Benchmark D: the VAG scheme Robert Eymard, Cindy Guichard and Raphaèle Herbin Presentation of the scheme Let Ω be a bounded open domain of R, let f L 2 Ω) and let Λ be a measurable function from Ω to the set M R) of matrices, such that for a.e. x Ω, Λx) is symmetric, and such that the set of its eigenvalues is included in [λ,λ], where 0 < λ λ. We wish to approximate the function u solution of u H0 Ω) and v H0 Ω), Λx) ux) vx)dx = f x)vx)dx, ) Ω Ω by the approximate gradient scheme [4, 2] which reads: U X D, V X D, Λx) D Ux) D V x)dx = f x)π D V x)dx, 2) Ω Ω where Π D is a reconstruction operator and D a discrete gradient operator which act on the discrete functional space X D, where the index D denotes the discretization; these operators are defined as defined as follows:. M is the set of control volumes, that are disjoint open subsets of Ω such that K = Ω, V = V int V ext is the set of vertices of the mesh; any element K of M is defined by its vertices s V K, its faces σ F K ; each face is also defined by the set of its vertices s V σ, using a suitable geometric definition for R. Eymard Université Paris-Est, e-mail: robert.eymard@univ-mlv.fr C. Guichard IFP Energies nouvelles and Université Paris-Est, e-mail: guichard@ifpenergiesnouvelles.fr R. Herbin Université Aix-Marseille, e-mail: Raphaele.Herbin@latp.univ-mrs.fr Work supported by Groupement MOMAS and ANR VFSitCom
2 R. Eymard, C. Guichard and R. Herbin the resulting surface in the case of non-planar faces; we assume that Λ is constant on all K M, and we denote by Λ K its value in K; 2. the set of discrete unknowns X D is the finite dimensional vector space on R, containing all real families U = u K ),u s ) s V ), such that u s = 0 if s V ext ;. the mapping Π D : X D L 2 Ω) maps U = u K ),u s ) s V ) X D to the piecewise constant function u D L 2 Ω) equal to u K on each cell K M ; 4. the mapping D : X D L 2 Ω) is the reconstruction of a gradient from the values U = u K ),u s ) s V ) X D ; different expressions for this reconstruction are proposed below, which all lead to convergent gradient schemes in the sense of [4, 2]. Their theoretical analysis is related to that of the SUSHI scheme [, ]. Detailed numerical results are given in this paper only using the method described in subsection.2; the differences obtained using the other expressions are commented in the last section. The exterior faces are those of the form K Ω for any boundary control volume K, and the interior faces are those of the form K L for two neighbouring control volumes K and L. For any face σ, we define a point x σ, which is a barycentre with non-negative weights β σ,s of the elements of the set V σ including all the vertices of the face, and the value u σ is defined by u σ = s V σ β σ,s u s with s V σ β σ,s =. In the next three subsections, we describe three ways of defining a gradient operator which satisfies the VAG requirements. The first gradient is constructed from the Stokes formula on the cells of the mesh we call it the primal cell to distinguish it from further constructed cells), and requires a stabilization. The second gradient and third gradients are constructed on tetrahedral or octahedral sub cells of the primal mesh, and are natively stable.. Stabilised gradient on the primal mesh cells For a face σ F, we denote by τ any triangular sub-face with vertices x σ, s and s, where s and s are two consecutive vertices of σ. The barycentre x τ of each sub-face τ may thus be expressed by the following barycentric combination: x τ = s V σ β τ,s s with s V σ β τ,s =, where β τ,s 0 for all s V σ. We then define β τ,s = 0 for all s V \ V σ. Next, we reconstruct a value u τ at the point x τ, by u τ = s Vσ β τ,s u s. Let K M be an element of the mesh. We denote by T K the set of all sub-faces of the faces of K. We first define, for U = u K ),u s ) s V ), an approximation of the gradient on cell K:
Benchmark D: the VAG scheme K U = K where we denote by τ u τ u K )n K,τ = u s u K )v K,s, ) τ T K s V K v K,s = K β τ,s τ n K,τ, τ T K where n K,τ is the unit normal vector to τ, outward to K, and τ, K are respectively the area and the volume of τ and K. We then define a partition M K,s of K there is no need to define this partition precisely), such that M K,s = K /N K, where N K is the number of vertices of K and we introduce R K,s U = u s u K K U s x K ). We then define, for a given γ > 0, the constant value K,s U in M K,s : K,s U = K u + γr K,s Uv K,s. We finally define a piecewise constant gradient by D Ux) = K,s U for a.e. x M K,s. This scheme is denoted by VAG in [5]..2 Piecewise constant gradient on octahedral sub-cells For a given face σ of a control volume K and for any vertex s of σ, we respectively denote by s and s + the preceding and the following vertices of s in the face σ defining any orientation on σ), and we consider the degenerate) octahedron, denoted by V K,σ,s and depicted in Fig., whose vertices are A = x K, A 2 = x σ, A = 2 s +s), A 5 = s, A 6 = 2 s+ +s) and A 4 = 2 x σ +s) note that all these octahedra are disjoint, and that the union of their closure is Ω). The approximate values of U at the vertices of V K,σ,s are respectively u = u K, u 2 = u σ, u = 2 u s + u s), u 5 = u s, u 6 = 2 u s + + u s) and u 4 = 2 u σ + u s ) the main diagonals of V K,σ,s are therefore A,A 4 ), A 2,A 5 ) and A,A 6. We then define the following approximate gradient: K,σ,s U = i= u i+ u i ) A i+ A i+4 A i+2 A i+5 Det A i+ A i+4, A i+2 A i+5, A i A i+ ), 4) setting A 7 = A and A 8 = A 2. We finally define a piecewise constant gradient by D Ux) = K,σ,s U for a.e. x V K,σ,s. Remark that, denoting for simplicity V instead of V K,σ,s, defining F V as the set of the 8 triangular faces of V and V τ as the set of the vertices of each triangular face τ of V, one may check that K,σ,s U = V τ n V,τ τ F ) u s. 5) V s V τ
4 R. Eymard, C. Guichard and R. Herbin A 5 = s A = s A 6 A 4 s + A = x K A A 0 = x K A 2 = x σ A = x σ s A 2 = s Fig. The octahedral left) and tetrahedral right) cells for the definition of the gradient We then set define K U, used in the tables below, by K K U = σ F K This scheme is denoted by VAGR in [5]. s V σ V K,σ,s K,σ,s U.. Piecewise constant gradient on tetrahedral sub cells For a given face σ of a control volume K and for any pair of consecutive vertices s,s ) of σ, we consider the tetrahedron, denoted by V K,σ,s,s and depicted in Fig., whose vertices are A 0 = x K, A = x σ, A 2 = s and A = s note that all these tetrahedra are disjoint, and that the union of their closure is Ω). The approximate values of U at the vertices of V K,σ,s,s are respectively u 0 = u K, u = u σ, u 2 = u s and u = u s. We then define the following approximate gradient: K,σ,s,s U = i= u i u 0 ) A 0 A i+ A 0 A i+2 Det A 0 A i+, A 0 A i+2, A 0 A i ), 6) where A 4 = A and A 5 = A 2. We finally define a piecewise constant gradient by D Ux) = K,σ,s,s U for a.e. x V K,σ,s,s. Remark that 5) also holds in this case, denoting V instead of V K,σ,s,s.
