality: we can restrict the statement to linear continuous operators, as B.
ku0 kV ≤
Theorem 1 [8] If B is a continuous linear operator B ∈ L(V, Q∗ ), the following statements are equivalent:
1. Im(B) is closed in Q∗ ;
2. Im(B t ) is closed in V ∗ ;
3. (ker B)0 = {v 0 ∈ V ∗ : hv 0 , viV ∗ ×V = 0 ∀v ∈ ker B} = Im(B t );
0 4. ker B t = q 0 ∈ Q∗ : hq 0 , qiQ∗ ×Q = 0 ∀q ∈ ker B t = Im(B);
5. there exists β > 0 such that for any v ∈ V :
kBvkQ∗ ≥ β
inf
v0 ∈ker B
kv + v0 kV =: kvkV / ker B ;
6. there exists β > 0 such that for any q ∈ Q:
kB t qkV ∗ ≥ β
inf
q0 ∈ker B t
kq + q0 kQ := kqkQ/ ker B t .
56
See Section 4.1.3 for a sketch of a proof [16]. The equivalence between statements 1
and 5 leads to the inf-sup condition for the saddle point problem: we have seen that for
all v ∈ V :
kBvkQ∗ ≥ βkvkV / ker B , β > 0 .
Since B is an operator B : V −→ Q∗ , Bv =: gv ∈ Q∗ and its norm is:
hgv , qiQ∗ ×Q
hBv, qiQ∗ ×Q
b(v, q)
= sup
= sup
.
kqkQ
kqkQ
gv ∈Q
q∈Q
q∈Q kqkQ
kgv kQ∗ = sup
Then:
b(v, q)
≥ βkvkV / ker B ,
q∈Q kqkQ
sup
and
∀v ∈ V ,
b(v, q)
=β.
q∈Q kvkV / ker B kqkQ
inf sup
v∈V
The same procedure applies for the adjoint operator B t :
inf sup
q∈Q v∈V
b(v, q)
=β,
kqkQ/ ker B t kvkV
which is exactly the inf-sup condition we already found in (2.10). We can now study the
complete problem:
Proposition 5 [6] Under the assumptions of Proposition 4, let g ∈ Im(B) and let
u ∈ V be the solution of problem (4.5) and (4.6). Then, if Im(B) is closed in Q∗ (which
means inf-sup condition (2.10) holds), there exists p ∈ Q such that (u, p) ∈ V × Q is a
solution of problem (2.7). Furthermore, estimate (2.12) holds.
Proof :
The linear form L is defined as
L(v) = hf, viV ∗ ×V − a(u, v),
∀v ∈ V .
By definition of u,
L(v0 ) = hf, v0 iV ∗ ×V − a(u, v0 ) = 0 ∀v0 ∈ ker B
That means
hL, v0 iV ∗ ×V = 0,
∀v0 ∈ ker B ,
and L belongs to the polar space of ker B, (ker B)0 . We have seen that if Im(B) is closed
in Q∗ , then (ker B)0 ≡ Im(B t ). If L ∈ Im(B t ), there exists p ∈ Q such that L = B t p
and
b(v, p) = hB t p, viV ∗ ×V = hL, viV ∗ ×V = L(v), ∀v ∈ V ,
and (u, p) ∈ V × Q is the solution we were looking for.
57
Now, using (2.10), we see that
kpkQ/ ker B t
≤
1
b(v, p)
sup
β v∈V kvkV
≤
L(v)
1
sup
β v∈V kvkV
≤
1
hf, viV ∗ ×V − a(u, v)
sup
β v∈V
kvkV
≤
1
kf kV ∗ kvkV + kakkukV kvkV
sup
β v∈V
kvkV
≤
1
(kf kV ∗ + kakkukV ) .
β
then by definition of L :
finally by continuity of a and f
We now have to provide a bound for ug , which completes expression (4.8) and gives a
bound for u. It is easy as ug must satisfy only Bu = g: we can choose ug such that, by
the closed range of B
kgkQ∗
kug kV ≤
.
β
Then, remembering that u = u0 + ug with u0 ∈ ker B and by relation (4.8), we have:
kukV
≤ ku0 kV + kug kV
1
1
(kf kV ∗ + kakkug kV ) + kgkQ∗
α
β
1
kak
1
≤
kf kV ∗ +
kgkQ∗ + kgkQ∗ ,
α
β
β
≤
which can be used in the bound for p above in order to obtain expressions (2.12). Recalling Section 2.2, we notice that the invertibility of a(·, ·) on ker B is a necessary and
sufficient condition for equation (4.7) to admit a unique solution u0 ∈ ker B. Therefore,
an inf-sup condition for a on ker B, equation (2.13), is a weaker necessary requirement
which can replace coercivity over ker B.
4.1.3
General ideas for the proof of the Closed Range Theorem for
linear continuous operators
In this Subsection some concepts, e.g. the polar space and the norm over the quotient
space, although already introduced, will be defined more precisely.
58
A continuous linear form between Hilbert spaces is defined:
A : U −→ V,
A ∈ L(U, V ) ,
and its adjoint or transpose At ∈ L(V ∗ , U ∗ ):
At : V ∗ −→ U ∗ :
hv 0 , AuiV ∗ ×V = hAt v 0 , uiU ∗ ×U ,
∀u ∈ U, v 0 ∈ V ∗ .
Properties for the adjoint operator (2, 4, 6 in Theorem 1) follow naturally from those of
the original one (1, 3, 5 in Theorem 1). The range of the adjoint is in fact always closed:
taking a convergent sequence {vn0 }n∈N ∈ V ∗ :
hvn0 , AuiV ∗ ×V = hAt vn0 , uiU ∗ ×U ,
∀u ∈ D(A) ,
if vn0 → v 0 and At vn0 → u0 :
hv 0 , AuiV ∗ ×V = hu0 , uiU ∗ ×U ,
∀u ∈ D(A) ,
and u0 := At v 0 .
We can then deal with statements 1, 3 and 5, using the following Propositions. First of
all, we can define the polar space of a subspace X ⊂ U :
X 0 = x0 ∈ U ∗ : hx0 , xiU ∗ ×U = 0 ∀x ∈ X .
Proposition 6 [16] Im(A) is closed if and only if Im(A) = ker(At )0 .
Proof :
By definition of the adjoint operator At :
hv 0 , AuiV ∗ ×V = hAt v 0 , uiU ∗ ×U ,
∀u ∈ U, v 0 ∈ V ∗ ,
and v 0 ∈ ker At if and only if v 0 ∈ Im(A)0 .
We recall that for any subspace Z of a normed space X, Z is closed if and only if
0
Z 0 ≡ Z. Thus,
0
0
Im(A) = Im(A)0 = ker At ,
if and only if Im(A) is closed.
It is straightforward to apply this result to prove equivalence of statements 1 and 3
of the Theorem.
