A finite volume method for approximating 3D diffusion operators on general meshes

Journal of Computational Physics

Volumn 228, Issue 16, 2009, Pages 5763-5786

  Hermeline, F ^a  

^a CEA DAM DIF   (France)

Author keywords

Anisotropic diffusion; Discrete duality finite volume method; Distorted meshes; Elliptic equations

Indexed keywords

ANISOTROPY; DIFFUSION; GEOMETRY; TENSORS;

3D DIFFUSION; ANISOTROPIC DIFFUSION; DIFFUSION OPERATORS; DISCRETE DUALITY FINITE VOLUME METHODS; DISTORTED MESH; ELLIPTIC EQUATIONS; FINITE-VOLUME METHOD; FULL TENSOR COEFFICIENTS; SYMMETRICS; TENSOR COEFFICIENTS;

FINITE VOLUME METHOD;

EID: 67649340811     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.05.002     Document Type: Article

References (59)

1. Hermeline F. Une méthode de volumes finis pour leséquations elliptiques du second ordre (in French). C.R. Acad. Sci. Paris Ser. 1 326 (1998) 1433-1436
2. Hermeline F. A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481-499
3. Hermeline F. Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Meth. Appl. Mech. Eng. 192 (2003) 1939-1959
4. Hermeline F. A finite volume method for solving Maxwell equations in inhomogeneous media on arbitrary meshes. C.R. Acad. Sci. Paris Ser. 1 339 (2004) 893-898
5. Domelevo K., and Omnes P. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. Math. Model. Numer. Anal. 39 (2005) 1203-1249
6. Delcourte S., Domelevo K., and Omnes P. Discrete duality finite volume method for second order elliptic problems. In: Benkhaldoun F., Ouazar D., and Raghay S. (Eds). Finite Volumes for Complex Applications (IV) (2005), Hermes Science Publishing 447-458
7. Andreianov B., Boyer F., and Hubert F. Duplex finite volume schemes for nonlinear elliptic problems on general 2D meshes. In: Benkhaldoun F., Ouazar D., and Raghay S. (Eds). Finite Volumes for Complex Applications (IV) (2005), Hermes Science Publishing 365-375
8. C. Pierre, Modélisation et simulation de l'activité électrique du cœur dans le thorax, analyse numérique et méthodes de volumes finis (in French), Ph.D. Thesis, University of Nantes, France, 2005.
9. Njifenjou A., and Nguena I.M. A finite volume method for second order elliptic problems. Int. J. Finite Volume 3 2 (2006)
10. Hermeline F. Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes. Comput. Meth. Appl. Mech. Eng. 196 (2007) 2497-2526
11. Andreianov B., Boyer F., and Hubert F. Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Meth. Part. Differ. Equations 23 1 (2007) 145-195
12. Yuan G., and Sheng Z. Analysis of accuracy of a finite volume scheme for diffusion equations on distorted meshes. J. Comput. Phys. 220 (2007) 751-771
13. Delcourte S., Domelevo K., and Omnes P. A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes. SIAM J. Numer. Anal. 45 3 (2007) 1142-1174
14. S. Delcourte, Développement de méthodes de volumes finis pour la mécanique des fluides (in French), Ph.D. Thesis, University of Toulouse, France, 2007.
15. Hermeline F. Approximating second order vector differential operators on distorted meshes in two space dimensions. Int. J. Numer. Meth. Eng. 76 (2008) 1065-1089
16. Hermeline F., Layouni S., and Omnes P. A finite volume method for the approximation of Maxwell's equations in two space dimensions on arbitrary meshes. J. Comput. Phys. 227 (2008) 9365-9388
17. Boyer F., and Hubert F. Finite volume method for non linear transmission problems. Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering vol. 60 (2008), Springer, Berlin, Heidelberg 443-450
18. Njifenjou A., and Kinfack A.J. Convergence analysis of an MPFA method for flow problems in anisotropic heterogeneous porous media. Int. J. Finite Volume 5 1 (2008)
19. Coudière Y., Pierre C., Rousseau O., and Turpault R. 2D/3D discrete duality finite volume scheme (DDFV) applied to ECG simulation. In: Eymard R., and Herard J.M. (Eds). Finite Volumes for Complex Applications (V) (2008), Wiley 313-320
20. Boyer F., and Hubert F. The DDFV "discrete duality finite volumes" and m-DDFV schemes. In: Eymard R., and Herard J.M. (Eds). Finite Volumes for Complex Applications (V) (2008), Wiley 735-750
21. D. Calhoun, A finite volume discretization of the surface Laplacian for general curved, logically Cartesian meshes, 2008, personal communication.
22. Chainais-Hillairet C. Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. Int. J. Numer. Meth. Fluids 59 (2009) 239-257
23. Y. Coudière, F. Hubert, A 3D discrete duality finite volume method for non linear elliptic problem, in: A. Handlovicova, P. Frolkovic, K. Mikula, D. Sevcovic (Eds.), Proceedings of Algoritmy 2009, 2009, pp. 51-60.
24. B. Andreianov, M. Bendahmane, On finite volume discretizations of gradient and divergence operators in the 3D case, HAL:hal00355212, 2009.
25. Herbin R., and Hubert F. Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Eymard R., and Herard J.M. (Eds). Finite Volumes for Complex Applications (V) (2008), Wiley 659-692
