SIAM J. NUMER. ANAL.
Vol. 43, No. 4, pp. 1481–1503
c 2005 Society for Industrial and Applied Mathematics
A POSTERIORI ERROR ESTIMATIONS OF SOME
CELL-CENTERED FINITE VOLUME METHODS∗
SERGE NICAISE†
Abstract. This paper presents the natural framework to residual based a posteriori error
estimation of some cell-centered finite volume methods for the Laplace equation in Rd , d = 2 or
3. For that purpose we associate with the finite volume solution a reconstructed approximation,
which is a kind of Morley interpolant. The error is then the difference between the exact solution
and this Morley interpolant. The residual error estimator is based on the jump of normal and
tangential derivatives of the Morley interpolant. We then prove the equivalence between the discrete
H 1 seminorm of the error and the residual error estimator. Numerical tests confirm our theoretical
results.
Key words. finite volume method, cell-centered method, a posteriori error estimates
AMS subject classifications. 65N30, 65N15
DOI. 10.1137/S0036142903437787
1. Introduction. The finite volume method is a well-adapted method for the
discretization of various partial differential equations and is very popular in the engineering community [24]. The mathematical community recently started to analyze it
in detail. Presently, existence and uniqueness results as well as a priori error estimates
are available for a quite large class of problems; we refer to [10] and the references
cited there. Contrary to the finite element methods [26], a posteriori error estimates
for finite volume methods are less developed, and until now only a few results have
been obtained in that direction. See [14, 22, 1, 12, 13] for cell-centered finite volume
methods, [17, 19, 25, 23] for vertex-centered methods, and [2, 3, 15, 16] for finite
volume element methods. Since finite volume methods have some similarities with
the finite element methods, we may hope that this gap will be filled soon.
The goal of our paper is to present the natural framework to residual based a
posteriori (efficient and reliable) error estimation of some cell-centered finite volume
methods for linear elliptic equations. In a first attempt we restrict ourselves to the
Laplace equation in Rd , d = 2 or 3. The case of diffusion–convection–reaction equations will be only sketched; for details, we refer to a forthcoming paper [20]. The key
idea is the reconstruction of a piecewise polynomial approximation of the finite volume
solution, its principal property being that the mean of its flux through any edge/face
of the mesh is equal to the numerical flux through that edge/face (this interpolant is
consequently smoother than the approximated solution). This reconstructed approximation is then a kind of Morley interpolant of the finite volume solution. In general
a Morley interpolant is not in H 1 , and therefore the Morley interpolant may be considered as a nonconforming approximation of the exact solution. The second key idea
is to use the Helmholtz decomposition of the error, the difference between the exact
solution and this Morley interpolant, as was done in [7] for the a posteriori error analysis of a nonconforming finite element approximation of the Laplace equation. As in
∗ Received by the editors November 17, 2003; accepted for publication (in revised form) March
24, 2005; published electronically October 19, 2005.
http://www.siam.org/journals/sinum/43-4/43778.html
† Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, F-59313 Valenciennes Cedex
9, France ([email protected]).
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A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
[7] the residual error estimator is then naturally based on the jump of normal and tangential derivatives of the Morley interpolant. We finally show the equivalence between
the discrete H 1 seminorm of the error and the residual error estimator. The proof of
the upper error bound uses the Helmholtz decomposition of the error and some quasiorthogonality relations obtained using the above-mentioned property of the Morley
interpolant. The proof of the lower error bound is more standard and simply uses
some Green’s formulas and inverse inequalities as for finite element methods [26].
Note that our purposes also require the introduction of new finite elements of
Morley type on rectangles and tetrahedra.
We further give explicitly the size of the constants appearing in the error estimates by estimating the constants involved in the interpolation error estimates (using
some related eigenvalue problems and extension techniques) and in some inverse inequalities. In particular, we obtain constants in the upper error bound that are quite
close to unity.
The idea to interpolate the finite volume solution by a smoother function having
the above-mentioned property on the flux was presented in [12] in an L1 framework for
time-dependent nonlinear convection–diffusion equations in Rd × R+ . In that paper
the authors obtain a reliable estimator in an L1 -norm, instead of the energy norm.
Furthermore, their interpolant is a piecewise linear Lagrange interpolant on a dual
mesh. As a consequence, to guarantee the property on the flux, the (primal) mesh
has to be admissible in the sense of [10, Def. 9.1], a deep obstacle for adaptivity. To
avoid this admissibility condition and use the energy norm framework, we need to
use the natural degrees of freedom on the mesh, namely, the mean of the flux on the
edges/faces, and consequently use higher order polynomials.
The outline of the paper is as follows: In section 2 we describe the so-called cellcentered method for the Laplace equation on a mesh made of triangles, rectangles,
or tetrahedra. Some standard inverse inequalities and interpolation error estimates
are recalled in section 3, where some constants are specified as explicitly as possible.
Section 4 is devoted to the introduction of some finite elements of Morley type. In
section 5 we introduce the Morley interpolant of the approximated solution and prove
its main properties. The upper and lower error bounds are then deduced in section 6.
The upper error bound is based on the properties of the Morley interpolant and the
use of the Helmholtz decomposition of the error, while the lower error bound is proved
in a quite standard way. In section 7 we briefly describe how to extend our results
to diffusion–convection–reaction equations. Finally, section 8 is devoted to numerical
experiments that confirm our theoretical considerations.
2. Discretization of the Laplace equation. Let Ω be an open subset of Rd ,
d = 2 or 3, with a polygonal (d = 2) or polyhedral (d = 3) boundary Γ.
As usual, we denote by L2 (·) the Lebesgue spaces and by H s (·), s ≥ 0, the
standard Sobolev spaces. If D is an open subset of Rd , d = 2 or 3, the usual norm
and seminorm of H s (D) are denoted by · s,D and | · |s,D . For brevity the L2 (D)norm will be denoted by · D and in the case D = Ω, we will drop the index Ω. The
space H01 (Ω) is defined, as usual, by H01 (Ω) := {v ∈ H 1 (Ω)/v = 0 on Γ}. In what
follows the symbol | · | will denote either the Euclidean norm in Rd , d = 2 or 3, or
the length of a line segment, or the area of a plane face, or finally the measure of a
domain of Rd .
We consider the standard elliptic problem: for f ∈ L2 (Ω) let u ∈ H01 (Ω) be the
variational solution of
(1)
−Δu = f in Ω,
SERGE NICAISE
1483
which means that u satisfies
(2)
∇u · ∇v dx =
f v dx ∀v ∈ H01 (Ω).
Ω
Ω
To approximate this problem by a finite volume scheme we fix a family of meshes
Th , h > 0, regular in Ciarlet’s sense [4, p. 124]. In two dimensions we assume that all
elements of Th are either triangles or rectangles, while in three dimensions the mesh
consists only of tetrahedra. For K ∈ Th we recall that hK is the diameter of K and
h = maxK∈Th hK .
For any edge/face E of K, we denote by hE,K its height in K, namely, hE,K = d|K|
|E|
if K is a triangle or a tetrahedron and hE,K = |K|
|E| if K is a rectangle. For an edge/face
E, its mean height is hE = 12 (hE,K + hE,L ), when E is the edge/face of K and L. The
regularity of the mesh implies in particular that for any edge/face E of K one has
σ1 hE,K ≤ hK ≤ σ2 hE,K ,
σ3 hE,K ≤ hE ≤ σ4 hE,K
(3)
(4)
for some positive constants σi , i = 1, . . . , 4, depending on the aspect ratio of Th .