We say that A ∈ L(U, V ) is bounded from below if there exists c > 0:
kAukV ≥ ckukU ,
∀u ∈ D(A) .
This implies that A has a continuous inverse on its range. The inverse exists, since A is
injective: if Au = 0 then
0 = kAukV ≥ ckukU = 0 ,
and u = 0. Therefore ker A is trivial and A is injective. For every u ∈ U there exists
v ∈ Im(A) such that u = A−1 v. Since A is bounded form below:
1
kA−1 vkU ≤ kvkV ,
c
and A−1 is continuous.
59
∀v ∈ Im(A) ,
Proposition 7 [16] Given A ∈ L(U, V ) injective, Im(A) is closed if and only if A is
bounded from below.
Proof :
• Hypothesis: A is bounded from below.
Let us define a convergent sequence {vn }n∈N ∈ Im(A). If vn → v, it must be
proved that v ∈ Im(A). There exists a sequence {un }n∈N ∈ U : Aun = vn . A is
bounded from below, then, for n, m ∈ N:
kvn − vm kV = kAun − Aum kV = kA(un − um )kV ≥ ckun − um kU .
Since {vn }n∈N is convergent, kun − um kU → 0. That means, for every ε > 0, there
exists N ∗ such that for n, m ≥ N ∗ :
kun − um kU ≤ ε ,
and {un }n∈N is a Cauchy sequence in a Banach space, then convergent. By continuity of A:
un → u ⇒ Aun → Au ,
and by uniqueness of the limit Au = v ∈ Im(A).
• Hypothesis: Im(A) is closed.
If Im(A) is closed, then the restriction
A : U −→ Im(A) ,
is a continuous and injective form between Banach spaces. Then, by the Banach
Theorem, there exists A−1 and it is continuous:
kA−1 vkU ≤ ckvkV ,
∀v ∈ Im(A),
0 < c := kA−1 k .
As there exists u = A−1 v, the condition above is equivalent to A being bounded
from below. Taking a closed subspace M ⊂ U , we consider the quotient space U/M . It contains
the equivalence classes [u] ∈ U/M of elements of U defined up to an element of M . We
we can define the new norm:
k[u]kU/M := inf kwkU = inf ku + mkU ,
m∈M
w∈[u]
where u is any representer of the class [u].
Proposition 8 [16] Given A ∈ L(U, V ) (not necessarily injective!), Im(A) is closed if
and only if
kAukV ≥ c inf ku + v0 kU = k[u]kV / ker A , c > 0 .
v0 ∈ker A
60
Proof :
If A is a continuous mapping, then its kernel is closed. As a matter of fact, if a
sequence {un }n∈N in ker A is convergent, then by continuity of A and uniqueness of the
limit, the sequence converges to an element in the kernel:
un ∈ ker A → u
⇒
Aun = 0 → Au = 0 ,
and u ∈ ker A.
The quotient space U/ ker A is a Banach space, since ker A is closed and U is a Banach
space. We can therefore define an injective operator Ã:
à : U/ ker A −→ V,
Ã[u] := Aw ,
where w ∈ U is any representer of the equivalence class [u] ∈ U/ ker A. We can give the
following upper bound:
kÃ[u]kV = kAwkV ≤ kAkkwkU ,
∀w ∈ [u] .
By arbitrarity of w ∈ [u]:
kÃ[u]kV = inf kAwkV ≤ kAk inf kwkU = kAkk[u]kU/ ker A .
w∈[u]
w∈[u]
We observe that à is continuous (from the inequality above) and injective and that
Im(A) ≡ Im(Ã). Proposition 7 can then be applied: Im(A) is closed if and only if à is
bounded from below:
kÃ[u]kV ≥ ck[u]kU/ ker A ,
∀[u] ∈ U/ ker A,
c>0.
That is equivalent to:
kAukV ≥ c
whence the thesis.
inf
v0 ∈ker A
ku + v0 kU ,
c>0,
Everywhere else, equivalence classes and representers will be identified, as long as
this will not lead to any confusion.
4.2
Comparison between Babuška and Brezzi bounds
Babuška and Brezzi Theories originate two different bounds for the solution of the same
problem. Is there a relation between them? Actually in [10] a proof is given that any
of the two conditions implies the other. We can then spend some time in showing this
equivalence.
Proposition 9 Given conditions (2.10), inf-sup for the form b, and (2.13), ker-inf-sup
for a, then (4.2) follows, in the l1-norm (but also in the l2-norm, [9]).
61
Proof :
We have proved in the previous Section that, under these assumptions, the saddle
point problem has a solution (u, p) ∈ V × Q which satisfies
1
kak
1
kf kV ∗ +
1+
kgkQ∗ ,
kukV
≤
α0
β
α0
kak
kak
kak
1
1+
kf kV ∗ + 2 1 +
kgkQ∗ ,
kpkQ\ker B t ≤
β
α0
β
α0
where α0 is the constant appearing in (2.13). Taking the l1 norm in the product space
V × Q we see that
k(u, p)kl1−V ×Q := kukV + kpkQ/ ker B t
≤ kf kV
∗
≤ max
1
1
+
α0 β
1
1
+
α0 β
kak
1+
α0
+ kgk
Q∗
kak
1
kak
1+
+ 2
α0
β
β
kak
kak
1
kak
1+
+ 2
(kf kV ∗ + kgkQ∗ ) .
, 1+
α0
α0
β
β
So, there exists a γl1 like in (4.2)
1
1
1
kak
kak
kak
1
≤ max
+
1+
+ 2
.
, 1+
γl1
α0 β
α0
α0
β
β
Following a completely different procedure [9], we could actually find another result, this
time in the l2 norm, such that
1
≤K,
(4.9)
γl2
where
p
2
K2 ≤ k11 + k22 + (k11 − k22 )2 + 4k12
≤ k12 + max(k11 , k22 ) ,
(4.10)
k=
β −1 kak,
k11 =
α0−2 (1
+
k 2 ),
k22 =
k2 k
11
+
β −2 ,
k12 = kk11 . Proposition 10 Given condition (4.1) in the l1-norm, conditions (2.10) and (2.13)
follow.
Proof :
We use (4.1) in the l1-norm:
a(u, v) + b(v, p) + b(u, q)
≥ γl1 k(u, p)kV ×Q ,
k(v, q)kV ×Q
v∈V q∈Q
sup
62
∀(u, p) ∈ V × Q .
Setting u = 0:
b(v, p)
≥ γl1 kpkQ/ ker B t ,
v∈V q∈Q k(v, q)kV ×Q
sup
∀p ∈ Q ,
and the sup is attained at q ≡ 0:
sup
v∈V
b(v, p)
≥ γl1 kpkQ/ ker B t ,
kvkV
∀p ∈ Q ,
which is exactly the condition (2.10).
Babuška condition (4.1) implies that the problem (2.7) has a solution for each righthand side f ∈ V ∗ , g ∈ Q∗ . We can then set g ≡ 0, which means that we are actually
solving the well-posed problem:
u0 ∈ V0 := ker B,
a(u0 , v0 ) = f (v0 ),
∀v0 ∈ V0 .
