The mortar element method for quasilinear elliptic boundary
value problems
Leszek Marcinkowski
1
Abstract
We consider a discretization of quasilinear elliptic boundary value problems by the mortar
version of finite element method. We show that the error estimate is of the same order as in
the standard conforming finite element method. We propose a method of solving the discrete
problem and prove its convergence. The method combines the Schwarz preconditioning technique
with the Richardson-Newton method.
Keywords. Nonlinear elliptic problem, mortar finite element, additive Schwarz method
AMS(MOS) subject classifications: 65N30,65N15,65N55
1. INTRODUCTION
Recently variational conforming and nonconforming decomposition methods are analyzed and used
to approximate differential equations. A large problem is split into some smaller ones that can, for
example, be solved independently. It was natural to consider a method that use locally in each
subdomain an independent discretization adapted to the local properties of the solution. One of the
invented methods was the mortar element method which ensures a good transmission of information
between adjacent subdomains. This transmission (in some sense) is optimal. We refer for a general
presentation of the mortar method for linear problems to [7], [5] and [6] and for a presentation of
the matching constraints in terms of Lagrange multipliers to [4].
Recently, there is a development of parallel algorithms devoted to solve systems of linear equations
arising from the mortar version of finite element discretization for linear elliptic problems, see [2, 3,
11, 13].
The goal of this paper is to give an analysis of the error of the mortar version of finite element
discretization applied to some nonlinear problems and discuss a domain decomposition method for
solving the discrete problem in geometrically conforming case. For our knowledge there is no results
devoted to such topics for nonlinear problems. Namely, in this paper we discuss the application of
the mortar element method to a second-order nonlinear elliptic boundary value problem with the
strongly monotone and Lipschitz continuous operator in a polygonal region Ω with the Lipschitz
boundary. We consider first the geometrically conforming case of the mortar element method, i.e.
the intersection of the closures of two subdomains can be the empty set, an edge or a vertex, and a
more general case: the geometrically nonconforming one, when we do not impose this condition. In
the first case, we see that the estimate of the error is of the same order as in the standard conforming
piecewise linear finite element discretization provided that the solution of the differential problem
is in H 2 (Ω). In the latter one we have to strengthen our regularity assumption. Namely, we shall
assume that the solution of the differential problem belongs to the space H 5/2 (Ω) in order to derive
the same error estimate. The technique that we use to obtain our estimates is a generalization of
that used for linear problems, cf. [7] and [5]. We propose an algorithm for solving discrete problem
in case of the geometrically conforming version of the mortar element method, combining Schwarz
methods with Newton’s one. Namely we construct a preconditioner in terms of Additive Schwarz
1 Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland.
Electronic mail address [email protected]. This work has been supported in part by Polish Scientific Grant
102/P03/95/09
1
Method for a linear problem and apply it to the Richardson iteration for the discretized nonlinear
problems considered in this paper. The implementation of our preconditioner can be done in parallel
and consists of solving one global linear problem on coarse grid and two kinds of local problems. The
global problem is the standard conforming finite element one defined for Poisson’s equation. The
first kind of local problems consists of ones that are one dimensional and are associated with the
vertices of subdomains. The second kind includes the ones that are defined on two substructures
with a common edge. We use only nodal basis functions of mortar method corresponding to nodes,
such that values of functions at these nodes are degrees of freedom of the method.
The outline of the paper is as follows. In Section 2, we formulate the differential problem.
Section 3 is devoted to presenting the mortar element method and studying the error estimate in
the geometrically conforming case. In Section 4, we present the geometrically nonconforming case
of the mortar method. In Section 5, we discuss an application of Richardson-Newton method to
the system of nonlinear equations which arise from discretization of the boundary value problem by
the geometrically conforming mortar element method. We show that this method is almost optimal
convergent.
2. DIFFERENTIAL PROBLEM
In this section we formulate the differential problem.
We consider the following differential equation
−
2
X
∂
ai (x, u∗ , ∇u∗ ) + a0 (x, u∗ , ∇u∗ ) = f (x) in Ω
∂x
i
i=1
(2.1)
with the homogeneous Dirichlet boundary condition, where Ω ⊂ R 2 is Lipschitz continuous bounded
polygonal region. The weak formulation is of the form:
Find u∗ ∈ H01 (Ω) such that
a(u∗ , v) = f (v) ∀ v ∈ H01 (Ω),
(2.2)
where
a(u, v) =
Z X
2
ai (x, u, ∇u ) Di v + a0 (x, u, ∇u) v) dx,
(
Ω i=1
f (v) =
Z
f v dx
(2.3)
Ω
∂
∂
∂u ∂u
∂
, f ∈ L2 (Ω), ∇u = ( ∂x
u, ∂x
u)T . Let ai (x, p0 , p1 , p2 ) = ai (x, u, ∂x
,
) and
Here Di = ∂x
i
1
2
1 ∂x2
3
p = (p0 , p1 , p2 ). We assume that the functions ai : Ω × R → R, i = 0, 1, 2 satisfy the following
conditions: For some positive constants M, µ0 ,
ai ∈ C 1 (Ω × R3 ),
max{|ai (x, 0, 0, 0)|, |
i = 0, 1, 2
(2.4)
∂ai
∂ai
(x, p)|, |
(x, p)|} ≤ M, f or i, j = 0, 1, 2; k = 1, 2;
∂xk
∂pj
(2.5)
2
2
X
X
∂
ξi2
ai (x, p)ξi ξj ≥ µ0
∂p
j
i=1
i,j=0
(2.6)
for any ξ = (ξ0 , ξ1 , ξ2 ) ∈ R3 , ξ 6= 0.