Let us define Eh as the set of edges (d = 2) or faces (d = 3) of the triangulation
and set Ehint = {E ∈ Eh /E ⊂ Ω} the set of interior edges/faces of Th , while Ehext =
Eh \ Ehint is the set of exterior edges/faces of Th .
For an edge E of a two-dimensional (2D) element K, introduce nK,E = (nx , ny )
the unit outward normal vector to K along E. Similarly for a face E of a tetrahedron
K, set nK,E = (nx , ny , nz ) the unit outward normal vector to K on E. Furthermore,
for each edge/face E, we fix one of the two normal vectors and denote it by nE . In
two dimensions additionally introduce the tangent vector tK,E = n⊥
K,E := (−ny , nx )
such that it is oriented positively (with respect to K); similarly set tE := n⊥
.
E
The jump of some function v across an edge/face E at a point y ∈ E is defined
as
v(y) E := lim v(y + αnE ) − v(y − αnE ) ∀E ∈ Ehint ,
α→+0
v(y) E := v(y) ∀E ∈ Ehext .
For any K ∈ Th or E ∈ Eh , we denote by MK χ and ME χ the mean of χ on K
and E, respectively, i.e.,
1
1
χ(x) dx ∀K ∈ Th ,
ME χ =
χ(x) ds(x) ∀E ∈ Eh .
MK χ =
|K| K
|E| E
Finally, we will need local subdomains (also called patches). As usual, let ωK be
the union of all elements having a common edge/face with K. Similarly let ωE be the
union of both elements having E as edge/face.
The finite volume approximation uh of u is piecewise constant on Th , i.e., uh :=
(uK )K∈Th (uK being the approximation of u(xK ) for K ∈ Th , xK being the “center”
of the box K). To deduce the approximated equation satisfied by uh , we first integrate
(1) on a control volume K and use the divergence formula to obtain
−
(5)
∇u · nK,E ds =
f (x) dx ∀K ∈ Th ,
E∈EK
E
K
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A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
where EK is the set of edges/faces of K. The diffusion flux E ∇u · nK,E is approximated by a numerical diffusion flux FK,E (uh ) obtained using quadrature rules and
finite differences (see, e.g., [10]) and is consequently a linear combination of some values of uh around E [10, 5, 6]. For our further uses we do not need its exact form but
the principle of conservation of flux is required: FK,E (uh ) = −FL,E (uh ) for E = K̄ ∩ L̄.
These approximations lead to the following system. Find a solution uh := (uK )K∈Th of
(6)
FK,E (uh ) =
f (x) dx ∀K ∈ Th .
−
K
E∈EK
K
× xK
nL,E
E
nK,E × xL
L
Fig. 1. The standard orthogonality condition.
If the mesh Th is admissible in the sense of [10, Def. 9.1], i.e., satisfies standard
orthogonality conditions (see Figure 1), then the numerical diffusion flux is defined
by
(7)
|E|(uL − uK )
if E = K ∩ L,
d(xK , xL )
|E|uK
if E ⊂ K ∩ ∂Ω.
FK,E (uh ) := −
d(xK , Γ)
FK,E (uh ) :=
For general meshes, a possible choice for FK,E (uh ) is proposed in [5, 6] using the
diamond cell method.
From now on we suppose that system (6) is well defined. This is the case if the
mesh Th is admissible in the sense of [10] and if FK,E (uh ) is given by (7) (see, for
instance, [10]); while for an arbitrary mesh and the choice of FK,E (uh ) from [5, 6],
system (6) is well defined under some geometrical conditions on the mesh [5, 6].
3. Some analytic tools.
3.1. Bubble functions, extension operator, and inverse inequalities. For
our further analysis we require standard bubble functions and extension operators that
satisfy certain properties recalled here for the sake of completeness.
We need two types of bubble functions, namely, bK and bE associated with an element K and an edge/face E, respectively. For a triangle or a tetrahedron K, denoting
by λaK
, i = 1, . . . , d + 1, the barycentric coordinates of K and by aE
i , i = 1, . . . , d,
i
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SERGE NICAISE
d+1
and
the vertices
of the edge/face E ⊂ ∂K, we recall that bK = (d + 1)d+1 i=1 λaK
i
d
bE = dd i=1 λaE
.
i
For a rectangle K we here enumerate its vertices in a clockwise sense. Denoting
by λaK
, i = 1, . . . , 4, the “barycentric” coordinates of K, namely, λaK
is the unique
i
i
K
element in Q1 (K) such that λaK
(a
)
=
δ
,
then
we
recall
that
b
=
8λ
λaK
and
i,j
K
aK
j
1
3
i
K
K
K
K
(λ
+
λ
)
if
the
endpoints
of
the
edge
E
are
a
and
a
.
bE = 4λaK
a2
a3
1
2
1
One recalls that bK = 0 on ∂K, bE = 0 on ∂ωE , bK ∞,K = bE ∞,ωE = 1.
In two dimensions for an edge E ⊂ ∂K using temporarily the local coordinates
system (x, y) such that E is included into the x-axis, then the extension Fext (vE ) of
vE ∈ C(E) to K is defined by Fext (vE )(x, y) = vE (x). We proceed similarly in three
dimensions.
Now we may recall the so-called inverse inequalities, whose proof uses classical
scaling techniques and the fact that all norms are equivalent in a finite-dimensional
space [26].
Lemma 3.1 (inverse inequalities). Let K ∈ Th , E ∈ EK , vK ∈ Pk0 (K), and vE ∈
Pk1 (E) for some nonnegative integers k0 and k1 . Then there exist positive constants
β0 , β1 (resp., α0 , α1 , and α2 ) depending on the form of K (triangle, rectangle, or
tetrahedron), on the aspect ratio of the mesh Th , and on the polynomial degree k0
(resp., k1 ) such that
(8)
(9)
(10)
(11)
(12)
1/2
1/2
vK bK 2K ≤ vK 2K ≤ β0 vK bK 2K ,
2
∇(vK bK )2K ≤ β1 h−2
K vK K ,
1/2
1/2
vE bE 2E ≤ vE 2E ≤ α0 vE bE 2E ,
Fext (vE )bE 2K ≤ α1 hK vE 2E ,
2
∇(Fext (vE )bE )2K ≤ α2 h−1
K vE E .
Remark 3.2. In the above
lemma, if K is a square
and k1 = 2, then α0 =
√
√
8(6+ 21)
8(56+ 881)
≈ 1.967, α1 =
≈ 0.269, α2 =
≈ 6.528. These numbers
315
105
are obtained by reducing estimates (10)–(12) to an eigenvalue problem. Namely using
the standard basis of P2 , estimate (12) is equivalent to (AX, X) ≤ α2 (BX, X) for all
X ∈ R3 , where A and B are two explicit 3 × 3 matrices. Therefore, α2 is the largest
eigenvalue of the matrix B −1/2 · A · B 1/2 , or equivalently the largest eigenvalue of the
matrix B −1 · A, since B is invertible. A direct calculation yields the value of α2 . The
other estimates are proved in the same manner.
√
10+ 30
4
3.2. Interpolation error estimates. Here we collect some standard interpolation error estimates but we specify as explicitly as possible the involved constants. As
usual we start with the reference elements, which are the unit triangle K̂ of vertices
(0, 0), (1, 0), (0, 1), the unit square K̂ of vertices (0, 0), (1, 0), (0, 1), (1, 1), or the unit
tetrahedron K̂ of vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1).
Lemma 3.3. Let Ê be the edge/face of K̂ included in the axis/plane xd = 0. Then
there exist two positive constants μ and α such that for all v ∈ H 1 (K̂), the following
estimates hold:
(13)
(14)
v − MK̂ vK̂ ≤ μ∇vK̂ ,
v − MÊ vÊ ≤ α∇vK̂ .