Using the linear algebra terminology, a is non-singular. Babuška condition becomes
a(u0 , v) + b(v, p)
≥ γl1 k(u0 , p)kV ×Q ,
k(v, q)kV ×Q
v∈V q∈Q
sup
which means
inf
sup
u0 ∈V0 ,p∈Q v∈V
∀u0 ∈ V0 , p ∈ Q ,
a(u0 , v) + b(v, p)
≥ γl1 .
kvkV k(u0 , p)kV ×Q
We have therefore restricted B to (V0 × Q) × (V × {0}). Like in (2.10) and the corresponding equation, the inf-sup condition holds on the adjoint operator if it is injective.
We still have no evidence of that, so every element of the domain must be considered
up to an element of the kernel:
a(u0 , v) + b(v, p)
≥ γl1 ,
v∈V \V00 u0 ∈ker B,p∈Q kvkV k(u0 , p)kV ×Q
inf
sup
where
V00 = {v ∈ V :
a(u0 , v) + b(p, v) = 0 ∀u0 ∈ V0 , p ∈ Q} =
,
= {v0 ∈ V0 :
a(u0 , v0 ) = 0
∀u0 ∈ V0 }
and as a is non-singular, then it must be v0 ≡ 0 and V00 = {0}.
Then:
a(u0 , v) + b(v, p)
≥ γl1 ,
inf
sup
v∈V u0 ∈ker B,p∈Q kvkV k(u0 , p)kV ×Q
and taking v = v0 ∈ V0 :
inf
sup
v0 ∈V0 u0 ∈ker B
a(u0 , v0 )
≥ γl1 ,
kv0 kV ku0 kV
where we can also invert inf and sup, obtaining condition (2.13).
63
4.3
Applications and results
4.3.1
The error equation
We consider again the parametrized saddle point problem (2.29) and, for the Stokes
problem and its RB approximation, the definitions of the errors (3.16) and of the residuals
(3.17). Replacing the errors euN (µ), epN (µ) as new unknown in problem (2.29), we get the
error equations:
p
u (v; µ), ∀v ∈ X ,
a(euN (µ), v; µ) + b(v, eN (µ); µ) = rN
(4.11)
p
u
b(eN (µ), q; µ) = rN (q; µ),
∀q ∈ Y .
We can then apply the Babuška and Brezzi (now parameter-dependent) bounds (4.2),
(2.12) to the errors, replacing the new right-hand sides:
• from Babuška :
k(euN (µ), epN (µ))kV ×Q ≤
1
p
(µ))kV ∗ ×Q∗ ;
k(ru (µ), rN
γ(µ) N
(4.12)
• from Brezzi:
keu
N (µ)kV
kepN (µ)kQ/ ker B t
≤
1
1
kru (µ)kV ∗ +
α(µ) N
β(µ)
≤
1
β(µ)
1+
ka(µ)k
α(µ)
p
krN
(µ)kQ∗ ,
ka(µ)k
ka(µ)k
ka(µ)k
p
u
1+
krN
(µ)kV ∗ + 2
1+
krN
(µ)kQ∗ .
α(µ)
β (µ)
α(µ)
(4.13)
The right hand sides of the inequalities above, (4.12) and (4.13), are the error
bounds for the RB approximation: they depend only on the RB solution (and not on
the FE truth solution) and therefore they are a posteriori bounds.
4.3.2
Tests
We will refer below to the Brezzi bounds for velocity and pressure (or their errors) and to
the Babuška combined bound, for each problem (parameter-dependent or -independent),
as to:
∆Br
u
1
p
k(ru (µ), rN
(µ))kV ∗ ×Q∗ ,
γ(µ) N
1
1
ka(µ)k
p
u
:=
krN
(µ)kV ∗ +
1+
krN
(µ)kQ∗ ,
α(µ)
β(µ)
α(µ)
∆Br
p
:=
∆Bab
u,p
:=
1
β(µ)
from (4.12) ,
from the 1.eq. of (4.13) ,
ka(µ)k
ka(µ)k
ka(µ)k
p
u
1+
krN
(µ)kV ∗ + 2
1+
krN
(µ)kQ∗ , from the 2.eq. of (4.13) ,
α(µ)
β (µ)
α(µ)
64
and
∆Br
u,p =
q
2
Br 2
(∆Br
u ) + (∆p ) .
Since obviously kuk2V ≤ kuk2V + kpk2Q and the same for kpkQ , we can define
∆Bab
:= ∆Bab
u
u,p ,
∆Bab
:= ∆Bab
p
u,p .
We furthermore define again, in the context of the parametrized saddle point problem
formulation:
kf(µ)kV ∗
,
ρ(µ) :=
kg(µ)kQ∗
and, in the context of the parametrized error equations for the approximated problem:
ρ(µ) :=
u (µ)k ∗
krN
V
,
p
krN (µ)kQ∗
as in the expression (3.18). Comparing the Babuška and Brezzi bound involves the
introduction of the quantity ρ, as we will see.
Referring to the Stokes problem (2.7) and its approximation, as anticipated, we are
going to make some simple tests, making different assumptions on the domain Ω. The
coercivity and continuity constants for A can be computed through a generalized eigenvalue problem, equations (3.13), (3.14). The stability of the approximation is guaranteed
by the positivity of the Brezzi inf-sup constant, equation (3.11). The Babuška constant
is computed over the whole system, equation (4.3).
In the following test-problems, the domain Ω is parameter-free, then A ≡ V and
α = kak = 1. In this case, the Brezzi bound becomes:
2
kukV
≤ kfkV ∗ + kgkQ∗ ,
β
kpkQ/ ker B t
2
2
kfkV ∗ + 2 kgkQ∗ .
β
β
≤
In order to see which is the better bound, we can then compare the following ratios,
which turn out to be ρ-dependent:
p
∆Bab
1 1 + ρ2
u
=
,
∆Br
γ (ρ + β2 )
u
p
∆Bab
β 2 1 + ρ2
p
,
=
∆Br
2γ (ρβ + 1)
p
∆Bab
1
u,p
=
Br
∆u,p
γ
s
(ρ +
1 + ρ2
+ 4( βρ +
2 2
β)
65
1 2
)
β2
.
We observe that the quantity ρ has a double purpose: it is the criterium for the greedy
surrogate for enrichment option 3 that we saw in Section 3.3.2, and a quantity which appears in the bounds comparison here, even for non-parametrized problems. We consider
then three cases:
1. Fully-periodic case.
Considering Ω = (0, 2π) × (0, 2π), we can write the periodic solution in terms of
Fourier coefficients:
X
X
u=
ûk eik◦x , p =
p̂k eikx .
k
k
The bilinear forms are:
Z
∂ui ∂v̄i
= (ûk )i (v̂k )i |k|2 (2π)2 ,
∂xj ∂xj
a(u, v) =
Ω
Z
b(u, q) = −
q̄
Ω
∂ui
= −(ûk )i q̂k iki (2π)2 ,
∂xi
and the L2 (Ω) norm is defined by the mass bilinear form:
Z
m(p, q) =
pq̄ = (2π)2 .