As a direct consequence of the above assumptions we obtain that for all p, q ∈ R 3 there is a
positive constant L such that in Ω
∀p, q ∈ R3 |ak (x, p) − ak (x, q)| ≤ L(
2
X
i=0
2
|pi − qi |2 )1/2 k = 0, 1, 2
(2.7)
Under the above assumptions it can be proven that the form a(·, ·) is strongly monotone and Lipschitz
continuous and that the problem (2.2) has the unique solution, see [16] or [18].
3. THE GEOMETRICALLY CONFORMING CASE
We now define a discrete space V h ⊂ L2 (Ω) that is not a subspace of H01 (Ω). In that sense our
method is nonconforming. In this section, we consider a geometrically conforming version of the
mortar method. In the next section, we consider the geometrically nonconforming one. We consider
a partition of Ω into polygonal subdomains i.e.
Ω=
N
[
Ωi with Ωi ∩ Ωj = ∅ if i 6= j
i=1
that are arrange in such a way that the intersection of Ωk ∩ Ωl for k 6= l is either the empty set, an
edge or a vertex. We call this partition geometrically conforming. A more general case is considered
in the next section. The mortar element method first deals with the union of all edges (interfaces).
Γ=
N
[
∂Ωk \ ∂Ω
(3.1)
k=1
and consists of choosing one of the decomposition of Γ, that is made of disjoint open segments (that
are edges of subdomains) called mortars, denoted by γk , 1 ≤ k ≤ M i.e.
Γ=
M
[
γ k γk ∩ γl = ∅ if k 6= l.
k=1
We denote an common, open edge to Ωi and Ωj by γij . By γk(i) we denote an edge of Ωi that is
a mortar (master) and by δk(j) an edge of Ωj that occupies geometrically the same place, called
nonmortar (slave). There is no rule in selecting γk(i) as a mortar.
Let us introduce some notation. We define a local space
1
HD
(Ωi ) = {v ∈ H 1 (Ωi ) : v|∂Ω∩∂Ωi = 0}
and a global one
X=
N
Y
1
HD
(Ωi )
i=1
PN
with a seminorm |v|X = (
and a norm kvkX = (kvk2L2 (Ω) + |v|2X )1/2 .
With each Ωk we associate a quasiuniform triangulation Th (Ωk ) made of elements that are triangles, cf. [12]. By hk we denote a parameter of this triangulation, that is the maximum diameter
of the triangles. As the triangulation Th (Ωk ) is chosen over each Ωk , so we can give the definition of
the finite element functions. Let us assume that we work with the simple generic case of linear finite
elements. We first define the finite element functions locally and introduce the space
1/2
2
i=1 |vi |H 1 (Ωi ) )
Xh (Ωk ) = {vk,h ∈ C(Ωk ) : vk,h|∂Ω = 0, ∀ t ∈ Th (Ωk ), vk,h|t ∈ P1 (t)}
where P1 (t) is the set of all linear polynomials over the t triangle in Th (Ωk ). Let W hj (γij ) be the
restriction of Xh (Ωj ) to γij .
We also introduce the global space Xh as
Xh (Ω) =
N
Y
k=1
3
Xh (Ωk )
that can be considered as a subspace of X consisting of functions whose restriction over each Ω k
belongs to Xh (Ωk ). Note that, since the triangulations on two adjacent subdomains are independent,
the interface γij = γk(i) = δk(j) is provided with two different and independent (1D) triangulations
the hi and hj ones and two different spaces W hi (γk(i) ) and W hj (δk(j) ). Additionally we define an
auxiliary test space M hj (δk(j) ) being a subspace of the nonmortar space W hj (δk(j) ) such that its
functions are constant on elements which intersect the ends of δk(j) . The dimension of M hj (δk(j) )
is equal to dimension of W hj (δk(j) ) minus two. In what follows we express the matching condition
that is sufficient to ensure the optimality of the global approximation and define our discrete space
V h:
V h = {uh ∈ Xh (Ω) : ∀ δm(j) ⊂ Γ,
Z
∀ψ ∈ M hj (δm(j) )
(ui,h − uj,h )|δm(j) ψ ds = 0 }
(3.2)
δm(j)
where ui,h , uj,h in the integral are the traces of uh onto δm(j) = γij , the common edge to Ωi and Ωj .
The condition (3.2) is called mortar.
Our discrete problem is to find uh ∈ V h such that
b(uh , vh ) = f (vh ) ∀vh ∈ V h
where
b(u, v) =
N Z
X
k=1
and
(
2
X
(3.3)
ai (x, uk , ∇uk ) Di vk + a0 (x, uk , ∇uk ) vk ) dx
Ωk i=1
f (v) =
N Z
X
k=1
f vk dx
Ωk
Remark 3.1 If u, v ∈ H01 (Ω) then a(u, v) = b(u, v).
With the help of the Lemma 3.2 and Lemma 3.3, which are proven below in this section, we can
prove that under our assumptions (2.4)- (2.6) there exists the unique solution of (3.3), cf [18].
Now we state the main theorem of this section.