1
≈ 0.565244. If K̂ is
If K̂ is the reference square, then μ = π1 and α = √π tanh
π
1
1
2
the reference triangle, then μ = π and α = μ1 , where μ1 is the first positive root of
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A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
the transcendental equation
(15)
sinh x + tan x = 0.
√
1/4
(2(11+4 6))
√
This means that α≈0.730276.
√ If K̂ is the reference tetrahedron, then μ ≤
3π
(2(11+4 6))1/4
≈ 0.466715 and α≤ √π tanh π ≈ 1.43549.
Proof. The two estimates are Poincaré-like inequalities and follow from the
Bramble–Hilbert lemma. But this argument does not give an estimate for μ and α.
Therefore, we argue as follows. For the first estimate, denote by λ21 the first positive
eigenvalue of the Laplace operator on K̂ with Neumann boundary2 conditions. Then
∇v
by the min-max principle, we know that λ21 = min v∈H 1 (K̂) v
2K̂ . This identity is
v=0,M v=0
K̂
K̂
equivalent to
λ21 v2K̂ ≤ ∇v2K̂
∀v ∈ H 1 (K̂) : MK̂ v = 0.
We then obtain (13) with μ = λ−1
1 .
If K̂ is the unit square, it is well known that λ21 = π 2 and consequently μ = π1 .
If K̂ is the reference triangle, we use the following extension operator from K̂ to the
unit square (0, 1)2 , temporarily denoted by Ŝ. Namely, for v ∈ H 1 (K̂), we define its
extension Ev to Ŝ by
Ev(y1 , y2 ) = v(y1 , y2 ) if (y1 , y2 ) ∈ K̂,
Ev(y1 , y2 ) = v(1 − y2 , 1 − y1 ) if (y1 , y2 ) ∈ Ŝ \ K̂.
Note that Ev ∈ H 1 (Ŝ) and from v2K̂ = v2Ŝ , ∇v2K̂ = ∇v2Ŝ , we easily get
min
1
∇v2K̂
v∈H (K̂)
v=0,M v=0
K̂
v2K̂
≥
min
1
v∈H (Ŝ)
v=0,M v=0
Ŝ
∇v2Ŝ
v2Ŝ
= π2 .
√
On the other hand, one readily checks that the function ψ(x, y) = 2(cos(πx) +
cos(π(1 − y)) is an eigenvector of the eigenvalue π 2 of the Laplace operator with
Neumann boundary conditions in K̂. Therefore, we actually have λ21 = π 2 .
We use a similar argument for the reference tetrahedron. Namely we use an
extension operator from K̂ to the standard reference pentahedron P̂ = Ê × (0, 1). For
that purpose denote by K̂2 and K̂3 the tetrahedra of vertices (1, 0, 0), (0, 1, 0), (0, 0, 1),
(1, 0, 1) and (0, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1), respectively. We remark that P̂ = K̂ ∪
K̂2 ∪ K̂3 , that K̂ and K̂2 have a common face, and similarly that K̂2 and K̂3 have
a common face. Note further that |K̂| = |K̂2 | = |K̂3 | = 16 . Therefore, as before there
exists an affine transformation F1 which maps K̂ onto K̂2 and let their common face
be invariant. Similarly denote by F2 the affine transformation which maps K̂2 onto
K̂3 and let their common face be invariant. Denote by Ai , i = 1, 2, the 3 × 3 matrices
and by bi ∈ R3 , i = 1, 2, such that Fi (x) = Ai x + bi for all x ∈ R3 . Now we are able
to define the extension operator E: for v ∈ H 1 (K̂), we define
Ev(y) = v(y),
Ev(y) =
Ev(y) = v(F1−1 (y))
if y ∈ K̂,
v(F1−1 (F2−1 (y)))
if y ∈ K̂2 ,
if y ∈ K̂3 .
Using the above properties between the tetrahedra K̂, K̂2 , and
K̂3 and somechanges of
variables, we readily check that Ev ∈ H 1 (P̂ ) and satisfies P̂ Ev(y) dy = 3 K̂ v(x) dx,
and
|Ev(y)|2 dy = 3
|v(x)|2 dx,
|∇Ev(y)|2 dy =
∇v(x) · T · ∇v(x) dx,
P̂
K̂
P̂
K̂
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SERGE NICAISE
where the matrix T is given by
⎛
−1
−1 −
T = Id + (A
+ A−1
A1
1 A1 )
1 (A2 A2 )
−2
3
−1
5
= ⎝−2
0
⎞
0
−1⎠ .
5
√
An easy calculation yields T 2 = 2(11 + 4 6) ≈ 6.44949.
These identities directly lead to
∇v2K̂
min
1
v∈H (K̂)
v=0,M v=0
K̂
v2K̂
≥
3
T 2
min
1
∇v2P̂
v∈H (P̂ )
v=0,M v=0
P̂
v2P̂
=
3π 2
.
T 2
2
We then conclude that μ ≤ T
3π 2 .
For the second estimate, we start with the following nonstandard eigenvalue problem in the unit square K̂ = (0, 1)2 . Find λ2 and v ∈ H 1 (K̂) solution of
∇v · ∇w = λ2
vw ∀w ∈ H 1 (K̂).
(16)
K̂
Ê
For this eigenvalue problem, let us show that the min-max principle holds at least for
the first positive eigenvalue λ̃21 . Namely λ̃21 is characterized by
λ̃21 =
(17)
∇v2K̂
min
1
v∈H (K̂)
=0,M v=0
Ê
Ê
v
v2Ê
.
Denote by m the above right-hand side. Consider a minimizing sequence vn of the
above minimum, namely, for all n ∈ N, let vn ∈ H 1 (K̂) be such that
MÊ vn = 0,
vn Ê = 1,
∇vn 2K̂ → m as n → ∞.
Since ∇vK̂ + vÊ is a norm on H 1 (K̂) equivalent to the standard norm, the
sequence (vn )n is bounded in H 1 (K̂). Therefore, there exists a subsequence, still
denoted by (vn )n , such that
vn → v in H 1 (K̂) as n → ∞.
From the above properties of the sequence (vn )n , we deduce that v satisfies
MÊ v = 0,
vÊ = 1,
∇v2K̂ = m.
It remains to show that v is an eigenvector of problem (16) corresponding to the
eigenvalue m. For that purpose let us fix z ∈ H 1 (K̂) such that
MÊ z = 0 and
(18)
vz = 0.
Ê
Consider the mapping
Φ:R→R:α→
∇(v + αz)2K̂
1 + α2 z2K̂
.
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A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
From the above minimization problem and the properties of v, the mapping Φ hits
its minimum at α = 0. Since Φ is smooth, we deduce that Φ (0) = 0, or
(19)
∇v · ∇z = 0.
K̂
Since any w ∈ H 1 (K̂) such that MÊ w = 0 may be written in the form w = βv + z,
with β ∈ R and z ∈ H 1 (K̂) satisfying (18), we deduce that
∇v · ∇w = β
∇v · ∇v +
∇v · ∇z.
K̂
K̂
K̂
By the properties of v and identity (19), we conclude that
∇v · ∇w = βm = m
vw;
K̂
Ê
this last identity follows from (18).
To find the first eigenvalue of problem (16), we remark that its strong form is
⎧
⎪
⎨−Δv = 0 in K̂,
∂v
= λ2 v on Ê,
∂n
⎪
⎩ ∂v
= 0 on ∂ K̂ \ Ê.