Ω
The relations above are a consequence of the orthogonality relation:
Z 2π
2π, if k = k 0 ,
0
ei(k−k ) =
0
0, otherwise.
The matrices for the problem, where the 2πs cancel out, are then:
|k|2
0
,
A≡X=
2
0 |k|
B=
−ik1 −ik2
,
Y =1.
There are as many identical linear systems as kmax we choose for approximating
the solution. A condition on the wavenumber kmax must be set so that u and p
have bounded energy. First of all, as our domain is periodic, then it must be:
Z
Z
u=0,
p=0,
Ω
Ω
66
which means
û0 = 0 ,
p̂0 = 0 .
By Parseval identity, the total energy can be computed as the sum of the energies of
each coefficient. We compute the norm with the X, Y matrices for each component
and we sum up:
X
X
|p̂k |2 < ∞ .
|k|2 |ûk |2 < ∞,
k\{0}
k\{0}
Given these preliminar assumptions, for each component:
β=
γ=
inf
q∈Q/ ker B t
q̂k BX −1 B 0 q̂k
=1,
q̂k (2π)2 q̂k
(v̂k , qk )BX −1 B 0 (v̂k , qk )
= 0.6180 .
(v̂k , qk )X (v̂k , qk )
(v,q)∈V ×Q
inf
In Figure 4.1 the trends of the three ∆ ratios, as a function of ρ in a [0.1, 100]
interval, are shown. The curve being above one means that
∆Bab ≥ ∆Br ,
and therefore that the Brezzi bound is sharper. In this first case, that happens for
the velocity only, for ρ ≥ 1, while Babuška stays sharper both for the pressure and
the combined bound.
67
Pressure Error Bound ratio
Velocity Error Bound ratio
1.8
1
1.7
0.9
1.6
∆Bab
/ ∆Br
p
p
∆Bab
/ ∆Br
u
u
1.5
1.4
1.3
1.2
0.8
0.7
1.1
0.6
1
0.9
0.8 −1
10
0
1
10
ρ
(a)
0.5 −1
10
2
10
10
0
10
∆Bab
u
(ρ) .
∆Br
u
(b)
1
ρ
10
∆Bab
p
(ρ) .
∆Br
p
Combined Velocity and Pressure Error Bound ratio
1
0.9
∆Bab
/ ∆Br
u,p
u,p
0.8
0.7
0.6
0.5
0.4 −1
10
0
10
(c)
1
ρ
10
2
10
∆Bab
u,p
(ρ) .
∆Br
u,p
Figure 4.1: Ratio of the Babuška and Brezzi error bound for the Stokes problem on a
square with fully-periodic b.c. (velocity, pressure and combined variables).
68
2
10
2. Semi-periodic case.
Periodicity is just set upon the y-component: we take Ω = (−1, 1) × (0, 2π), and
the Fourier decomposition of the solution is:
X
X
ûk (x)eiky , p =
p̂k (x)eiky .
u=
k
k
The Fourier coefficients are now x-dependent: we solve the system by mean of a 1D
FE approximation (with the usual Taylor-Hood element and homogenous Dirichlet
boundary conditions).
We can define the bilinear forms:
Z
∂ui ∂v̄i
a(u, v) =
=
Ω ∂xj ∂xj
Z
1
= 2π
−1
(û0k )x (v̂k0 )x + k 2 (ûk )x (v̂k )x + (û0k )y (v̂k0 )y + k 2 (ûk )y (v̂k )y dx ,
Z
b(u, q)
∂ui
= − q̄
= −2π
Ω ∂xi
Z
m(p, q) =
Z
Z
1
−1
p̂k (v̂k0 )x + ik p̂k (v̂k )y dx ,
1
pq̄ = 2π
p̂k q̂k .
−1
Ω
The 2π-free matrices of the system are then:
2
k Muu + Mu0 u0
0
, B = −Mu0 p −ikMup ,
A=
0
k 2 Muu + Mu0 u0
where M(·,·) are matrices corresponding to the quantities defined by the bilinear
forms in the 1D FE approximation. We perform the usual computation of the
inf-sup constants. This time the result is k-dependent. As both constants increase
with k, we take the lowest value, which occurs at k = 1.
β=
and
γ=
inf
q∈Q/ ker B t
q̂k (x)BX −1 B 0 q̂k (x)
= 0.4736 ,
q̂k (x)Y q̂k (x)
(v̂k (x), qk (x))BX −1 B 0 (v̂k (x), qk (x))
= 0.1887 .
(v̂k (x), qk (x))X (v̂k (x), qk (x))
(v,q)∈V ×Q
inf
These are the ρ-dependent trends of the ∆s. We observe that the Brezzi bound
is now sharper than the Babuška one for every value of ρ, for the velocity, while
for both the pressure and the combined variables, Brezzi becomes slightly (by a
factor ' 1.2) sharper than Babuška for ρ ≥ 10.
69
Pressure Error Bound ratio
Velocity Error Bound ratio
5.5
1.3
5
1.2
4.5
1.1
1
∆Bab
/ ∆Br
p
p
∆Bab
/ ∆Br
u
u
4
3.5
3
0.9
0.8
2.5
0.7
2
0.6
1.5
1 −1
10
0
1
10
(a)
ρ
0.5 −1
10
2
10
10
0
10
∆Bab
u
(ρ) .
∆Br
u
(b)
1
ρ
10
∆Bab
p
(ρ) .
∆Br
p
Combined Velocity and Pressure Error Bound ratio
1.3
1.2
1.1
∆Bab
/ ∆Br
u,p
u,p
1
0.9
0.8
0.7
0.6
0.5
0.4 −1
10
0
10
(c)
1
ρ
10
2
10
∆Bab
u,p
(ρ) .
∆Br
u,p
Figure 4.2: Ratio of the Babuška and Brezzi error bound for the Stokes problem on a
square with semi-periodic b.c. (velocity, pressure and combined variables).
70
2
10
3. Square.
We take a square Ω = (−1, 1) × (−1, 1) with homogeneous Dirichlet b.c. on all
edges, apply the 2D FE approximation and compute the matrices A, B, X, Y .