Theorem 3.1 Let u∗ and uh be the solutions of (2.2) and (3.3), respectively and let u∗ ∈ H 2 (Ω).
Then we have
ku∗ − uh kX ≤ c h
where c is a constant independent of hk and h = maxk {hk }.
To prove this theorem we first state and prove some auxiliary lemmas. First we define an auxiliary
operator associated with the form (2.3), which is a generalization of normal derivative in case of the
linear operators.
Definition 3.1 Let γ ⊂ ∂Ωi be an segment. Then let li : H 3/2 (Ωi ) → L2 (γ) be defined as
(li u)(x) =
2
X
ai (x, u(x), ∇u(x))ni
on γ
i=1
where n = (n1 , n2 ) is the normal vector to γ (in Ωi ).
It is easy to show that under assumptions (2.4) - (2.5) li u is well defined in L2 (γ) , cf. [18].
Next lemma states the continuity of li across γ, the part of the interface.
4
(3.4)
Lemma 3.1 Let γ ⊂ ∂Ωi ∩ ∂Ωj be an segment and Ωij = Ωi ∪ Ωj . li : H 3/2 (Ωi ) → L2 (γ) and
lj : H 3/2 (Ωj ) → L2 (γ) be defined by (3.4). Then for u ∈ H 3/2 (Ωij ) we have
lj uj = −li ui a.e. on γ
i.e.
2
X
ak (x, ui (x), ∇ui (x))nk =
k=1
2
X
ak (x, uj (x), ∇uj (x))nk a.e. on γ
k=1
where n = (n1 , n2 ) is the normal vector to γ and ui , uj are the restrictions of u to Ωi and Ωj
respectively.
Proof. First we prove that li is Lipschitz continuous. We may assume that γ is parallel to the x
∂
axis and then we have li u(x) = a2 (x, u(x), ∇u(x)). We use ∂x
u(x) = u(x) to simplify the notation.
0
Let now u, v ∈ H 3/2 (Ωij ), then we conclude that
Z
2
|a2 (x, u, ∇u) − a2 (x, v, ∇v)| dx ≤ L
2
Z X
2
γ i=0
γ
|
∂
(u − v)|2 dx ≤ Cku − vk2H 3/2 (Ωij )
∂xi
We have used the standard trace theorem and (2.7). As the statement of the lemma obviously
is satisfied for u ∈ C ∞ (Ωij ), we finish the proof using the density of C ∞ (Ωij ) in H 3/2 (Ωij ) and
Lipschitz continuity of li and lj .
The next corollary can be proven using the same ideas as in [18].
Corollary 3.1 Let γ ⊂ ∂Ωi be an segment, li : H 3/2 (Ωi ) → L2 (γ) be defined by (3.4) and u ∈
H 2 (Ωi ) then under our assumptions (2.4)- (2.5) we have li u ∈ H 1/2 (γ) with |li u|H 1/2 (γ) ≤ c (|Ωi | +
kuk2H 2 (Ωi ) )1/2 .
Now we define the restriction of the form a(·, ·) to H 1 (Ωi ).
Definition 3.2 Let a bilinear form bi (·, ·) : H 1 (Ωi ) × H 1 (Ωi ) → R be defined as
∀u, v ∈ H 1 (Ωi ) bi (u, v) =
Z
(
2
X
ai (x, u, ∇u ) Di v + a0 (x, u, ∇u) v) dx
Ωi i=1
Remark 3.2 Note that for all u, v ∈ X b(u, v) =
PN
i=1 bi (ui , vi ).
Using the assumptions (2.4) - (2.6) and the results of [6] we can prove that the form b(u h , vh ) is
strongly monotone and Lipschitz continuous in V h . We state that in two lemmas
Lemma 3.2 The form b(·, ·) is strongly monotone in V h , i.e. for all uh , vh ∈ V h we have
b(uh , uh − vh ) − b(vh , uh − vh ) ≥ c kuh − vh k2X
where c is a positive constant independent of hk .
Proof. Let uh , vh ∈ V h . As V h ⊂ X we get that restrictions of uh , vh to Ωk denoted by uk,h , vk,h
1
are in HD
(Ωk ). In [18] was proven that
∀u, v ∈ H 1 (Ωi ) bi (u, u − v) − bi (v, u − v) ≥ c|u − v|2H 1 (Ωi )
So we can deduce that
b(uh , uh − vh ) − b(vh , uh − vh ) =
N
X
bk (uk,h , uk,h − vk,h ) − bk (vk,h , uk,h − vk,h )
k=1
5
≥c
N
X
|uk,h − vk,h |2H 1 (Ωk ) = c |uh − vh |2X ≥ c kuh − vh k2X
k=1
The last estimate follows from the fact that for uh ∈ V h |uh |X is equivalent to kuh kX , with a constant
independent of all hk , what was proved in [6].
Lemma 3.3 The form b(·, ·) is Lipschitz continuous in X (and thus in V h ), i.e. for all u, v, w ∈ X
we have
|b(u, w) − b(v, w)| ≤ M ku − vkX kwkX
where M is a positive constant.
Proof. In [18] was proven that
∀ u, v, w ∈ H 1 (Ωi ) |bi (u, w) − bi (u, v)| ≤ M ku − vk2H 1 (Ωi ) kwk2H 1 (Ωi )
Summing over all subdomains, using the above result for u, v, w ∈ X and Schwarz inequality ( for
the standard inner product in RN ) we end the proof.