26. Klausen R.A., and Russel T.F. Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci. 8 (2004) 341-377
27. Friis H.A., Edwards M.G., and Mykkeltveit J. Symmetric positive definite flux-continuous full-tensor finite-volume schemes on unstructured cell-centered triangular grids. SIAM J. Sci. Comput. 31 (2008) 1192-1220
28. Younes A., and Fontaine V. Efficiency of mixed hybrid finite element and MPFA methods on quadrangular grids and highly anisotropic media. Int. J. Numer. Meth. Eng. (2008) 10.1002/nme.2327
29. Younes A., and Fontaine V. Hybrid and multi-point formulations of the lowest-order mixed methods for Darcy's flow on triangles. Int. J. Numer. Meth. Fluids (2008) 10.1002/fld.785
30. Kuznetsov Y., and Repin S. New mixed finite element method on polygonal and polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 18 3 (2003) 261-278
31. Bermudez A., Gamallo P., Nogueiras M.R., and Rodriguez R. Approximation properties of lowest-order hexahedral Raviart-Thomas finite elements. C.R. Acad. Sci. Paris Ser. 1 340 (2005) 687-692
32. Lipnikov K., Shashkov M., and Svyatskiy D. The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys. 211 (2006) 473-491
33. Brezzi F., Lipnikov K., Shashkov M., and Simoncini V. A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Meth. Appl. Mech. Eng. 196 (2007) 3682-3692
34. Aarnes J.E., Krogstad S., and Lie K.A. Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12 (2008) 297-315
35. Chen Q.Y., Wan J., Yang Y., and Mifflin R.T. Enriched multi-point flux approximation for general grids. J. Comput. Phys. 227 (2008) 1701-1721
36. Fatenejad M., and Moses G.A. Extension of Kershaw diffusion scheme to hexahedral meshes. J. Comput. Phys. 227 (2008) 2184-2187
37. Bailey T.S., Adams M.L., Yang B., and Zika M.R. A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids. J. Comput. Phys. 227 (2008) 3738-3757
38. Voronoi G. Nouvelles applications des paramètres continusà la théorie des formes quadratiques: recherche sur les parallélloèdres primitifs (in french). J. Reine Angew. Math. 134 (1908) 97-178
39. Delaunay B. Sur la sphère vide (in french). Bul. Acad. Sci. URSS Class. Sci. Nat. (1934) 793-800
40. F. Hermeline, Une méthode automatique de maillage en dimension N (in french), Ph.D. Thesis, University P. and M. Curie Paris 6, Paris, 1980.
41. Hermeline F. Triangulation automatique d'un polyèdre en dimension N (in french). RAIRO Anal. Numér. 16 (1982) 211-242
42. George P.L., and Hermeline F. Delaunay's mesh of a convex polyhedron in dimension d. Application to arbitrary polyhedra. Int. J. Numer. Meth. Eng. 33 (1992) 975-995
43. George P.L. Automatic mesh generation and finite element computation. In: Ciarlet P.G., and Lions J.L. (Eds). Handbook of Numerical Analysis vol. 7 (1996), North Holland, Elsevier 713-1018
44. Frey P.J., and George P.L. Mesh Generation (2000), Hermes Science Publishing
45. Eymard R., Gallouet T., and Herbin R. Finite volume methods. In: Ciarlet P.G., and Lions J.L. (Eds). Handbook of Numerical Analysis vol. 7 (2000), North Holland
46. W.J. Coirier, An adaptively-refined Cartesian, cell-based scheme for the Euler and Navier-Stokes equations, Ph.D. Thesis, Michigan University, 1994.
47. Coudière Y., Vila J.P., and Villedieu P. Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. M2AN 33 3 (1999) 493-516
48. Coudière Y., and Villedieu P. Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. M2AN 34 6 (2000) 1123-1149
49. Bertolazzi E., and Manzini G. On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems. Math. Models Meth. Appl. Sci. 17 1 (2007) 1-32
50. Manzini G., and Russo A. A finite volume method for advection-diffusion problems in convection dominated regimes. Comput. Meth. Appl. Mech. Eng. 197 (2008) 1242-1261
51. Sheng Z., and Yuan G. A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes. SIAM J. Sci. Comput. 30 3 (2008) 1341-1361
52. Le Potier C. Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés (in french). C.R. Acad. Sci. Paris Ser. 1 341 (2005) 787-792
53. Kapyrin I.V. A family of monotone methods for the numerical solution of three-dimensional diffusion problems on unstructured tetrahedral meshes. Doklady Math. 76 2 (2007) 734-738
54. Lipnikov K., Shashkov M., Svyatskiy D., and Vassilevski Y. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227 (2007) 492-512
55. Yuan G., and Sheng Z. Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227 (2008) 6288-6312
56. Meurant G. Computer Solutions of Large Linear Systems (1999), Elsevier
57. Guillaume P., and Latocha V. Numerical convergence of a parametrisation method for the solution of highly anisotropic two-dimensional elliptic problem. J. Scient. Comp. 25 (2005) 423-444
58. Crumpton P.I., Shaw G.J., and Ware A.F. Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. J. Comput. Phys. 116 (1995) 343-358
59. Hermeline F. Numerical experiments with the DDFV method. In: Eymard R., and Herard J.M. (Eds). Finite Volumes for Complex Applications (V) (2008), Wiley 851-863