Computing the eigenvalue problems required for β as in (3.11) and for γ as in (4.3),
we notice that the first eigenvalue is zero: ker B t is not trivial and the pressure
is defined up to a constant. As we take p ∈ Q/ ker B t , the first zero-eigenvalue is
descarded and the second lowest one taken. We increase the degrees of freedom in
order to check the convergence of β and γ (in Chapter 2 we have seen that with a
good FE space they have to be h-uniform). We stop when the increase or decrease
rate between a constant value and the previous one is sufficiently small.
velocity d.o.f.s
13
41
145
545
2113
8321
33052
131585
β
0.5000
0.5005
0.4779
0.4660
0.4579
0.4522
0.4480
0.4447
β = 0.4447,
rate
+0.0010
-0.0451
-0.0250
-0.0172
-0.0125
-0.0094
-0.0073
γ
0.2071
0.2075
0.1917
0.1835
0.1780
0.1742
0.1713
0.1691
rate
+0.0017
-0.0762
-0.0428
-0.0297
-0.0216
-0.0163
-0.0127
γ = 0.1691 .
Observing Figures 4.3, we remark that the trends are definitely similar to the ones
obtained for the semi-periodic case.
71
Velocity Error Bound ratio
Pressure Error Bound ratio
6
1.3
5.5
1.2
5
1.1
1
∆Bab
/ ∆Br
p
p
4
u
∆Bab / ∆Br
4.5
u
3.5
3
0.9
0.8
2.5
0.7
2
0.6
1.5
1 −1
10
0
1
10
ρ
(a)
2
10
10
0.5 −1
10
∆Bab
u
(ρ) .
∆Br
u
0
10
(b)
1
ρ
10
∆Bab
p
(ρ) .
∆Br
p
Combined Velocity and Pressure Error Bound ratio
1.3
1.2
1.1
∆Bab
/ ∆Br
u,p
u,p
1
0.9
0.8
0.7
0.6
0.5
0.4 −1
10
0
1
10
(c)
ρ
10
2
10
∆Bab
u,p
(ρ) .
∆Br
u,p
Figure 4.3: Ratio of the Babuška and Brezzi error bound for the Stokes problem on a
square with fully-homogeneous Dirichlet b.c. (velocity, pressure and combined variables).
72
2
10
We can briefly summarize the results:
• the Brezzi bound for the velocity is usually slightly sharper (1 to 5 times) than the
Babuška one;
• the Brezzi bound for the pressure and the combined variables is worse than the
Babuška one for fully periodic (by a factor 0.6 − 0.7). For semiperiodic and square,
Brezzi becomes slightly (1.1 to 1.3 times) sharper than Babuška from ρ ' 10 on.
4.3.3
The channel
We have analyzed the Stokes problem on a simple square with different boundary conditions. The tests on the error bound we have run so far finally find an application if we
focus on our case-study, the channel with a µ-parametrized (width × height) obstacle.
In this case, the bounds will provide the a posteriori estimation for the RB approximation. We must first of all check the convergence of β(µ) and γ(µ), taking for instance
µ = µref = (0.3, 0.3):
velocity d.o.f.s
4977
11774
16244
18260
32834
43278
46607
130103
β(µref )
0.0960
0.0958
0.0958
0.0958
0.0958
0.0958
0.0958
0.0958
rate
-0.0021
0
0
0
0
0
0
γ(µref )
0.0091
0.0091
0.0091
0.0091
0.0091
0.0091
0.0091
0.0091
rate
0
0
0
0
0
0
0
Since convergence is immediately verified, all computations can be made with the
first mesh (as it has been done in the previous Chapters).The inf-sup constants and the
continuity and coercivity constants, ka(µ)k and α(µ) are now parameter-dependent. We
can therefore compute the Babuška and Brezzi bounds varying the first and then the
second parameter (the other being fixed at the reference value), and on the corners of
the parameters space D. We can observe the following plots:
• for the first varying parameter, Figures 4.4;
• for the second varying parameter, Figures 4.5;
• on the corners of D, Figures 4.6.
73
Velocity Error Bound ratio
Pressure Error Bound ratio
100
5
µ=(0.1,0.3)
µ=(0.1,0.3)
µ=(0.2,0.3)
80
4
µ=(0.2,0.3)
µ=(0.3,0.3)
µ=(0.4,0.3)
60
∆Bab
/ ∆Br
p
p
∆Bab
/ ∆Br
u
u
µ=(0.3,0.3)
µ=(0.5,0.3)
40
20
0 −1
10
µ=(0.4,0.3)
3
µ=(0.5,0.3)
2
1
0
1
10
ρ
0 −1
10
2
10
10
∆Bab
(a) uBr (ρ; (µ1 , µ2−ref ) .
∆u
0
10
1
ρ
10
∆Bab
p
(b)
(ρ; (µ1 , µ2−ref ) .
∆Br
p
Combined Velocity and Pressure Error Bound ratio
5
µ=(0.1,0.3)
4
µ=(0.2,0.3)
∆Bab
/ ∆Br
u,p
u,p
µ=(0.3,0.3)
µ=(0.4,0.3)
3
µ=(0.5,0.3)
2
1
0 −1
10
0
10
(c)
1
ρ
10
2
10
∆Bab
u,p
(ρ; (µ1 , µ2−ref ) .
∆Br
u,p
Figure 4.4: Ratio of the Babuška and Brezzi error bound for the Stokes problem (velocity,
pressure and combined variables) at sample values of the parameters: varying of µ1 and
reference value for µ2 .
74
2
10
Velocity Error Bound ratio
Pressure Error Bound ratio
120
5
µ=(0.3,0.2)
µ=(0.3,0.2)
4
µ=(0.3,0.3)
µ=(0.3,0.3)
80
µ=(0.3,0.4)
∆Bab
/ ∆Br
p
p
∆Bab
/ ∆Br
u
u
µ=(0.3,0.1)
µ=(0.3,0.1)
100
µ=(0.3,0.5)
60
µ=(0.3,0.4)
3
µ=(0.3,0.5)
2
40
1
20
0 −1
10
0
1
10
ρ
0 −1
10
2
10
10
∆Bab
(a) uBr (ρ; (µ1−ref , µ2 ) .
∆u
0
10
1
ρ
10
∆Bab
p
(b)
(ρ; (µ1−ref , µ2 ) .
∆Br
p
Combined Velocity and Pressure Error Bound ratio
5
µ=(0.3,0.1)
µ=(0.3,0.2)
4
∆Bab
/ ∆Br
u,p
u,p
µ=(0.3,0.3)
µ=(0.3,0.4)
3
µ=(0.3,0.5)
2
1
0 −1
10
0
10
(c)
1
ρ
10
2
10
∆Bab
u,p
(ρ; (µ1−ref , µ2 ) .
∆Br
u,p
Figure 4.5: Ratio of the Babuška and Brezzi error bound for the Stokes problem (velocity,
pressure and combined variables) at sample values of the parameters: varying of µ2 and
reference value for µ1 .
75
2
10
Velocity Error Bound ratio
Pressure Error Bound ratio
100
2.5
µ=(0.1,0.1)
µ=(0.1,0.1)
µ=(0.5,0.1)
80
2
µ=(0.5,0.1)
µ=(0.5,0.5)
µ=(0.1,0.5)
60
∆Bab
/ ∆Br
p
p
∆Bab
/ ∆Br
u
u
µ=(0.5,0.5)
40
1
20
0 −1
10
µ=(0.1,0.5)
1.5
0.5
0
1
10
(a)
ρ
0 −1
10
2
10
10
0
10
∆Bab
u
(ρ; µ) .