We now formulate and prove the lemma which is a generalization of the second Strang lemma
for the boundary value problem considered in this paper, cf. [12] for the proof in the linear case.
Lemma 3.4 Let u∗ and uh be the solutions of (2.2) and (3.3), respectively. Let u∗ ∈ H 2 (Ω). Under
assumptions (2.4) - (2.6) we have


R

X γ lm u∗ [wh ]ds 
inf
sup
m
ku∗ − uh kX ≤ C
ku∗ − vh kX +
(3.5)
wh ∈ V h
 vh ∈ V h

kwh kX
γm ⊂Γ
where [wh ] is a jump of wh across γm , lm u∗|γm = li(m) u∗ is defined in (3.4), the sum is taken over
all mortars γm and C is a constant independent of hi .
Proof. We have for all vh ∈ V h
ku∗ − uh kX ≤ ku∗ − vh kX + kuh − vh kX
(3.6)
Let denote wh = uh − vh , then Lemma 3.2 and (3.3) yield that
(1/c)kuh − vh k2X ≤ b(uh , wh ) − b(vh , wh ) = f (wh ) − b(vh , wh )
We now can deduce that
!
N Z
2
X
X
∂
∗
∗
∗
∗
ai (x, u , ∇u )) + a0 (x, u , ∇u ) wk,h dx = b(u∗ , wh )+
f (wh ) =
(−
∂x
i
Ωk
i=1
k=1
+
N Z
X
k=1
2
X
(−ai (s, u∗ , ∇u∗ )ni wk,h ) ds = b(u∗ , wh ) +
∂Ωk i=1
X Z
γm ⊂Γ
lm u∗ [wh ]ds
γm
The last equality follows from Lemma 3.1. From this, Lemma 3.3 and (3.7) it follows that
X Z
2
∗
lm u∗ [wh ]ds
(1/c)kwh kX ≤ b(u , wh ) − b(vh , wh ) +
γm ⊂Γ
≤ M ku∗ − vh kX kwh kX +
X Z
γm ⊂Γ
6
γm
lm u∗ [wh ]ds
γm
(3.7)
Dividing by kwh kX and substituting in (3.6) we complete the proof.
The first term in (3.5) is known as the approximation error and the second one we call the
consistency error which is a consequence of the discontinuities of the elements of V h through the
interface. We can now turn to the proof of Theorem 3.1.
Proof of Theorem 3.1. Let us first consider the approximation error. In [7] or in [5] for 3D
case it was proven that if v ∈ H01 (Ω) with v|Ωk ∈ H 2 (Ωk ) then there exists constant c independent
of hk such that
N
X
inf
(3.8)
kv − vh k2X ≤ c
h2k |v|Ωk |2H 2 (Ωk )
vh ∈ Vh
k=1
Let us turn to the consistency term. We now prove that
|
XZ
δm
lm u∗ [wh ]ds | ≤ c
δm
N
X
k=1
hk
n
|Ωk | + ku∗ k2H 2 (Ωk )
o1/2
kwk,h kH 1 (Ωk )
(3.9)
where c is a constant independent of hk . It follows the same lines as in [5]. Let us consider one
interface γij common to Ωi and Ωj . Assume that γm(i) is a mortar, then δm(j) is a nonmortar. From
(3.2) we have
Z
Z
hj
∗
∀ψ ∈ M (δm(j) )
lm u [wh ] ds =
(lm u∗ − ψ) (wj,h − wi,h ) ds
δm(j)
δm(j)
Hence
|
Z
lm u∗ [wh ]ds | ≤
δm(j)
inf
kl u∗ − ψk[H 1/2 (δm(j) )]0 kwj,h − wi,h kH 1/2 (δm(j) )
ψ ∈ M hj (δm(j) ) m
Using the trace theorem, cf. [1], we have
Z
inf
|
lm u∗ [wh ]ds | ≤ c
kl u∗ − ψk[H 1/2 (δm(j) )]0 (|wi,h |H 1 (Ωi ) + |wj,h |H 1 (Ωj ) )
ψ
∈
M hj m
δm(j)
It can be proven, e.g. see [5], that
inf
kl u∗ − ψk[H 1/2 (δm(j) )]0 ≤ c hj |lm u∗ |H 1/2 (δm(j) )
ψ ∈ M hj (δm(j) ) m
(3.10)
Now we sum over all nonmortars δm(j) = γij and use Corollary 3.1 what proves (3.9). Combining
(3.9) and (3.8) completes the proof of the theorem.
4. THE GEOMETRICALLY NONCONFORMING CASE
In this section we consider the geometrically nonconforming version of the mortar finite element
method. As in the previous section we assume that Ω is divided into disjoint, polygonal subregions
Ωk . As in the third section we introduce in each subdomain a quasi-uniform triangulation. Let
γij = Ωi ∩ Ωj . The local spaces Xh (Ωk ) and the global spaces X, Xh (Ω) be defined as in the
previous section. The mortar element method consists of choosing one of the decomposition of Γ
defined in (3.1), made of mortars γm that are disjoint i.e.
Γ=
M
[
γ m , γm ∩ γn = ∅, m 6= n
m=1
7
and that satisfy the assumption that each mortar is an edge of one subdomain i.e. γm = γm(i) is an
edge of Ωi . We assume that there is at least one such decomposition.
