Benchmark on Anisotropic Problems A symmetric finite volume scheme for anisotropic heterogeneous second-order elliptic problems Leo Agelas and Daniele A. Di Pietro
Institut Français du Pétrole 1 & 4, avenue de Bois-Préau 92852 Rueil-Malmaison Cedex - France
Institut Français du Pétrole 1 & 4, avenue de Bois-Préau 92852 Rueil-Malmaison Cedex - France
[email protected],
[email protected] ABSTRACT. In this paper, we assess a new family of finite volume discretization schemes introduced in [AE 07] on benchmark test cases. These are based on the discrete variational formulation framework developped in [EGH 08], [EH 07], [EGH 07]. The use of a subgrid for each cell of the mesh enables us to obtain fluxes only between cells sharing an edge as opposed to the cell centered finite volume scheme [EGH 07] for which fluxes are also defined between cells sharing only a vertex. The resulting finite volume schemes are cell centered, symmetric and coercive on general polygonal and polyhedral meshes and anisotropic heterogeneous media and can be proved to be convergent even for diffusion coefficients under usual shape regularity assumptions. Using L type interpolation from [AEMN 07], [AAV 07] for the intermediate subgrid unknowns enable us to take into account large jumps of the diffusion coefficients. KEYWORDS:
Anisotropy benchmark, finite volumes
1. Introduction In this work, we use a new family of finite volume discretization schemes for diffusion fluxes (see [AE 07]), based on discrete variational formulations together with L scheme ideas, in order to build cell centered finite volume approximations of diffusion fluxes which satisfy the following properties : consistency of the fluxes for all regular functions and permeability fields, stability: the norm of the solution should be bounded by the norms of the data (in the framework defined in [EGH 08], [EH 07] it is derived from the coercivity of the bilinear form, ensuring existence, uniqueness and stability of the solution); convergence: the convergence should hold on general meshes with usual shape regularity assumptions, and for anisotropic heterogeneous permeability fields with eigenvalues uniformly bounded from below and above.
Unfortunately, these properties on difficult anisotropic problems are obtained at the expense of larger scheme and flux stencils. For example, for a cartesian grid in 2D, the stencil by control volume is 21 and in 3D, 81.
We consider the following problem: find an approximation of , weak solution to the following equation:
div
! $
& # " % ('
in
under the following assumptions:
is an open bounded connected polygonal subset of )+*, -
where : and by l ¡ respectively. The function space [k l is equipped with the discrete scalar product
b b ¤ 2 e [ l ¥ [k lx¦ ,§ ¨ W j M¥ W ¥ j M' ZW Y\[ j Y\lnm - W\ j 1 ?©~ ,§ ¨ satisfies the following property: The associated seminorm [ ªP«¬ [k l ,§ ¨ ' ¨ ¡ C p®]^j t j Y\lnm , The subgrid of each cell ] b of the b mesh is defined by the set of pyramids ] j W of - -dimensional measure , joining the face e to the cell center . We denote 8f -dimensional by Q W the set of 1faces ¯ of ] j interior to the cell ] such that %],j±°P² Y\l m ¯ U e . The 1- If -dimensional measure of ¯ is denoted by b ¯ b , its ² ² barycenter by , and its unit normal vector outward to ] j by W³\ . The distance j ] # ³ W P µ U p r W L ¶ t from the barycenter of to ´ Q e is denoted by . The following property holds for any polygonal or polyhedral cell ] j : f b b b b Wr\ ² ² W b ],j br· e W\ j j W S¸ ¹² 1 ¸Sº » Y\lnm ¯ A constant consistent discrete gradient is defined on each subcell ]^j by
f b b b b ² [ l ¡ W W ²|¿ ¼ k [ l W b ] j b\½ e j W S WZ j ¹ 4T º^ ' ² Y\l m ¯ ·¾ 7 ² is a linear interpolation operator which uses the cell unknowns The operator ¾ for cells { sharing a face with the cell ] , ¯ #!] j and, possibly, local boundary face
²
unknowns in order to give an approximation of the face unknowns , ¯ #h] j . A ² possible choice for the operator ¾ is given in [EGH 08, page 5 §2.2]. More generally, this operator must verify the following condition : for all smooth functions , there exists a constant ÀP12 depending only on such that
b ² b ¾ 7 1H W 3 ]# OÁM 1H1 j M 7e # Q w>y3z 7 H1 ² EµÀP12D
[$Â 4
[ 3 ' [ Ã" . where  4
[ converges to zero as
_ Consider the following bilinear form on [2 l 5& [2 l : ½ b ] j b 4 [k l Wr W @¼ [2 l Wr Ä~Å [k l 7 [k l WZY\[ j Y\l m b b ´ : W\ j Wr\ ¶ [k l WrL ¶ k [ l ¿ Ç ¶|Y\lnm dÆ j - W ¶dÇ
where : W\ j is a positive real, are defined as follows:
Î
W ² k [ l W j 1 [k l Ç Ç
W
the matrix defined by
È
É(Í W ÊËÍ Ì * Ë
¾ ² 1 [k l ¡ W j W ¸ 4T [k l W j W j W ¸ @T [k l W
and the residuals for all ¯ for all e
# Q W # QX'
ÄuÅ The stabilization term L©1 involving the residuals allows us to prove the coercivity of J§ ¨ seminorm under the usual shape regularity assumptions. In with respect to the discrete variational form the problem reads: find ^[2 lÏ# [k l such that lM¡ XÐ2l3¡ C and, for all [k l6# [2 l , Ä~Å [k l [2 l ÑdÒ B [ -\D C
where [k l is the subspace of [k l with vanishing boundary face values. The above formulation is equivalent to the following hybrid finite volume scheme
Õ
ÓÔ
j Y\lnm2Ø W\ j W \ ÖuÔ Ø× j [ l
j Ð j
[
l Ø ¡ j
[
b] b W
l
for all ]&# OP b for all e ] { for all e # Q w>y3z
# dQ } z
where the fluxes are such that
Ä Å [k l 7 [k l V Ø W\ j [ l M1 W j 3 ÚÙ [2 l # [kC l ' \W Y\[ j \Y l m
[1]
Since the fluxes derive from a symmetric bilinear form, the resulting scheme will be symmetric. Indeed, thanks to the conservation of the fluxes, we can express the face unknowns l according to cell unknowns ^[ , then the system j Y\lnm