∆Br
u
(b)
1
ρ
10
∆Bab
p
(ρ; µ) .
∆Br
p
Combined Velocity and Pressure Error Bound ratio
2.5
µ=(0.1,0.1)
µ=(0.5,0.1)
2
∆Bab
/ ∆Br
u,p
u,p
µ=(0.5,0.5)
µ=(0.1,0.5)
1.5
1
0.5
0 −1
10
0
10
(c)
1
ρ
10
2
10
∆Bab
u,p
(ρ; µ) .
∆Br
u,p
Figure 4.6: Ratio of the Babuška and Brezzi error bound for the Stokes problem (velocity,
pressure and combined variables) at sample values of the parameters: the edges of D .
76
2
10
As we have seen in the previous tests, the ratios have a σ-trend, where Brezzi becomes
sharper as ρ increases. In particular, the best situation for the Brezzi bound is µref : in
this case it overtakes Babuška for pressure and combined variables at ρ = 11, while it is
better for any values of ρ in the velocity case. These differences are due to the parameterdependence of the constants. We can remark that, although α(µ) and ka(µ)k have the
same trend when either the width or the height is changing, the inf-sup constants have
rather peculiar trends in the two cases, Figures 4.7.
β dependence on parameters values
0.12
−3
12
µ1 varying
x 10
γ dependence on parameters values
µ1 varying
11
µ varying
µ varying
2
0.11
2
10
9
γ
β
0.1
0.09
8
7
6
0.08
5
0.1
0.2
0.3
µ1,µ2
0.4
4
0.1
0.5
(a) β(µ1 , µ2−ref ), β(µ1−ref , µ2 ) .
0.2
0.3
µ1,µ2
0.4
0.5
(b) γ(µ1 , µ2−ref ), γ(µ1−ref , µ2 ) .
α dependence on parameters values
||a|| dependence on parameters values
3
1
µ1 varying
µ2 varying
0.9
2.5
0.8
0.6
µ1 varying
0.5
µ2 varying
γa
α
0.7
2
1.5
0.4
0.3
0.2
0.1
0.2
0.3
µ ,µ
1
0.4
1
0.1
0.5
2
0.2
0.3
µ ,µ
1
0.4
0.5
2
(d) ka(µ1 , µ2−ref )k, ka(µ1−ref , µ2 )k .
(c) α(µ1 , µ2−ref ), α(µ1−ref , µ2 ) .
Figure 4.7: Variation of the Brezzi and Babuška inf-sup constants, coercivity and continuity constants for a(·, ·; µ) with respect to the dimensions of the obstacle in the Stokes
Problem over a parametrized channel.
From a different perspective, we can vary continuously the parameters and plot out
the ratios for three values of ρ, Figures 4.8.
77
Velocity Error Bound ratio
50
ρ = 0.1
ρ=1
ρ = 10
30
40
∆Bab
/ ∆Br
u
u
Bab
/ ∆u
Br
40
∆u
Velocity Error Bound ratio
50
20
10
ρ = 0.1
30
ρ=1
ρ = 10
20
10
0
0.1
0.2
0.3
µ1
0.4
0
0.1
0.5
(a) ∆Bab
/∆Br
u
u (ρ) (µ1 , µ2−ref ) .
2.5
∆Bab
/ ∆Br
p
p
1.5
∆p
Bab
/ ∆p
Br
2
1
0.5
0.5
0.2
0.3
µ1
0.4
ρ = 0.1
1.5
1
0
0.1
ρ=1
ρ = 10
0
0.1
0.5
(c) ∆Bab
/∆Br
p
p (ρ) (µ1 , µ2−ref ) .
0.5
Combined Velocity and Pressure Error Bound ratio
2.5
2
Br
∆Bab
/ ∆Br
u,p
u,p
2
1.5
ρ = 0.1
1.5
1
1
0.5
0.5
0
0.1
0.4
3
ρ = 0.1
ρ=1
ρ = 10
2.5
0.3
µ2
Combined Velocity and Pressure Error Bound ratio
3
0.2
(d) ∆Bab
/∆Br
p
p (ρ) (µ1−ref , µ2 ) .
Bab
0.5
Pressure Error Bound ratio
2
∆u,p / ∆u,p
0.4
3
ρ = 0.1
ρ=1
ρ = 10
2.5
0.3
µ2
(b) ∆Bab
/∆Br
u
u (ρ) (µ1−ref , µ2 ) .
Pressure Error Bound ratio
3
0.2
0.2
0.3
µ
0.4
0
0.1
0.5
1
ρ=1
ρ = 10
0.2
0.3
µ
0.4
0.5
2
Br
(e) ∆Bab
u,p /∆u,p (ρ) (µ1 , µ2−ref ) .
Br
(f) ∆Bab
u,p /∆u,p (ρ) (µ1−ref , µ2 ) .
Figure 4.8: Ratio for three fixed values of ρ of the Babuška and Brezzi error bounds
for the velocity, the pressure and the combined variables in the Stokes Problem over a
parametrized channel. Dependence on the dimensions of the obstacle.
78
We conclude that it is worth using the Brezzi bound only if ρ takes sufficiently high
values. Since our goal is applying the bound to the error equations, like in (4.12) and
(4.13), in this case ρ is defined as in equation (3.18). It corresponds to the ratio of the
dual norm of the residuals and depends on the RB solution (uN (µ), pM (µ)), and on the
RB spaces.
We can then apply the results from the previous Chapter. Here evidence is given
that, for option 1 and 2 and for the kind of tests implemented, the values taken by ρ(µ)
(computed, at each step M ∗ , as a maximum among 60 test-parameters) are generally
within [1, 4]. We conclude that, at least in the framework described throughout this
work, as the values of ρ are small, then applying the Babuška error is the best choice,
although it could be expensive in terms of computational effort to compute the eigenvalue
problem over the whole system, (4.3).
We present the errors for velocity, pressure and combined variables for the channel
problem with their Brezzi and Babuška bounds. Option 1 has been used. For each
value of M ∗ , dimension of the pressure RB space, the maximum error (normalized with
respect to the norm of u(µ), p(µ) or (u(µ), p(µ))) is evaluated on a set of 60 random
parameters. In Figures 4.9, 4.10, 4.11 we can then see the errors, their bounds and the
ratios between them.