The mortar sides of Ωi we denote by γm(i) and the slave sides (the edges that are not mortars)
we denote by δk(i) . For each edge that is not a mortar we define a space of traces W hj (δk(j) ) and a
test space M hj (δk(j) ) in the same way as in the previous section, i.e. W hj (δk(j) ) is the restriction of
Xh (Ωj ) to δk(j) and M hj (δk(j) ) is a subspace of W hj (δk(j) ) such that its functions are constant on
elements which intersect the ends of δk(j) .
Now we define our discrete space as
V h = { vh ∈ Xh (Ω) : ∀δk(j) − not a mortar
Z
X
vi,h|δk(j) ) ψ ds }
(vj,h −
∀ψ ∈ M hj (δk(j) )
δk(j)
(4.1)
m(i)
where the sum is taken over m such that γm(i) ∩ δk(j) 6= ∅.
The discrete problem is to find uh ∈ V h such that
b(uh , vh ) = f (vh ) ∀vh ∈ V h
(4.2)
where the form b(·, ·) is defined as in the third section.
We have as in the previous section that norm k · kX and seminorm | · kX are equivalent and that
analogous lemmas to Lemma 3.2 and 3.3, are also valid in this case, cf. [7]. Thus there exists the
unique solution of (4.2), see [18].
Now we state the main theorem of this section.
Theorem 4.2 Let u∗ and uh be the solutions of (2.2) and (4.2), respectively. Let u∗ ∈ H 5/2 (Ω).
Then under the hypotheses of (2.4) - (2.6) we have
ku∗ − uh kX ≤ c h
where c is a positive constant independent of hk and h = maxk (hk ).
To prove Theorem 4.2 we state an analogue of Lemma 3.4 for the geometrically nonconforming
case. The proof is similar to that of Lemma 3.4.
Lemma 4.5 Let u∗ and uh be the solutions of (2.2) and (4.2), respectively. Let u∗ ∈ H 2 (Ω). Under
assumptions (2.4) - (2.6) we have
R
X δ lm u∗ [wh ]ds
inf
sup
∗
∗
m
(4.3)
ku − uh kX ≤ C
ku − vh kX +
vh ∈ V h
wh ∈ V h
kwh kX
δm ⊂Γ
where [wh ] is a jump of wh across δm , lm u∗|δm = li(m) u∗ that is defined in (3.4). The sum is taken
over all δm , sides of substructures that are not mortars.
Proof of Theorem 4.2. The first term, the approximation error, can be estimated from (3.8)
whose statement is also true in this case, cf. [7] or [5].
It remains to estimate the second term, the consistency error. First we prove that
Z
X
kwk,h kH 1 (Ωk ) )
(4.4)
|
lm u∗ [wh ]ds | ≤ c hj |lm u∗ |H 1 (δm(j) ) (kwj,h kH 1 (Ωj ) +
δm(j)
k
8
that is an analogue to (3.9). The sum is taken over all Ωk such that ∂Ωk ∩ δm(j) 6= ∅. Proceeding as
in the proof of (3.9) we see that
Z
Z
X
wk,h|δm ) ds|
|
lm u∗ [wh ]ds | = |
lm u∗ (wj,h −
δm(j)
δm(j)
∂Ωk ∩δm 6=∅
From (4.1) we can deduce that for all ψ ∈ M hj (δm(j) )
Z
Z
∗
|
lm u [wh ]ds | = |
(lm u∗ − ψ) (wj,h −
δm(j)
δm(j)
X
wk,h|δm ) ds|
∂Ωk ∩δm 6=∅
Using Schwarz inequality we have
Z
inf
klm u∗ − ψkL2 (δm(j) ) kwj,h −
|
lm u∗ [wh ]ds | ≤
hj
(δ
)
ψ
∈
M
m(j)
δm(j)
X
wk,h|δm kL2 (δm(j) )
∂Ωk ∩δm 6=∅
Combining the trace theorem and the standard approximation result for lm u∗ ∈ H 1 (δm(j) ) we prove
(4.4). As u∗ ∈ H 5/2 (Ω), we can obtain a result similar to that of Corollary 3.1. Combining this with
(4.4) and summing over all δm(j) we finish the proof of Theorem 4.2.
5. THE RICHARDSON-NEWTON METHOD
In this section we propose a method for solving the problem (3.3) arising from discretization of the
boundary value problem (2.2) by the geometrically conforming mortar finite element method. We
do not know, how to generalize this method to the geometrically nonconforming case.
For simplicity of presentation we describe the method with one additional assumption that the
subdomains Ωi are triangles and form a quasiuniform triangulation with a parameter H, cf. [12].
To define our method, we first have to introduce some special functions and subspaces of V h . We
will primarily work with nodal basis of the mortar finite element space associated with the following
sets of nodes:
• all nodes interior to the substructures
• all nodes interior to the mortars
• all nodes of vertices of subregions except those on ∂Ω
We associate a basis function with each node of these sets. The functions corresponding to nodes
in the interiors of the substructures are standard nodal basis functions as in the conforming finite
element discretization. A function associated with a node xk interior to the mortar γm(i) , we define
as follows. It is one at xk and zero at the remaining nodes defined above, i.e. nodes interior to
all substructures, vertices of all substructures and all nodes in the interiors of the mortars except
xk . The values of this function at the interior nodes of nonmortars δm(j) = γm(i) are determined
by the mortar condition (3.2), with zero values at the ends of δm(j) . We now define basis functions
associated with the vertices of the substructures. We first denote by ν the set of vertices of the
substructures that are associated with degrees of freedom of V h , i.e. those which are not on ∂Ω.