∂n
Using the standard argument of separation of variables, one finds a family of eigenvalues, its smallest one being π tanh π ≈ 3.12988. In order to be sure that this value
is the smallest eigenvalue of problem (16), we penalize it by an integral term in K̂.
Namely, for any > 0, we consider the problem
2
(20)
∇v · ∇w = λ
vw + vw
∀w ∈ H 1 (K̂).
K̂
Ê
K̂
This problem is an eigenvalue problem related to a selfadjoint nonnegative operator.
For that problem one can find all the eigenvalues by separation of variables. Since the
eigenvalues of (20) depend continously on , the first positive eigenvalue λ̃21, tends to
the first eigenvalue of problem (16). By direct calculations one shows that
λ̃21, → π tanh π.
We therefore conclude that λ̃21 = π tanh π. By the above “min-max” principle (17),
we deduce that α = 1/λ̃1 .
For the unit triangle, we start with the minization problem (17) (as before λ̃21 is
the first positive eigenvalue of problem (16)). Using the extension operator E from
K̂ to Ŝ, we deduce that
λ̃21 =
min
1
v∈H (K̂)
=0,M v=0
Ê
Ê
v
∇v2K̂
v2Ê
≥
min
1
v∈H (Ŝ)
+v =0,M
v=0
Ê
F̂
Ê∪F̂
v
∇v2Ŝ
v2Ê + v2F̂
,
where F̂ is the edge of Ŝ included into the line x1 = 1. The right-hand side is related
to the eigenvalue problem: find λ2 and v ∈ H 1 (Ŝ) solution of
(21)
∇v · ∇w = λ2
vw ∀w ∈ H 1 (Ŝ).
Ŝ
Ê∪F̂
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SERGE NICAISE
The same arguments as before give, as first positive eigenvalue μ21 , the first positive
root of the transcendental equation (15).
As for the first estimate we deduce that μ21 is the first positive eigenvalue of
problem (16). Indeed, if w(x1 , x2 ) is the eigenvector of problem (21) associated with
the eigenvalue μ21 , then one readily checks that v(x1 , x2 ) = w(x1 , x2 )−w(1−x2 , 1−x1 )
is an eigenvector of problem (16) associated with the eigenvalue μ21 .
As before the situation is not so convenient for the unit tetrahedron Therefore,
we first state the following estimate on the reference prism P̂ :
v − MÊ vÊ ≤ √
1
∇vP̂
π tanh π
∀v ∈ H 1 (P̂ ),
obtained as for the unit square. Now, using the extension operator E and this estimate, we may write
v − MÊ vÊ = Ev − MÊ EvÊ ≤ √
1
1
T 2 ∇vK̂ ;
∇EvP̂ ≤ √
π tanh π
π tanh π
this last estimate follows from the above properties of Ev.
Remark 3.4. To our knowledge, the exact value of μ is not explicitly known
for the unit tetrahedron. Numerical tests give for λ21 the approximated value λ21 ≈
14.444208445. This gives for μ the approximated value μ ≈ 0.26312, which is relatively
smaller than our theoretical upper bound. Similarly the exact value of α is not
explicitly known for the unit tetrahedron; an approximated value is 0.340355, and
therefore our theoretical upper bound is far from being optimal.
In the above arguments, the main difference between the unit triangle and the unit
tetrahedron concerns the extension operator. For the triangle, the extension operator
uses an orthogonal transformation, which is impossible for the unit tetrahedron. That
last case still requires more investigations.
The above lemma and scaling arguments lead to the following lemma.
Lemma 3.5. There exist two positive constants μ and α depending on K̂ such
that for all K ∈ Th and v ∈ H 1 (K), the following estimates hold:
(22)
v − MK vK ≤ μρ̂−1 hK ∇vK ,
(23)
v − ME vE ≤ αρ̂−1 hE,K hK ∇vK ,
−1/2
where E is an edge/face of K, and ρ̂ is the diameter of the inscribed ball of K̂.
4. Some finite elements of Morley type. As already mentioned the main
idea of our a posteriori error analysis is to use an interpolant p satisfying
∂p
ds = FK,E (uh ) ∀E ∈ EK .
E ∂nK,E
This means that we need to use finite elements having as degrees of freedom the mean
of the normal derivative of p on each edge/face. The simplest element is the so-called
Morley triangle [18, 4] usually used for the approximation of the plate problem. For
our further uses we extend this kind of elements to rectangles and tetrahedra. We
start by recalling the Morley triangle as well as a recent extension due to Nilssen, Tai,
and Winther [21] and then introduce our new elements.
1490
A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
4.1. Triangles. Here K is a (nondegenerate) triangle with vertices aK
i , i =
1, 2, Nf := 3.
The standard Morley triangle is defined by the triple (K, PK , ΣK ) [18, 4], where
PK = P2 (K) and
(24)
ΣK = {p(ai )}i=1,...,Nf ∪
E
∂p
ds
∂nK,E
.
E∈EK
Note that this element is not a C 0 -element; an extension which has this property was
recently built in [21, sect. 4], where they take
PK = P2 (K) ⊕ P1 (K)bK = {q + pbK : q ∈ P2 (K), p ∈ P1 (K)},
∂p
ΣK = {p(aK
)}
∪
{p(m
)}
∪
ds
.
i=1,2,3
E
E∈E
i
K
E ∂nK,E
E∈EK
4.2. Rectangles. Here K is a (nondegenerate) rectangle of vertices aK
i , i =
1, . . . , Nf := 4.
The first element is defined by PK = P2 (K) ⊕ Span {x3 − 3xy 2 , y 3 − 3yx2 } with
degrees of freedom ΣK defined by (24). We readily check that the triple (K, PK , ΣK )
is a finite element. The above choice is motivated by the fact that Δq ∈ R ∀q ∈ PK ,
since x3 − 3xy 2 and y 3 − 3yx2 are the unique homogeneous polynomials of degree 3
which are harmonic.
The second example is to take PK = Q2 (K) and ΣK := {p(aK
i )}i=1,...,5 ∪
K
{ E ∂n∂p
ds}
,
where
a
is
the
center
of
gravity
of
K.
E∈E
5
K
K,E
4.3. Tetrahedra. Here K is a (nondegenerate) tetrahedron with vertices aK
i ,
i = 1, 2, 3, Nf := 4.
Inspired from the second triangular example from [21] we choose PK = P1 (K) ⊕
P1 (K)bK = {q + pbK : p, q ∈ P1 (K)}, and ΣK defined by (24).
Similar to Lemma 4.1 of [21] (adapted to our setting) we can prove the following
lemma.
Lemma 4.1. The above triple (K, PK , ΣK ) is a C 0 -finite element.
5. The Morley interpolant.
5.1. Definition. For any vertex a of the triangulation we fix (wK (a))K∈Th :a∈K
suitable weights of interpolation around K. Since our analysis below is independent
of their choice, we do not describe them. They may be obtained using a discrete
projection of piecewise constant functions over affine functions on ωa [5, 6], a standard
technique to get a recovered gradient at the vertex a, leading further to the P1 exactness.
Namely for any vertex a the weights wK (a) may be fixed such that w(a) =
w
(a)uK , where w ∈ P1 (ωa ) is the discrete projection of uh on P1 (ωa ), i.e.,
K
K⊂ωa
w ∈ P1 (ωa ) is the unique minimizer of
|q(xK ) − uK |2 ,
q ∈ P1 (ωa ).
K⊂ωa
This choice implies that if uh were P1 (ωa ), then we would have w = uh in ωa . For
instance, if ωa is made of four squares, then this choice yields wK (a) = 1/4.