79
Normalized Velocity Error and Bound
2
10
error
Babuska bound
Brezzi bound
1
10
0
eu and ∆u
10
−1
10
−2
10
−3
10
−4
10
5
10
15
M MAX
(a) The relative error (red line)
(black) and
∆u
Br (µ)
(green).
ku(µ)kQ
20
25
keu
∆u
N (µ)kQ
Bab (µ)
, and the bounds:
ku(µ)kQ
ku(µ)kQ
Ratio Bound / Error (Velocity)
3
10
∆u / eu
Babuska
Brezzi
2
10
1
10
5
(b) The ratios
10
15
M MAX
20
25
∆u
∆u (µ)
Bab (µ)
(blue) and uBr
(green).
u
keN (µ)kQ
keN (µ)kQ
Figure 4.9: The velocity error between the truth solution of the Stokes problem and
the RB approximation (option 1) and its a posteriori Brezzi and Babuška upper bound;
ratio of the upper bound and the error. At each M ∗ , µ = arg 0max keuN (µ0 )kV .
80
µ ∈Stest
Normalized Pressure Error and Bound
1
10
error
Babuska bound
Brezzi bound
0
10
−1
ep and ∆p
10
−2
10
−3
10
−4
10
−5
10
5
10
15
M MAX
(a) The relative error (red line)
(black) and
∆pBr (µ)
(green).
kp(µ)kQ
20
25
∆pBab (µ)
kepN (µ)kQ
, and the bounds:
kp(µ)kQ
kp(µ)kQ
Ratio Bound / Error (Pressure)
4
10
Babuska
Brezzi
3
∆p / ep
10
2
10
1
10
5
(b) The ratios
10
15
M MAX
20
25
∆pBab (µ)
∆p (µ)
(blue) and pBr
(green).
p
keN (µ)kQ
keN (µ)kQ
Figure 4.10: The pressure error between the truth solution of the Stokes problem and
the RB approximation (option 1) and its a posteriori Brezzi and Babuška upper bound;
ratio of the upper bound and the error. At each M ∗ , µ = arg 0max kepN (µ0 )kV .
µ ∈Stest
81
Normalized Combined Error and Bound
1
10
error
Babuska bound
Brezzi bound
0
10
−1
eu,p and ∆u,p
10
−2
10
−3
10
−4
10
−5
10
5
10
15
M MAX
q
(a) The relative error (red line)
u,p
∆Bab (µ)
q
ku(µ)k2
+kp(µ)k2
V
Q
(black) and
20
25
p
keu
(µ)k2
+keN (µ)k2
N
V
Q
q
ku(µ)k2
+kp(µ)k2
V
Q
u,p
∆Br (µ)
q
ku(µ)k2
+kp(µ)k2
V
Q
, and the bounds
(green).
Ratio Bound / Error (Combined)
4
10
Babuska
Brezzi
3
∆u,p / eu,p
10
2
10
1
10
5
(b) The ratios
10
q
15
M MAX
u,p
∆Bab (µ)
p
u
keN (µ)k2
+keQ (µ)k2
V
V
(blue) and
20
q
25
u,p
∆Br (µ)
p
u
keN (µ)k2
+keN (µ)k2
V
Q
(green).
Figure 4.11: The combined (velocity and pressure) error between the truth solution of
the Stokes problem and the RB approximation (option 1) and its a posteriori Brezzi and
Babuška upperq
bound; ratio of the upper
bound and the error. At each M ∗ , µ =
82
p
u
2
2
0
0
keN (µ )kV + keN (µ )kQ .
arg maxµ0 ∈Stest
The Babuška bound is better than Brezzi for the pressure and the combined error,
while Brezzi is slightly better than Babuška for the velocity. This trend actually agrees
with the former results. We can plot out ρ at each M ∗ , evaluated at the parameters
considered for the three quantities, Figure 4.12. We can also plot out the values of the
parameters for each M ∗ , at which the maximum error, e.g. for the velocity, is reached:
Figures 4.13.
We compare these last plots with those at Figures 4.4, 4.5, 4.6 and 4.8 investigating
a relation between Babuška and Brezzi bound. Both the low values of ρ (' 1) and of µ
(far from µref = [0.3, 0.3]) confirm that, in general, the Babuška bound gives a better
approximation than the Brezzi one.
Residuals Ratio: Momentum / Continuity
2
velocity
pressure
combined
1.8
1.6
1.4
ρ
1.2
1
0.8
0.6
0.4
0.2
5
10
15
M MAX
20
25
Figure 4.12:
Ratio of the dual norm of the residuals of the momentum
and continuity equations, ρ(µ), for the option 1-RB approximation.
For
∗
each enrichment
step M , the parameter considered is respectively µ =
q
u
0
arg 0max keN (µ )kV , kepN (µ0 )kV , keuN (µ0 )k2V + kepN (µ0 )k2V .
µ ∈Stest
83
µ1 and µ2 (Velocity)
0.5
µ1
0.45
µ2
0.4
µ
0.35
0.3
0.25
0.2
0.15
0.1
5
10
15
M MAX
20
25
Figure 4.13: Values of the parameters where the error between the truth solution
and the option 1-RB approximation is maximum, at each enrichment step M ∗ : µ =
arg 0max keuN (µ0 )kV . Comparison with the µref value (0.3, 0.3) .
µ ∈Stest
84
4.4
Appendix: the Penalty Approach
The ∆ bounds so far considered can be useful in their application into the RB context
when their computation does not depend on the FE architecture, but rather only on the
low RB degrees of freedom N, M . An upper bound for ka(µ)k and lower bounds for
α(µ), β(µ) and γ(µ) are therefore required. Since the procedure is relatively easy only for
the first two quantities (SCM, [15]), we describe here an example which circumvents the
computation of lower bounds for β(µ) and γ(µ). This example is taken from [1] to show
how a good a posteriori error bound can be actually applied. The saddle point problem
is considered, introducing a penalty approximation. For the Stokes problem this means
relaxing the incompressibility constraint with almost - incompressibility. The new - now
coercive - problem consists in finding velocity and pressure (uε (µ), pε (µ)) ∈ V × Q such
that:
a(uε (µ), v; µ) + b(v, pε (µ); µ) = f (v; µ), ∀v ∈ V ,
b(uε (µ), q; µ) − εc(pε (µ), q; µ) = g(q; µ),
∀q ∈ Q ,
where, identifying the exact problem and FE (truth) approximation:
• a(·, ·; µ) is a continuous (ka(µ)k) and coercive (αa (µ)) bilinear form;
• c(·, ·; µ) is a continuous (kc(µ)k) and coercive (αc (µ)) bilinear form;
• b(·, ·; µ) is a continuous bilinear form satisfying the inf-sup stability condition
(β(µ) > 0).
The penalty formulation approximates the standard-formulation: for each ε > 0, there
exists C > 0 such that:
ku(µ) − uε (µ)kV ≤ Cε ,
kp(µ) − pε (µ)kQ ≤
C
ε.
β
We consider a new joint error for the velocity and the pressure:
ε
EN
M
=
2
keu,ε
N M (µ)kV
αc (µ) p,ε
+ε a
(µ)k2Q
ke
α (µ) N M
1
2
,
u,ε
p,ε
where eN
M (µ) and eN M (µ) are the differences between the penalty truth solution and
the penalty RB solution.