Each crosspoint of Γ belongs to numbers of subdomains and therefore corresponds to several nodes
of ν, to one degree of freedom for each of subregions that meet at that point. These nodes are in the
same geometrically position, but are assigned to different subdomains. Let vn ∈ ν be a vertex of Ωi .
Then a basis function associated with vn ∈ ν we denote as φvn . It is defined as one at vn and zero
at all other vertices of ν and at all interior nodes of all substructures. We now define this function
9
on Γ, i.e. on all mortars and nonmortars. There are three possible situations: the vertex v n can be
a common end of two mortars γn(i) and γm(i) , a common end of two nonmortars δl(i) and δk(i) or
a common end of a nonmortar δs(i) and a mortar γp(i) . In the first case φvn restricted to γn(i) and
γm(i) is a standard nodal function corresponding to vn , i.e. is one at vn and zero at the remaining
nodes of the both mortars. On nonmortars δn = γn(i) and δm = γm(i) this function is determined
by the mortar condition (3.2) with zero values at the ends of δn and δm , respectively. In the second
case φvn restricted to the mortars γl = δl(i) and γk = δk(i) is zero and on δl(i) and δk(i) is determined
by the mortar condition with one at vn and zero at the other ends of δl(i) and δk(i) . In the last case
φvn is defined on the mortar γp(i) (and δp = γp(i) ) as in the first case while on the nonmortar δs(i)
(and γs = δs(i) ) analogously to the second case. In all cases φvn is defined as zero on the remaining
mortars and nonmortars.
Using these P
basis functions we have V h = span{φ1 , . . . , φn }. Let the solution of (3.3) be repren
sented as uh = i=1 ui φi and introduce


n
X
ki (u1 , . . . , un ) = b 
uj φj , φi  , fi = (f, φi )L2 (Ω)
j=1
Let B = (k1 , . . . , kn )T , u = (u1 , . . . , un )T and f˜ = (f1 , . . . , fn )T . With these notations we rewrite
the problem (3.3) as the system of nonlinear algebraic equations
B(u) = f˜
(5.1)
h
Here and below we use u either as a function
Pn in V or as a vector representation in terms of the
T
nodal basis i.e. u = (u1 , . . . , un ) or u = i=1 ui φi .
Additionally we introduce a bilinear form on V h × V h as
a4 (u, v) =
N Z
X
i=1
∇ui ∇vi dx
Ωi
1/2
Let D be its matrix representation. We should point out that a4 (u, u)1/2 = (Du, u)Rn = |u|X ,
therefore a4 (·, ·) is positive definite in V h .
We solve (5.1) by a method that combines the additive Schwarz preconditioning technique with
Newton’s method. The Schwarz method is determined by subspaces of V h , and bilinear forms defined
on these subspaces, cf. [15]. In our case those forms are equal to a4 (·, ·). We now define subspaces
that form the decomposition of V h . Let V0 be the coarse space of continuous, piecewise linear
functions on the coarse triangulation with zero on ∂Ω. Next we define one dimensional vertex spaces
Vvn , that are associated with vn ∈ ν: Vvn = span{φvn }. Finally we introduce Vij spaces which are
associated with all pairs of two subdomains Ωi and Ωj that have a common edge γij which is the
mortar γm(i) ⊂ ∂Ωi and the nonmortar δm(j) ⊂ ∂Ωj . We define Vij as a subspace of V h such that
its functions can be nonzero at the interior nodes of Ωi and Ωj and at the interior nodes of γm(i)
and δm(j) . It is easy to see that
X
X
V h = V0 +
Vv n +
Vij
vn ∈ν
γij ⊂Γ
We then define operators T0 : V h → V0 , Tvn : V h → Vvn and Tij : V h → Vij , by
a4 (T0 (u), v) = b(u, v) ∀v ∈ V0
(5.2)
a4 (Tvn (u), v) = b(u, v) ∀v ∈ Vvn
(5.3)
10
and
a4 (Tij (u), v) = b(u, v) ∀v ∈ Vij
(5.4)
The matrix representation of these operators are denoted by the same symbols. They have the
−1
T
following form: T0 = R0T D0−1 R0 B(u), Tvn = RvTn Dv−1
Rvn B(u) and Tij = Rij
Dij
Rij B(u) where
n
D0 , Dvn and Dij are the matrix representations of a4 (·, ·) in the corresponding subspaces and
R0 : V h → V0 , Rvn : V h → Vvn and Rij : V h → Vij are the restrictions operators defined as in [8].
We note that T0 , Tvn and Tij are nonlinear in general. To define an additive Schwarz method ,
let
X
X
T = T0 +
Tv n +
Tij
(5.5)
vn ∈ν
γij ⊂Γ
We replace the problem (5.1) by the problem of finding u ∈ V h such that
T (u) = g
P
(5.6)
P
where g = g0 + vn ∈ν gvn + γij ⊂Γ gij with g0 = T0 (u), gvn = Tvn (u) and gij = Tij (u). Here u is
the solution of (5.1). We show that problems (5.6) and (5.1) have the same unique solution. These
gi can be pre-computed without knowing the exact solution u.