Benchmark D: the VAG scheme 5 2 Numerical results We provide the detailed numerical results obtained, using the scheme VAGR for computing the discrete gradient. In the numerical implementation, the values u K are locally eliminated, and the unknowns of the linear solver are the values u s. Denoting by denotes the Euclidean norm, the norms used in the bench tables have been computed using the following formulae: erl2 = normg = K K U, K u K ux K )/ 2 /2 K ux K ) 2, /2 ergrad = K K U ux K ) )/ 2 K ux K ) 2, ) /2 ener = K K U ux K ) Λ 2 / K ux K ) Λ 2, setting, for any K M, ξ 2 Λ = Λ Kξ ξ for all ξ R. Test Mild anisotropy, ux,y,z) = + sinπx)sin π y + 2 min = 0, max = 2, Tetrahedral meshes sin π z + 488 6072 5.77E-02 2.0E-02.95E+00.99E+00.77E+00 2 857 269.88E-02 6.84E-0.97E+00.99E+00.78E+00 60 2675 2.9E-02 9.E-0.98E+00.99E+00.79E+00 4 2997 489.E-02 5.52E-0.99E+00 2.00E+00.79E+00 5 5692 8688 8.7E-0.49E-0.99E+00 2.00E+00.79E+00 6 0994 60852.6E-0.8E-0.99E+00 2.00E+00.80E+00 488.76E-02-2.0E-0-2.28E-0-2 857.02E-02 2.9E+00.79E-0.5E+00.77E-0.5E+00 60 6.79E-0.94E+00.44E-0.05E+00.42E-0.08E+00 4 2997 4.44E-0 2.0E+00.E-0.4E+00.E-0.7E+00 5 5692 2.79E-0 2.8E+00 9.02E-02.06E+00 8.89E-02.0E+00 6 0994.75E-0 2.E+00 7.04E-02.E+00 6.92E-02.5E+00
6 R. Eymard, C. Guichard and R. Herbin Test Mild anisotropy, ux,y,z) = + sinπx)sin π y + 2 sin π z + min = 0, max = 2, Voronoi meshes 46 596 7.54E-02.56E-0 2.5E+00.86E+00.4E+00 2 9 6267 -.42E-0.79E-0.95E+00.8E+00.4E+00 684 794 8.40E-04 2.67E-02 2.02E+00.9E+00.60E+00 4 227 7069-9.54E-02.20E-02 2.06E+00.9E+00.66E+00 5 202 2788 -.78E-02.85E-0 2.06E+00.97E+00.70E+00 46.82E-0 -.96E-0-4.05E-0-2 9.87E-0-8.4E-02 2.49E-0.65E+00 2.5E-0.68E+00 684 9.92E-02 2.70E+00.55E-0 2.02E+00.62E-0.90E+00 4 227 7.5E-02.68E+00.9E-0.5E+00.2E-0.42E+00 5 202 4.74E-02 2.47E+00 9.56E-02.E+00 9.92E-02.29E+00 Test Mild anisotropy, ux,y,z) = + sinπx)sin π y + 2 min = 0, max = 2, Kershaw meshes sin π z + 729 5625 7.80E-02.0E-02.96E+00.96E+00.56E+00 2 49 7649.72E-02.06E-02.98E+00.99E+00.68E+00 597 9267-2.58E-04.75E-0.99E+00 2.00E+00.74E+00 4 274625 789057-2.64E-04 7.4E-04 2.00E+00 2.00E+00.78E+00 729 9.7E-02-4.9E-0-4.84E-0-2 49 5.5E-02 7.96E-0.09E-0 7.28E-0 2.84E-0 8.40E-0 597 2.97E-02 9.8E-0.74E-0 8.70E-0.54E-0 9.22E-0 4 274625.22E-02.E+00 7.40E-02.26E+00 6.44E-02.29E+00 Test Mild anisotropy, ux,y,z) = + sinπx)sin π y + 2 min = 0, max = 2, Checkerboard meshes sin π z + 97 24-9.8E-02.54E-0 2.08E+00.85E+00.4E+00 2 625 22585 -.90E-0 4.0E-02 2.9E+00.96E+00.70E+00 447 8864-6.2E-02.0E-02 2.06E+00.99E+00.78E+00 4 025 52967 -.70E-02 2.54E-0 2.02E+00 2.00E+00.79E+00 5 254977 22955-4.E-0 6.6E-04 2.00E+00 2.00E+00.80E+00