1491
SERGE NICAISE
We now introduce the Morley finite element space
Vh := vh ∈ L2 (Ω) : vh|K ∈ PK ∀K ∈ Th ,
L
K
L
vh|K (aK
i ) = vh|L (aj ) ∀K, L ∈ Th , i, j ∈ {1, . . . , Nf } : ai = aj ,
K
vh|K (aK
i ) = 0 ∀K, L ∈ Th , i ∈ {1, . . . , Nf } : ai ∈ Γ,
∂vh|K
∂vh|L
ds =
ds ∀E ∈ Eh , K, L ∈ Th : E = K ∩ L .
E ∂nE
E ∂nE
Since Vh is not necessarily included into H01 (Ω), the space Vh is equipped with
the norm | · |1,h := ( K∈Th | · |21,K )1/2 . Notice that Vh is indeed included into H01 (Ω)
for the second-triangular example and for our three-dimensional (3D) example.
Definition 5.1. For uh = (uK )K∈Th , we define its Morley interpolant IM uh as
the unique element vh in Vh satisfying
(25) vh|K (aK
wL (aK
∀K ∈ Th , i ∈ {1, . . . , Nf } : aK
i )=
i )uL
i ∈ Ω,
L∈Th :aK
∈L
i
(26)
vh|K (aK
i )
(27)
E
=0
∀K ∈ Th ,
∂vh|K
ds = FK,E (uh )
∂nK,E
i ∈ {1, . . . , Nf } : aK
i ∈ Γ,
∀E ∈ EK ,
K ∈ Th .
For the second triangular element we have to add the conditions
vh|K (mE ) = vh|L (mE ) = 12 (uK + uL ) ∀E ∈ Eh , K, L ∈ Th : E = K ∩ L,
vh|K (mE ) = 0 ∀E ∈ Eh , K ∈ Th : E ⊂ K ∩ Γ.
Similarly for the first-rectangular element we must add vh|K (aK
5 ) = uK for all K ∈ Th .
5.2. Some useful properties. We first prove a basic property of the Morley
interpolant.
Lemma 5.2. If uh is solution of (6), then IM uh satisfies
(28)
Δ(IM uh ) dx = −
f (x) dx ∀K ∈ Th .
K
K
Proof. By Green’s formula and property (27) satisfied by IM uh , we have
∂(IM uh )
Δ(IM uh ) dx =
ds =
FK,E (uh ),
K
E ∂nK,E
E∈EK
E∈EK
and we conclude by (6).
Now we prove some quasi-orthogonality relations that will be used for the upper
error bound. We first define the gradient jump of IM uh in the normal and tangential
direction by
JE,n (uh ) = ∂n∂E (IM uh ) E ∀E ∈ Ehint ,
∂
∀E ∈ Eh for nonconforming 2D cases,
∂tE (IM uh ) E
JE,t (uh ) =
0 ∀E ∈ Eh
for conforming cases.
Lemma 5.3. If u is a solution of (2) and uh is a solution of (6), then
(29)
∇(u − IM uh ) · ∇χ dx =
(f + ΔIM uh )(χ − MK χ) dx
K
K∈Th K
K∈T
h
−
JE,n (uh )(χ − ME χ) ds ∀χ ∈ H01 (Ω).
int
E∈Eh
E
1492
A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
Proof. For brevity denote the left-hand side of (29) by I1 (χ). By (2) and Green’s
formula on each triangle K, and recalling that χ ∈ H01 (Ω), we may write
∂(IM uh )
I1 (χ) =
f χ dx +
Δ(IM uh )χ dx −
χ ds
∂nK
Ω
K∈Th K
K∈Th ∂K
=
(f + Δ(IM uh ))χ dx −
JE,n (uh )χ ds.
K
K∈Th
int
E∈Eh
E
Using identity (28), we arrive at
I1 (χ) =
(f + Δ(IM uh ))(χ − MK χ) dx −
JE,n (uh )χ ds.
K∈Th
K
int
E∈Eh
E
The conclusion now follows from the fact that E JE,n (uh ) ds = 0, for all E ∈ Ehint ,
due to (27) and the principle of conservation of flux, FK,E (uh ) = −FL,E (uh ), if
E = K ∩ L, K, L ∈ Th .
Corollary 5.4. Under the assumptions of Lemma 5.3 the next estimate holds
(30)
|I1 (χ)| ≤
√ μ2 2
2 2
hK f + Δ(IM uh )2K
ρ̂
K∈Th
α Nf 4ρ̂2
2
+
2
2
h−1
E,K hK JE,n (uh )E
1/2
|χ|1,Ω .
K∈Th E∈E int ∩EK
h
Proof. Identity (29) and Cauchy–Schwarz’s inequality yield
f + Δ(IM uh )K χ − MK χK +
JE,n (uh )E χ − ME χE
|I1 (χ)| ≤
int
E∈Eh
K∈Th
≤
f + Δ(IM uh )K χ − MK χK
K∈Th
+
1 2
JE,n (uh )E χ − ME χE .
K∈Th E∈E int ∩EK
h
By the interpolation error estimates (22) and (23), we obtain
μ
α
−1/2
|I1 (χ)| ≤
hK
hE,K JE,n (uh )E |χ|1,K .
f + Δ(IM uh )K +
ρ̂
2ρ̂
int
E∈Eh ∩EK
K∈Th
We
the discrete Cauchy–Schwarz’s inequality and the well-known estimate
lconclude by
l
( i=1 ai )2 ≤ l i=1 a2i , valid for l = 2, 3, 4 and all real numbers ai .
Lemma 5.5. Assume that d = 2. If u is the solution of (2) and uh is the solution
of (6), then
(31)
∇(u−IM uh )·curl g dx =
JE,t (uh )(g−ME g) ds ∀g ∈ H 1 (Ω),
K∈Th
K
E∈Eh
E
where curl g = (∂2 g, −∂1 g) is the vectorial curl of g.
1493
SERGE NICAISE
Proof. Denote the left-hand side of (31) by I2 (g). Green’s formula on each element
K leads to (see Theorem I.2.11 of [11])
∂
(u − IM uh )g ds =
JE,t (uh )g ds,
I2 (g) = −
∂K ∂t
E
K∈Th
E∈Eh
since u ∈ H01 (Ω) and g ∈ H 1 (Ω). The conclusion follows from the property
(32)
JE,t (uh ) ds = 0.
E
Indeed, if aiE , i = 1, 2, are the two extremities of E, we have E JE,t (uh ) ds = uh E
(a1E ) − uh E (a2E ). Using properties (25) and (26), we have uh E (aiE ) = 0, i = 1, 2,
and therefore (32) holds.
Corollary 5.6. Under the assumptions of Lemma 5.5 the following estimate
holds:
1/2
α
−1
2
2
|I2 (g)| ≤
(33)
Nf
hE,K hK JE,t (uh )E
|g|1,Ω .
ρ̂
K∈Th E∈EK
Remark 5.7. The above fundamental properties are only based on the definition
of the scheme (6), the continuity of the interpolant at the interior nodes, the property
(26), and the interpolation property (27). Therefore, our further analysis works for
any finite element (K, PK , ΣK ) such that the associated interpolant satisfies these
properties. But the finite element and the definition of the interpolant should be well
chosen in order to guarantee the convergence of IM uh to the exact solution u. That
is the reason of the introduction of the weights wK (a) in (25) since it was shown in
[5, 6] that for a triangulation made of rectangles, the choice of the weights described
at the beginning of section 5.1 guarantees the convergence of uh to u. Convergence
analysis for arbitrary triangulations and appropriate weights is still to be done, but
it is outside the scope of this paper.