Introducing
X
X
−1
T
Rij
Dij
Rij
(5.7)
RvTn Dv−1
R vn +
M −1 = R0T D0−1 R0 +
n
vn ∈ν
we have
γij ⊂Γ
T (u) = M −1 B(u)
For solving (5.6) we use the following algorithm:
Algorithm 5.1 For τ defined in Theorem 5.4, see below, iterate for n = 0, 1, . . . until convergence.
un+1 = un − τ (T (un ) − g)
To prove the convergence of the algorithm we need the following auxiliary result.
Theorem 5.3 For any u ∈ V h
c(1 + log(H/h))−2 (Du, u)Rn ≤ (DM −1 Du, u)Rn ≤ C(Du, u)Rn
where c, C are positive constants independent of H, hi and h = infi hi .
This theorem is proved in the last part of this section. Theorem 5.3 yields that (5.6) has the unique
solution equal to the solution of (5.1) and that M −1 is invertible. The next corollary plays an
important role in the proof of convergence of the algorithm.
Corollary 5.2 There exists constants δ0 and δ1 such that
(B(u) − B(v), u − v)Rn ≥ δ0 ku − vk2M
and
kB(u) − B(v)kM −1 ≤ δ1 ku − vkM
where M was defined in (5.7), δ0 = C (1 + log(H/h))−2 , and C, δ1 are constants independent of
H, hi .
The proof of the corollary follows from Theorem 5.3, Lemma 3.2 and Lemma 3.3.
We now state the main theorem of this section that can be proven in the standard way using
Corollary 5.2, cf. [10] or [17].
11
Theorem 5.4 If we choose 0 < τ < 2δ0 /δ1 , where δ0 and δ1 are defined in Corollary 5.2, then
Algorithm 5.1 is convergent in the sense that
kun − ukM ≤ ρ(τ )n ku0 − ukM
where ρ2 = 1 − τ (2δ0 /δ1 − τ ) < 1. The optimal τopt = δ0 /δ1 and ρ2opt = 1 − δ02 /δ1 .
We now prove Theorem 5.3 using the general theory of ASM, cf. [15]. It reduces to check three key
assumptions.
Proof of Theorem 5.3.
Assumption (i)
We want to prove that there is a positive constant c independent of hi and HP
such that forP
all u ∈ V h
there exist functions u0 ∈ V0 , uvn ∈ Vvn and uij ∈ Vij such that u = u0 + vn ∈ν uvn + γij ⊂Γ uij
and
X
X
a4 (u0 , u0 ) +
a4 (uvn , uvn ) +
a4 (uij , uij ) ≤ c(1 + log(H/h))2 a4 (u, u)
(5.8)
vn ∈ν
γij ⊂Γ
We first select u0 ∈ V0 = V H by making u0 (cr ) = ucr , where cr ∈ Γ is a crosspoint and ucr is
the average value of u at the vertices of ν that coincide geometrically with c r . We further denote
ν(cr ) as the set of these vertices and ν(i) as the set of
Pvertices of Ωi . Let N (cr ) be the number of
vertices in ν(cr ). Thus we have that ucr = (1/N (cr )) vn ∈ν(cr ) u(vn ). Let now define uvn ∈ Vvn by
the pointwise interpolation of u − u0 at vn , i.e.
uvn = (u − u0 )(vn ) φvn
Note that w defined as
w = u − u0 −
X
uv n
vn ∈ν
vanishes at all vertices vn ∈ ν.
We now decompose w in Ωi as
wi = w|Ωi = Pi wi + Hi wi
where Hi wi is the discrete harmonic part of wi and Pi wi is the H01 (Ωi ) projection on Xh (Ωi )∩H01 (Ωi )
of wi , i.e. Hi wi = wi on ∂Ωi and
∀ψ ∈ Xh (Ωi ) ∩ H01 (Ωi ) a4 (Hi wi , ψ) = 0,
a4 (Pi wi , ψ) = a4 (wi , ψ)
(5.9)
Both Pi wi and Hi wi we extend as zero off Ωi .
With each space Vij we associate the common edge γij = Ωi ∩ Ωj which is the mortar γm(i) ⊂ ∂Ωi
and the nonmortar δm(j) ⊂ ∂Ωj . Then let wij ∈ Vij be equal to w on γm(i) and δm(j) , be zero on
∂Ωi \ γm(i) , ∂Ωj \ δm(j) and be extended as the discrete harmonic function in Ωi and Ωj . Note that
wij is zero off Ωi ∪ Ωj because wij ∈ Vij .