ε
A good a posteriori upper bound ∆εN M (µ) for EN
M must be:
• rigorous:
∆εN M (µ)
≥1;
ε
EN
M
• sharp: the above ratio must not be too big;
• efficient: the computation of ∆εN M (µ) must be N -independent.
85
It is possible to prove that this is a good choice:
∆εN M (µ)
:=
1
a
αLB
1
a (µ)
2
αLB
p
u
2
2
krN (µ)kV ∗ + c
krN (µ)kQ∗
.
εαLB (µ)
It depends on:
• the dual norms of the residuals: we have seen that they can be computed with
N -dependent computational costs only in the offline phase;
• the lower bounds (LB) of the coercivity constants: the Successive Constraints
Method (SCM) allows us to compute them online, with the FE quantities already
stored offline;
As a matter of fact, a priori strategies providing rapid convergence of the RB space
exist only for 1D parameters. For the 2D or higher case, we have to build adaptive
spaces, connected to the particular problem, for instance through a Greedy algorithm.
A big train parameters space is generated, a surrogate of the global space:
Dtrain ⊂ D .
Then an initial sample space is defined, e.g. randomly: S = {µ1 , . . . , µN0 }. At each step
∗
M ∗ ≥ N0 of the enrichment, the solution corresponding to the parameter µM enters the
RB spaces. This parameter is chosen as the argument which maximizes the normalized
error bound over the whole train-set:
∗
µM = arg max
µ∈Dtrain
∆εN M (µ)
.
k (uε (µ), pε (µ)) kV ×Q
The nature of this operation explains why the error bound must be efficiently computed:
∆εN M (µ) must be evaluated over the whole train-set (O(106 ) elements). If the maximum
error bound is low enough (up to a prescribed tolerance εtol )
∗
∆N M (µM ) ≤ εtol ,
∗
then M = M ∗ and the final RB spaces are built upon S = {µ1 , . . . , µM }. If the
∗
requested precision is not yet reached, then µM is picked as new parameter and the
procedure is re-iterated.
86
Conclusion
Wrapping everything up, this work can be split into two main studies, both focused on
the Stokes problem as a saddle point problem: a good employment of the supremizer
operator for getting stable RB spaces and a sharp a posteriori error bound for the RB
approximation.
As far as the first problem is concerned, we witnessed that using the supremizer
is indespensable for avoiding instability. In particular the results from options 2 and
3, for enriching the RB velocity space, are significant. Option 2 gives good results of
approximation, even though the Brezzi stability condition for saddle point problems
cannot be proved. A further investigation could be carried out, for differently proving
stability in an analytical way. Option 3 exploits a hybrid enriching method, including a
partially greedy procedure. It turns out to be stable and to give very good approximation
results. The bases are enriched with twice the number of elements with respect to option
1: this is a good characteristic of greedy approaches. Option 3 arises from the evaluation
of the ratio of the dual norms of the equations residuals, which is not a classically
considered parameter for enhancing stability.
The a posteriori error bound originated by the Babuška Theory is trustworthy for
noncoercive problems. It actually is the best bound for the examined Stokes problem too,
even if in this case Brezzi Theory can be applied for getting a bound. As a matter of fact,
an efficient and effective sampling strategy for the parameters in the RB spaces of the
solutions requires an efficient and sharp a posteriori error bound procedure. The Babuška
bound is certainly not efficient, although sharp. The Brezzi bound we tested in this work
is not sharp enough: it gives slightly better results than Babuška for the velocity error,
but, at least in our application, it produces a worse bound for the pressure and the
combined variables. We have seen how the geometry of the problem and the choice
of the enriching option differenctly affect the behavior of the two bounds, favoring the
Babuška bound in the considered workspace. Finally, the so far witnessed best approach
consists in circumventing the computation of the Babuška and Brezzi inf-sup constants,
for instance through a penalty approach for the Stokes problem, which nevertheless
implies a modification in the formulation of the problem itself.
87
Bibliography
[1] A.-L. Gerner, K. Veroy. Reduced Basis a posteriori Error Bounds for the Stokes
Equations in parametrized domains: a Penalty Approach. 2010;
[2] G. Rozza, K. Veroy. On the stability of the reduced basis method for Stokes equations
in parametrized domains. Comput. Methods Appl. Mech. Engrg. 196 1244-1260, 2007;
[3] K. Veroy. Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and
Applications. PhD Thesis, Massachusetts Institute of Technology, 2003;
[4] D. V. Rovas. Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. PhD Thesis, Massachusetts Institute of Technology, 2003;
[5] I. Babuška. Error Bounds for Finite Element Method. Numer. Math. 16, 322-333,
1971;
[6] F. Brezzi. On the Existence, Uniqueness and Approximation of Saddle-Point Problems arising from Lagrangian Multipliers. Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique, tome 8, n. 2, p. 129-151, 1974;
[7] A. Patera, G. Rozza. Reduced Basis Approximation and a posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate
Monographs in Mechanical Engineering, 2007;
[8] F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag,
1991;
[9] J. Xu, L. Zikatanov. Some observations on Babuška and Brezzi theories. PennstateDepartment of Mathematics, 2000;
[10] L. Demkowicz. Babuška ⇔ Brezzi ?? Texas Institute for Computational and Applied Mathematics;
[11] Y. Maday, A.T. Patera, and G. Turinici. A priori Convergence Theory for ReducedBasis Approximations of Single-Parameter Elliptic Partial Differential Equations.
Journal of Scientific Computing, Vol. 17, Nos. 1-4, 2002;
88
[12] A. Buffa, Y. Maday, A.T. Patera, C, Prud’homme and G. Turinici. A priori Convergence of the Greedy Algorithm for the parametrized Reduced Basis. Mathematical
Modelling and Numerical Analysis, 2009;
[13] Y. Maday, D. Meiron, A.T. Patera, and E.M. Ronquist. Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element
discretizations. SIAM J. Sci. Comput., Vol. 14, No. 2, pp. 310-337, 1993;
[14] D.B.P. Huynh, D.J. Knezevic, Y. Chen, J.S. Hesthaven, A.T. Patera. A NaturalNorm Successive Constraint Method for Inf-Sup Lower Bounds.2009;
[15] D.B.P. Huynh, G. Rozza, S. Sen, A.T. Patera. A successive constraint linear optimization method for lower bounds of parametric coercivity and infsup stability constants. C. R. Acad. Sci. Paris, Ser. I 345 pp. 473-478, 2007;
[16] J. Tinsley Oden, L.F. Demkowicz. Applied Functional Analysis. CRC, 1996;
[17] A. Quarteroni, A. Valli. Numerical Approximation of Partial Differential Equations.
Springer, 2008;
[18] C.C. Mei. Mathematical Analysis in Engineering. Cambridge, 1997;
[19] http://en.wikipedia.org/wiki/File:Saddle point.png .
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