6. Error estimators.
6.1. Residual error estimators. The exact element residual is defined by
RK := f + ΔIM uh on K. As usual this exact residual is replaced by some finitedimensional approximation called approximate element residual rK ∈ Pk (K). A realistic choice is to take rK = MK f + ΔIM uh since in the case ΔIM uh ∈ R we have
(thanks to Lemma 5.2) rK = 0.
Definition 6.1 (residual error estimator). The local and global residual error
estimators and approximation terms are defined by
−1
−1
2
2
2
2
2
hE,K JE,n (uh )E +
hE,K JE,t (uh )E ,
ηK := hK rK K +
η 2 :=
int
E∈EK ∩Eh
E∈EK
2
ηK
,
K∈Th
2
ζK
:=
K ⊂ωK
h2K RK − rK 2K ,
ζ 2 :=
K∈Th
2
ζK
.
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A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
6.2. Upper error bound.
Theorem 6.2. Let u be a solution of (2) and let uh be a solution of (6). Then
the error e := u − IM uh is bounded by
|e|1,h ≤ cup (η + ζ),
(34)
1/2
where cup = ρ̂1 max{2μ, αNf }.
Proof. Denote by ∇h e the brokent gradient of e, namely,
(∇h e)|K = ∇e|K on K
∀K ∈ Th .
As in Theorem 3.1 of [7] we use its Helmholtz decomposition
∇h e = ∇χ + curl ψ,
(35)
with χ ∈ H01 (Ω) and ψ ∈ H 1 (Ω) if d = 2 and χ = e and ψ = 0 if d = 3 such that
|χ|21,Ω + |ψ|21,Ω ≤ |e|21,h .
(36)
This estimate is direct
in three dimensions, while in two dimensions it directly follows
from the identity Ω ∇χ · curl ψ = 0, a consequence of Green’s formula (recalling that
χ = 0 on the boundary).
Owing to identity (35) we may write (with the notation from Lemmas 5.3 and
5.5)
2
2
|∇h e| dx =
∇h e · (∇χ + curl ψ) dx = I1 (χ) + I2 (ψ).
|e|1,h =
Ω
Ω
Using estimates (30) and (33), we obtain
1/2
1/2
√ μ2
αNf
α 2 Nf 2
ηt |ψ|1,Ω ,
η
|χ|
+
|e|21,h ≤ 2 2 Ξ2 +
1,Ω
ρ̂
4ρ̂2 n
ρ̂
where for brevity we set Ξ2 = K∈Th h2K f + Δ(IM uh )2K and
2
2
2
2
2
h−1
h
J
(u
)
,
η
=
h−1
ηn2 =
E,n
h
K
E
t
E,K
E,K hK JE,t (uh )E .
K∈Th E∈E int ∩EK
h
K∈Th E∈EK
By the discrete Cauchy–Schwarz inequality and estimate (36), we obtain
|e|21,h ≤
2μ2 2 α2 Nf 2 α2 Nf 2
4μ2 2
Ξ
+
η
+
η
≤
ζ
n
t
ρ̂2
2ρ̂2
ρ̂2
ρ̂2
4μ2 2
α 2 Nf 2
+ 2
hK rK 2K +
(ηn + ηt2 );
ρ̂
ρ̂2
K∈Th
this last estimate follows from the well-known estimate (a + b)2 ≤ 2a2 + 2b2 , valid for
all real numbers a, b. By the definition of cup and η the above estimate implies that
|e|1,h ≤ cup (ξ 2 + η 2 )1/2 ≤ cup (η + ξ).
Remark 6.3. Thanks to Lemma 3.3, we can estimate the constant cup appearing in the above upper bound. For a triangulation made of rectangles, then
cup = 2 max{ π1 , α} = 2α ≈ 1.13049. For a triangulation made of triangles, then
√
√
cup = ρ̂1 max{ π2 , 3α} = ρ̂3α ≈ 2.15928. Finally for a mesh made of tetrahedra, one
√
1/4
6))
√
has cup ≤ 2 (2(11+4
≈ 9.27912. For that last case, by Remark 3.4, a numerical
π tanh π
upper bound for cup is 2.20009. In both cases, the exact value, or the numerical upper
bound for the tetrahedral case, of cup is quite close to unity.
1495
SERGE NICAISE
6.3. Lower error bound.
Theorem 6.4. For all elements K, the following local lower error bound holds
(37)
ηK ≤ clow (∇h eωK + ζK ),
where c2low = max{2β02 β1 + 2Nf α02 σ1−1 σ22 σ3 (3α2 σ4−1 + 8α1 β02 β1 σ1−1 σ4 ), 2β02
+ 8Nf α02 α1 (1 +2β02 )σ1−1 σ22 σ3 σ4 }.
Proof.
Element residual. For a fixed element K denote wK = rK bK which belongs to
H01 (K). From the definition of RK and integration by parts, we may write
rK wK =
(rK − RK )wK −
Δ(u − IM uh )wK
K
K
K
=
(rK − RK )wK +
∇e · ∇wK .
K
K
By Cauchy–Schwarz’s inequality and the inverse inequalities (8) and (9), we conclude
that
(38)
1/2
hK rK K ≤ β0 (β1 ∇eK + hK rK − RK K ).
Normal jump. Fix an arbitrary E ∈ Ehint . Recall that JE,n (uh ) ∈ Pk (E) for some
k ∈ N and set wE := Fext (JE,n (uh ))bE ∈ H01 (ωE ). By elementwise partial integration,
we get
∂e
JE,n (uh )wE = −
wE = −
(∇e · ∇wE + ΔewE ) dx
∂nK
E
K⊂ωE ∂K
K⊂ωE K
ΔeK wE K .
≤ ∇h eωE ∇wE ωE +
K⊂ωE
Inequalities (10)–(12) and properties (3) and (4) in the previous estimate lead to
−1
2
2 −1
2
2
2
hK RK K .
hE JE,n (uh )E ≤ 2α0 σ1 2α2 σ4 ∇h eωE + 2α1 σ4
K⊂ωE
By estimate (38) we arrive at
(39)
hE JE,n (uh )2E ≤ 4α02 σ1−1 (α2 σ4−1 + 4α1 σ4 β02 β1 )∇h e2ωE
h2K RK − rK 2K .
+ 8α02 α1 σ1−1 σ4 (1 + 2β02 )
K⊂ωE
Tangential jump (in two dimensions). For a fixed edge E set wE := Fext (JE,t (uh ))
bE ∈ H01 (ωE ). For u ∈ H 1 (ωE ) and wE ∈ H01 (ωE ), partial integration leads to
∂u
0=
∇u · curl wE .
wE = −
∂ωE ∂t
ωE
For IM uh we integrate elementwise and obtain using the above identity
∂(IM uh )
wE =
JE,t (uh )wE = −
∇(IM uh ) · curl wE
∂t
E
K⊂ωE ∂K
K⊂ωE K
=−
∇(u − IM uh ) · curl wE ≤ ∇h eωE ∇wE ωE .
K⊂ωE
K
1496
A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
The inverse inequalities (10) and (12), as well as (3) and (4), lead to
hE JE,t (uh )2E ≤ 2α02 α2 σ1−1 σ4−1 ∇h e2ωE .
(40)
The conclusion follows from estimates (38)–(40) and properties (3) and (4).
Remark 6.5. In the above proof, we see that if rK = 0, then the constant clow
reduces to c2low = 2Nf α02 σ1−1 σ22 σ3 max{3α2 σ4−1 , 2α1 σ4 }. Let us illustrate this constant
clow in the particular case considered in the next section. Take a triangulation made
of squares, build the Morley interpolant with the help of the first rectangular
element
from section 4, and choose rK = 0. Then by Remark 3.2, we have clow = 24α02 α2 ≈
24.622.