We finish the decomposition of u by setting
uij = wij + (1/Ne (i))Pi wi + (1/Ne (j))Pj wj
where Ne (k) is a number of edges γkl ⊂ Γ ∩ ∂Ωk and Ne (k) is equal to 3 if ∂Ωk ∩ ∂Ω = ∅ and 2 or 1
otherwise. Note that
X
X
uij
u = u0 +
uv n +
vn ∈ν
12
γij ⊂Γ
We first estimate a4 (u0 , u0 ). Let ui be the average value of u over Ωi . Using the inverse inequality
we have
a4 (u0 , u0 ) =
N
X
|u0 |2H 1 (Ωi ) =
i=1
N
X
|u0 − ui |2H 1 (Ωi ) ≤ c
i=1
N
X
X
|u0 (vn ) − ui |2
i=1 vn ∈ν(i)
We consider one vertex vn0 of Ωi , which geometrically coincides with a crosspoint cr and we have
that u0 (vn0 ) = u0 (cr ). Hence
|u0 (vn0 ) − ui |2 = |
1
N (cr )
X
u(vn ) − ui |2 ≤ c
X
|u(vn ) − ui |2
vn ∈ν(cr )
vn ∈ν(cr )
Note that average values of u over a mortar γm(i) and a nonmortar δm(j) that occupies geometrically
the same place, are equal to each other. Using this, the standard Sobolev-like inequality for finite
elements, see e.g. [14], and the Poincare inequality, we obtain
X
(1 + log(H/hn ))|un |2H 1 (Ωn )
(5.10)
|u0 (vn0 ) − ui |2 ≤ c
vn ∈ν(cr )
The sum is taken over all subdomains with the common vertex cr . Summing over all subregions and
all their vertices gives the estimate
a4 (u0 , u0 ) ≤ c
N
X
(1 + log(H/hi ))|ui |2H 1 (Ωi )
(5.11)
i=1
We now estimate
P
vn ∈ν
a4 (uvn , uvn ). We deduce that
a4 (uvn , uvn ) = |u(vn ) − u0 (vn )|2 |φvn |2X ≤ c (1 + log(H/h))|u(vn ) − u0 (vn )|2
We have used the fact that |φvn |2X ≤ c (1 + log(H/h)), cf. [11]. We further use the same arguments
as for the estimate of a4 (u0 , u0 ) to get
X
a4 (uvn , uvn ) ≤ c
(1 + log(H/h))2 |ui |2H 1 (Ωi )
(5.12)
vi ∈ν(cr )
where cr is a crosspoint geometrically coinciding with vn . Summing over all subdomains and their
vertices we obtain
N
X
X
a4 (uvn , uvn ) ≤ c
(1 + log(H/h))2 |ui |2H 1 (Ωi )
(5.13)
vn ∈ν
We now estimate
P
γij ⊂Γ
i=1
a4 (uij , uij ). We deduce that
a4 (uij , uij ) ≤ c
n
|wij |2H 1 (Ωj ) + |wij |2H 1 (Ωi ) + |Pi wi |2H 1 (Ωi ) + |Pj wj |2H 1 (Ωj )
o
We first estimate |Pi wi |2H 1 (Ωi ) (|Pj wj |2H 1 (Ωj ) can be estimated in the same way). Using (5.12) we get
|Pi wi |2H 1 (Ωi ) ≤ c (|ui |2H 1 (Ωi ) +
X
|uvn |2H 1 (Ωn ) ) ≤ c (|ui |2H 1 (Ωi ) +
X
(1 + log(H/h))2 |un |2H 1 (Ωn ) )
n
vn ∈ν(i)
where the last sum is taken over all substructures that have a common vertex with Ω i .
13
We now estimate the norms of wij the discrete harmonic part of uij . We first note that
|wij |2X = |Hi wi |2H 1 (Ωi ) + |Hj wj |2H 1 (Ωj ) ≤ c {kwi k2H 1/2 (γ
00
m(i) )
+ kwj k2H 1/2 (δ
00
m(j) )
} ≤ c kwi k2H 1/2 (γ
00
m(i) )
The first inequality follows from extension property of discrete harmonic functions, cf. [9], and the
1/2
second one from H00 stability of functions in V h over each edge γij ⊂ Γ, cf. [4]. Thus it remains to
prove the estimate of kwi k2 1/2
.
H00 (γm(i) )
P2
Let denote by vn1 , vn2 the ends of γm(i) . Then wi|γm(i) = z − i=1 z(vni )φvni − z0 +
P2
i=1 z0 (vni )φvni where z = ui − ui and z0 = u0 − ui . From this we obtain
)
(
2
2
X
X
2
2
2
z0 (vni )φvni kH 1/2 (γ )
z(vni )φvni kH 1/2 (γ ) + kz0 −
kwi kH 1/2 (γ ) ≤ c kz −
00
m(i)
00
i=1
m(i)
i=1
00
m(i)
The first term we can estimate by c (1 + log(H/hi ))2 |ui |2H 1 (Ωi ) , cf. [15] while the second one easily
by
2
2
X
X
kz0 −
z0 (vni )k2H 1/2 (γ ) ≤
c (1 + log(H/hi ))|u0 (vni ) − ui |2
00
i=1
m(i)
i=1
cf. [13]. Now using (5.10) we deduce that
kwi k2H 1/2 (γ
00
m(i) )
≤ c (1 + log(H/h))2
X
|un |2H 1 (Ωn )
n
where the sum is taken over all indices of subdomains that has a vertex that geometrically coincides
with one of the ends of γm(i) .
Summing over all subspaces Vij we have
X
a4 (uij , uij ) ≤ c (1 + log(H/h))2 |u|2X = c (1 + log(H/h))2 a4 (u, u)
γij ⊂Γ
Combining this, (5.11) and (5.13), we get (5.8) what ends the proof of Assumption (i).
Assumptions (ii)
It is obviously satisfied with ω = 1 as all local forms equals a4 (·, ·).
Assumption (iii)
It is satisfied with ρ() ≤ C.
ACKNOWLEDGMENTS
The author is greatly indebted to prof. Maksymilian Dryja for his encouragement and his invaluable
advises.
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