Benchmark D: the VAG scheme 7 97.25E-0-4.7E-0 -.97E-0-2 625.E-0.7E+00.50E-0.72E+00.52E-0.54E+00 447.0E-02 2.00E+00 5.7E-02.47E+00 6.09E-02.4E+00 4 025 7.92E-0.99E+00 2.5E-02.2E+00 2.77E-02.8E+00 5 254977 2.0E-0 2.00E+00.8E-02.E+00.2E-02.08E+00 Test 2 Heterogeneous anisotropy, ux,y,z) = x y 2 z + xsin2πxz)sin2πxy)sin2πz), min = 0.862, max =.048, Prism meshes 080 9964-8.7E-0-8.4E-0.0E+00 9.84E-0.5E+00 2 2060 70894-8.25E-0-8.9E-0.04E+00.0E+00.66E+00 6240 20754-8.52E-0-8.59E-0.05E+00.0E+00.69E+00 4 4420 54024-8.5E-0-8.57E-0.04E+00.0E+00.70E+00 080.66E-0 -.40E-0 -.8E-0-2 2060 4.26E-02 2.7E+00.7E-02 2.E+00.64E-02 2.E+00 6240.9E-02 2.08E+00.67E-02 2.0E+00.6E-02 2.0E+00 4 4420.0E-02 2.05E+00 9.44E-0 2.06E+00 9.25E-0 2.07E+00 Test Flow on random meshes, ux, y, z) = sin2πx) sin2πy) sin2πz), min =, max =, Random meshes 25 297 -.5E+00-7.55E-0.68E+00 6.98E-0.5E+00 2 729 5625 -.E+00-9.9E-0.2E+00 9.24E-0 2.99E+00 49 7649 -.08E+00-9.85E-0.06E+00 9.82E-0.44E+00 4 597 9267 -.0E+00-9.96E-0.0E+00 9.96E-0.56E+00 25.5E+00-6.9E-0-6.26E-0-2 729 2.56E-0 2.56E+00 2.02E-0.90E+00.8E-0 2.E+00 49 5.9E-02 2.0E+00 8.04E-02.45E+00 5.0E-02.9E+00 4 597.49E-02 2.09E+00.45E-02.28E+00.74E-02.68E+00
8 R. Eymard, C. Guichard and R. Herbin Test 4 Flow around a well, Well meshes 248 27072.89E-0 4.29E-0 5.2E+00 5.2E+00.68E+0 2 2800 6584 2.4E-0 2.50E-0 5.E+00 5.E+00.65E+0 5889 4079.55E-0.57E-0 5.E+00 5.E+00.64E+0 4 2582 4964.8E-0.20E-0 5.E+00 5.E+00.6E+0 5 2500 64520 9.0E-02 9.09E-02 5.4E+00 5.4E+00.6E+0 6 45668 78094 7.27E-02 7.0E-02 5.4E+00 5.5E+00.6E+0 7 79084 2055600 5.69E-02 5.68E-02 5.6E+00 5.6E+00.6E+0 248 6.47E-0-5.78E-02-5.5E-02-2 2800 2.7E-0.2E+00 2.54E-02.05E+00 2.4E-02.08E+00 5889.9E-0.E+00.2E-02 2.9E+00.5E-02 2.85E+00 4 2582 8.42E-04.7E+00 7.59E-0.9E+00 7.E-0.79E+00 5 2500 4.47E-04 2.72E+00 5.0E-0.7E+00 4.95E-0.68E+00 6 45668 2.02E-04 4.0E+00.55E-0.8E+00.47E-0.80E+00 7 79084.75E-04 7.84E-0.26E-0 4.76E-0.9E-0 4.56E-0 Test 5 Discontinuous permeability, ux,y,z) = α i sin2πx)sin2πy)sin2πz), min = 0, max =, Locally refined meshes 60 48-7.9E+02 -.00E+02 7.9E+02.00E+02.24E+0 2 05 6825-7.82E+0 -.54E+0 7.82E+0.54E+0 5.20E+0 88 46025-9.90E+0-7.89E+0 9.90E+0 7.89E+0 8.60E+0 4 07 560-9.99E+0-9.4E+0 9.99E+0 9.4E+0 9.56E+0 5 9785 255779 -.00E+02-9.86E+0.00E+02 9.86E+0 9.80E+0 60 6.9E+00 -.60E+00-8.27E+00-2 05.9E+00.0E+00 5.97E-0.82E+00 6.0E-0 4.84E+00 88 2.55E-0 2.55E+00.86E-0.92E+00.80E-0.99E+00 4 07 6.0E-02 2.2E+00 5.96E-02.76E+00 4.78E-02 2.05E+00 5 9785.52E-02 2.08E+00 2.24E-02.46E+00.26E-02 2.00E+00 Comments on the results The results obtained using ) VAG) instead of 4) VAGR) are systematically less precise, except in the test5 case, where we obtained the following tables:
Benchmark D: the VAG scheme 9 60 48-7.65E+02 -.00E+02 7.65E+02.00E+02 6.76E+0 2 05 6825-7.7E+0 -.54E+0 7.7E+0.54E+0 4.65E+0 88 46025-9.02E+0-7.89E+0 9.02E+0 7.89E+0 8.9E+0 4 07 560-9.72E+0-9.4E+0 9.72E+0 9.4E+0 9.4E+0 5 9785 255779-9.9E+0-9.86E+0 9.9E+0 9.86E+0 9.77E+0 60 6.7E+00-7.2E+00-2.90E+0-2 05 9.5E-0.60E+00 6.9E-0 4.6E+00 6.76E-0 6.9E+00 88.49E-0.05E+00 2.24E-0.85E+00 2.20E-0.85E+00 4 07.27E-02 2.5E+00 6.7E-02 2.00E+00 5.95E-02 2.0E+00 5 9785 7.98E-0 2.E+00.7E-02.90E+00.54E-02 2.02E+00 The results using 6) are very similar to those obtained using 4) VAGR). For both ) VAG) and 4) VAGR), we have chosen the conjugate gradient solver of the ISTL library with ILU0) preconditioning with tolerance or reduction factor) set to 0 0. The following observations have been made on the computing times, using ) VAG) we may expect that similar observations could be done with VAGR).. On the fourth Kershaw mesh and test, we obtain the following CPU times using the conjugate gradient solver of the PETSC library: with ILU2), s, with ILU), 7s, with ILU0), 0s, and with Jacobi, s, which shows that the ILU0) preconditioning seems the fastest one on this case. Note that this computing time is depending on the unknown orderings. For the bench computations, we used the recursive domain decomposition ordering, which is the most efficient for direct solvers, and the respective computing times with PETSC CG+ILU0) and with ISTL CG+ILU0) are 0. and.2 s. Using the reverse Cuthill - McKee ordering, we respectively obtain 4.4 s and 5. s with PETSC CG+ILU0) and ISTL CG+ILU0). 2. The computing times, for the conjugate gradient solver of the PETSC library with ILU) preconditioning, in the test case on tetrahedral meshes 2 to 5, have been approximately equal to 0.0, 0.0, 0.04, 0.08, and 0.6 s, showing the possibility to apply this method on much larger meshes. Finally, we may not exclude that the systematic choice of computing the L 2 error with respect to the values in the control volumes instead of the vertex values, makes all these results somewhat pessimistic. References. R. Eymard, T. Gallouët, and R. Herbin. Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, sushi: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal., 04):009 04, 200. see also http://hal.archivesouvertes.fr/.
0 R. Eymard, C. Guichard and R. Herbin 2. R. Eymard, C. Guichard, and R. Herbin. Small-stencil D schemes for diffusive flows in porous media. submitted, 200. see also http://hal.archives-ouvertes.fr/.. R. Eymard, T. Gallouët and R. Herbin. Benchmark D: the SUSHI scheme these proceedings, 20. 4. R. Eymard and R. Herbin. Gradient schemes for diffusion problem. these proceedings, 20. 5. R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klöfkorn and G. Manzini. D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. these proceedings, 20. The paper is in final form and no similar paper has been or is being submitted elsewhere.