7. Diffusion–convection–reaction equations. In this section we describe how
to extend the above results to diffusion–convection–reaction equations; for details we
refer the reader to [20].
Consider the linearized diffusion–convection–reaction problem: for f ∈ L2 (Ω) let
u ∈ H01 (Ω) be the unique solution of
(41)
Au := div (−∇u + vu) + bu = f in Ω,
where is a fixed positive constant, v ∈ Rd , and b is a nonnegative real number.
Integrating (41) on a control volume K and using the divergence formula, we
obtain
(−∇u + vu) · nK,E ds +
bu dx =
f (x) dx ∀K ∈ Th .
E∈EK
E
K
K
The continuous diffusion flux −∇u · nK,E is approximated as before,
the convection
flux vu · nK,E by a first order upwind scheme, and the reaction term K u by a simple
quadrature formula (see [10]). These approximations lead to the following system.
Find uh := (uK )K∈Th , the solution of
R
−FK,E (uh ) + vK,E FEC (uh ) + bFK
(42)
(uh ) =
f (x) dx ∀K ∈ Th ,
K
E∈EK
where vK,E = v · nK,E , the quantity FK,E (uh ) is supposed to satisfy the principle
R
of conservation of flux, while the quantities FEC (uh ) and FK
(uh ) are, respectively,
defined by
(43)
FEC (uh ) := |E|uE,+ ,
where for E ∈ Ehint , uE,+ = uKE,+ , KE,+ being the upstream control volume, i.e.,
vKE,+ ,E ≥ 0; while for E ∈ K̄ ∩ Γ, uE,+ = uK if vK,E ≥ 0, and uE,+ = 0 else.
R
FK
(uh ) = |K|uK .
For a restricted admissible mesh in the sense of [10, Def. 9.4], if the numerical
diffusion fluxes FK,E (uh ) are given by (7), then system (42) is well defined as proved
in [9]. For a general mesh as here, we simply assume that system (42) has a unique
solution.
As for the Laplace operator, we associate with the finite volume solution uh its
Morley interpolant IM uh . This interpolant is related to the quantities involved in
(42), namely, the diffusion and convection fluxes, and the reaction term. For that
purpose, for each element K, we build a C 0 -finite element (K, PK , ΣK ) having as
1497
SERGE NICAISE
degrees of freedom the mean of the normal derivative and of the function on each
edge, as well as the mean on K. For instance, if K is a triangle, we may take
PK = {q + (p + αbK )bK : q ∈ P2 (K), p ∈ P1 (K), α ∈ R},
K
ΣK = p(ai )
∪
p ∪
p ds
∪
K
i=1,2,3
E
E∈EK
E
∂p
ds
∂nK,E
.
E∈EK
Then for uh = (uK )K∈Th , we define its interpolant IM uh as the unique element
vh in Vh ∩ H01 (Ω) satisfying (25)–(27) and
R
vh ds = FEC (uh ) ∀E ∈ Ehint ,
vh dx = FK
(uh ) ∀K ∈ Th .
E
K
The key point of our a posteriori analysis is the following basic property of the
Morley interpolant, obtained using Green’s formula and the above properties of IM uh .
Lemma 7.1. If uh is a solution of (42), then IM uh satisfies
(A(IM uh ) − f ) dx = 0 ∀K ∈ Th : measd−1 (K ∩ Γ) = 0.
K
This property and similar arguments to those used before allow us to prove the
following error bounds.
Theorem 7.2. Let u be a solution of (41), and let uh be a solution of (42). Then
the error is bounded by
2
|∇e| +
(44)
Ω
1/2
1
div v + b |e|2
≤ c1 (η + ζ),
2
where c1 is a positive constant depending on the aspect ratio of the mesh and of the
size of .
For all elements K, the following local lower error bound holds:
(45)
ηK ≤ c2
|∇e|2 +
ωK
1/2
1
div v + b |e|2
+ ζK ,
2
where c2 is a positive constant depending on the aspect ratio of the mesh and of the
size of .
Remark 7.3. 1. By modifying appropriately the estimator ηK , we may skip the
dependence of c1 with respect to and give explicitly the dependence of c2 on this
parameter; see [20].
2. In the case of a large Peclet number P e ≡ −1 |v| and/or large number Γ ≡ −1 b,
problem (41) is singularly perturbed and the solution may generate sharp boundary
or interior layers, where the solution of the limit problem (corresponding to = 0)
is not smooth or does not satisfy the Dirichlet boundary condition. In that case, the
use of anisotropic meshes is recommended. This will be addressed in [20].
8. Numerical results. In this section we present two numerical tests that illustrate the efficiency and reliability of our estimator. The second example further
indicates that our estimator is appropriate for adaptivity. Additionally, for both examples we provide the order of convergence of the error |e|1,h ; both cases confirm that
IM uh is a good approximation of u.
1498
A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
8.1. A smooth solution. The first example is for a smooth solution in the unit
square ]0, 1[2 and quasi-uniform meshes made of squares. Namely, we consider problem
(1) in ]0, 1[2 with the following prescribed exact solution u(x, y) = xy(1−x)(1−y). The
meshes are uniform ones made of squares of size h = 1/n for n = 4, 8, . . . , 256 obtained
by dividing each segment in n subintervals. Since the meshes are made of squares we
use the scheme (6) with the numerical diffusion flux given by (7); furthermore, the
Morley interpolant is built using the first-rectangular element from subsection 4.2 and
the weights wK (a) = 1/4.
We first investigate the order of convergence of the approximated solution uh as
well as its interpolant IM uh to the exact solution u in different norms. Namely, we
present in Figure 2 the following norms: ū − uh (where ū is piecewise constant on
Th and is equal to u(xK ) on each K); |u − uh |1,Th (where the mesh-depending norm
| · |1,Th is defined in [10, Def. 9.3]); u − IM uh ; and |u − IM uh |1,h . These quantities
are illustrated in Figure 2 by lines 1–4, respectively, in a double logarithmic scale
so that the slope of the curve corresponds to the order of convergence. Theorem
9.3 of [10] yields the order of convergence 1 for |u − uh |1,Th . Figure 2 even reveals
a better order of convergence of about 1.5, probably due to the smoothness of u.
For the L2 -norms, we remark a quadratic order of convergence, a usual phenomenon.
On the other hand, for the discrete H 1 -norm of the reconstructed approximation,
we also see a quadratic order of convergence. This seems to be a superconvergence
effect, probably due to the smoothness of the solution and of the use of structured
meshes.
log(error)
−3.15
1
−4.35
1
−5.54
−6.74
−7.94
−9.13
(2)
−10.33
−11.52
2
(4)
−12.72
1
(3)
−13.91
log(n)
(1)
−15.11
0.39
1.00
1.62
2.23
2.85
3.47
4.08
4.70
5.31
5.93
6.55
Fig. 2. Illustration of different norms for test 1.
Now we investigate the main theoretical results which are the upper and lower
error bounds (34) and (37). In order to present them appropriately, we consider the
ratios
|u − IM uh |1,h
as a function of | log n|,
η+ξ
ηK
:= max
as a function of | log n|.
K∈Th ∇(u − IM uh )ωK + ξK
qup :=
qlow
The first ratio qup is frequently referred to as the effectivity index. It measures the
reliability of the estimator and is related to the global upper error bound. The second
1499
SERGE NICAISE
ratio is related to the local lower error bound and measures the efficiency of the
estimator. From our theoretical considerations, both ratios should be bounded from
above which is confirmed experimentally as shown in Figure 3. Hence our estimator
is reliable and efficient.
In Figure 4 we compare the discrete H 1 seminorm |u − IM uh |1,h and the global
error estimator η with respect to n. We remark that the orders of convergence are the
same (namely, 2). This figure further confirms the equivalence between |u − IM uh |1,h
and η. All related quantities are summarized in Table 1.
0.6
0.35
0.55
0.3
0.5
qup
low
0.4
q
0.25
0.45
0.2
0.4
0.15
0.35
0.1
0.5
1
1.5
| Log(n) |
2
0.3
0.5
2.5
1
1.5
| Log(n) |
2
2.5
Fig. 3. qup (left) and qlow (right) in dependence of | log n| for test 1.
log(error)
−2.87
(2)
−3.93
(1)
−4.99
−6.06
−7.12
−8.18
−9.24
−10.31
2
−11.37
1
−12.43
−13.49
0.39
log(n)
1.00
1.62
2.23
2.85
3.47
4.08
4.70
5.31
5.93
6.55
Fig. 4. Comparison between − ln |u − IM uh |1,h (line (1)) and − ln η (line (2)) with respect to
ln n for test 1.
8.2. A nonsmooth solution. For the second example we use the L-shaped
domain Ω :=] − 1, 1[2 \ ]0,1[×] − 1, 0[ with the exact singular solution given in polar
coordinates by u = r2/3 sin 2θ
3 considered as a solution of the Dirichlet problem with
nonhomogeneous Dirichlet boundary conditions. As before the domain is discretized
using uniform meshes made of squares of size h = 1/n for n = 4, 8, . . . , 256. Since
u presents singular behavior near the origin and uniform meshes are used, we have
a reduction of the order of convergence from 1 to 2/3 for the norm |u − uh |1,h (see
Theorem 2.4 of [8] and Figure 5). From Figure 5 we notice the same phenomenon of
reduction of the order of convergence for the other norms.
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A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
Table 1
Data for test 1.
n
4
8
16
32
64
128
256
ū − uh 0,Ω
2.96e−03
7.57e−04
1.91e−04
4.77e−05
1.19e−05
2.99e−06
7.47e−07
|u − uh |1,Th
1.57e−02
5.58e−03
1.99e−03
7.09e−04
2.51e−04
8.90e−05
3.15e−05
u − IM uh 0,Ω
2.84e−03
8.58e−04
2.25e−04
5.68e−05
1.42e−05
3.56e−06
8.90e−07
|u − IM uh |1,h
1.37e−02
3.72e−03
9.53e−04
2.40e−04
6.00e−05
1.50e−05
3.75e−06
η
2.09e−02
6.43e−03
1.74e−03
4.48e−04
1.14e−04
2.86e−05
7.17e−06
qlow
0.4174
0.4875
0.5357
0.5513
0.5569
0.5580
0.5584
qup
0.2753
0.2708
0.2665
0.2638
0.2636
0.2616
0.2612
qlow
2.4949
2.4661
2.4527
2.4471
2.4448
2.4438
2.4435
qup
0.2958
0.3003
0.3022
0.3030
0.3033
0.3035
0.3034
log(error)
−1.07
1
−2.02
2/3
−2.96
−3.91
−4.86
(4)
(2)
−5.80
−6.75
−7.69
4/3
−8.64
1
(3)
(1)
−9.59
−10.53
0.39
1.00
1.62
2.23
2.85
3.47
4.08
4.70
5.31
log(n)
5.93
6.55
Fig. 5. Illustration of different norms for test 2.
Table 2
Data for tests 2.
n
4
8
16
32
64
128
256
ū − uh 0,Ω
1.63e−02
6.86e−03
2.81e−03
1.14e−03
4.56e−04
1.82e−04
7.25e−05
|u − uh |1,Th
7.02e−02
4.52e−02
2.87e−02
1.82e−02
1.15e−02
7.23e−03
4.55e−03
u − IM uh 0,Ω
1.66e−02
8.11e−03
3.59e−03
1.51e−03
6.21e−04
2.52e−04
1.01e−04
|u − IM uh |1,h
1.26e−01
8.11e−02
5.15e−02
3.26e−02
2.06e−02
1.30e−02
8.16e−03
η
4.26e−01
2.70e−01
1.71e−01
1.07e−01
6.78e−02
4.27e−02
2.69e−02
Again we have tested the rate of convergence of ū − uh , |u − uh |1,Th , u −
IM uh , and |u − IM uh |1,h . Here we notice that both discrete H 1 -norms have a rate of
convergence of 2/3 and that the rate of convergence of the L2 -norms is twice, namely,
4/3.
As before we further check the boundedness of the ratios qup and qlow . These
quantities are given in Table 2 and illustrated in Figures 6 and 7. From these figures
we can draw the same conclusion as before, namely, the efficiency and reliability of
our estimator.
From Tables 1 and 2, we see that the experimental bounds for qup and qlow are
quite smaller than the theoretical ones. This is quite realistic since the experimental
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SERGE NICAISE
0.38
2.5
2.49
0.36
2.48
2.47
0.34
qup
low
2.46
q
0.32
2.45
2.44
0.3
2.43
2.42
0.28
2.41
0.26
0.5
1
1.5
| Log(n) |
2
2.4
0.5
2.5
1
1.5
| Log(n) |
2
2.5
Fig. 6. qup (left) and qlow (right) in dependence of | log n| for test 2.
log(error)
0.15
−0.45
(2)
−1.04
−1.64
(1)
−2.24
−2.83
−3.43
−4.02
2/3
−4.62
1
−5.21
log(n)
−5.81
0.39
1.00
1.62
2.23
2.85
3.47
4.08
4.70
5.31
5.93
6.55
Fig. 7. Comparison between − ln |u − IM uh |1,h (line (1)) and − ln η (line (2)) with respect to
ln n for test 2.
Z
0.046
0.023
0.000
−1
0
Y
−1
0
1
X
1
Fig. 8. Distribution of the local estimator for test 2 and n = 64.
values depend on the chosen solution, while the theoretical analysis always considers
the worse case.
Finally, in Figure 8 we give the distribution of the local residual error estimators
for our second example with the mesh corresponding to n = 64. From this figure we
1502
A POSTERIORI ESTIMATES FOR FINITE VOLUME METHODS
may conclude that our estimator is appropriate for adaptivity, since it detects the
region of large errors, namely, the neighborhood of the origin.
9. Conclusions. We have proposed and rigorously analyzed a new a posteriori
error estimator for a cell-centered finite volume method that is reliable and efficient.
This estimator is based on the construction of an appropriate interpolant of Morley
type and the use of a Helmholtz decomposition of the error. The size of the constants
appearing in the error estimates has been given as explicitly as possible, as a function
of the aspect ratio of the mesh and of the form of the elements (triangles, rectangles,
or tetrahedra). Some numerical experiments confirm our theoretical predictions and
show that our estimator is appropriate for adaptivity.
The extension of our method to diffusion–convection–reaction equations is briefly
described; the details are postponed to a forthcoming paper.
Adaptive algorithms are not considered here since they require more investigations. They will be considered in forthcoming works.
Acknowledgments. I am very grateful to Karim Djadel (University of Lille,
France) who conducted the numerical experiments presented in section 8. I further
thank Joachim Schöberl (University of Linz, Austria) and Philipp Frauenfelder (ETH
Zürich, Switzerland) who gave me the numerical values of μ and α presented in
Remark 3.4 and obtained using their hp-FEM code.
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