Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations

Journal of Computational Physics

Volumn 220, Issue 2, 2007, Pages 751-771

  Manzini, Gianmarco ^a   Putti, Mario ^b  

^a CNR   (Italy)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

Anisotropy; Finite volume; Mesh locking; Numerical solver; Partial differential equation; Poisson problem; Unstructured grid

Indexed keywords

BENCHMARKING; BOUNDARY CONDITIONS; DIFFUSION; LOCKS (FASTENERS); MESH GENERATION; OPTICAL ANISOTROPY; POISSON EQUATION;

ANISOTROPIC DIFFUSION EQUATIONS; ANISOTROPY RATIO; CONVERGENCE RATES; FINITE-VOLUME; LOCKING EFFECTS; MESH LOCKING; NUMERICAL SOLVERS; OPTIMAL CONVERGENCE; POISSON PROBLEM; UNSTRUCTURED GRID;

FINITE VOLUME METHOD;

EID: 33845637272     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.05.026     Document Type: Article

References (34)

1. Aavatsmark I., Barkve T., Boe O., and Mannseth T. Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput. 19 5 (1998) 1700-1716 ISSN 1064-8275
2. Aavatsmark I., Barkve T., Boe O., and Mannseth T. Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results. SIAM J. Sci. Comput. 19 5 (1998) 1717-1736 ISSN 1064-8275 (print), 1095-7197 (electronic)
3. Adams R.A. Sobolev Spaces (1975), Academic Press, New York
4. Amestoy P., and Puglisi C. An unsymmetrized LU multifrontal factorization. SIAM J. Matrix Anal. Appl. 24 2 (2002) 553-569
5. Babuška I., and Suri M. On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29 5 (1992) 1261-1293 ISSN 0036-1429
6. T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA Paper 89-0366. AIAA, January 1989.
7. Bear J. Dynamics of Fluids in Porous Media (1972), Dover, New York
8. Bertolazzi E., and Manzini G. Algorithm 817 P2MESH: generic object-oriented interface between 2-D unstructured meshes and FEM/FVM-based PDE solvers. ACM Trans. Math. Software 28 1 (2002) 101-132
9. Bertolazzi E., and Manzini G. A cell-centered second-order accurate finite volume method for convection-diffusion problems on unstructured meshes. Math. Mod. Meth. Appl. Sci. 8 (2004) 1235-1260
10. Bertolazzi E., and Manzini G. A second-order maximum principle preserving finite volume method for steady convection-diffusion problems. SIAM, J. Numer. Anal. 43 5 (2006) 2172-2199
11. Bertolazzi E., and Manzini G. A unified treatment of boundary conditions in least square-based finite volume methods. Comput. Math. Applications 49 11-12 (2005) 1755-1765
12. Bochev P., and Lehoucq R.B. On the finite element solution of the pure Neumann problem. SIAM Rev. 47 1 (2005) 50-66
13. Brezzi F. Stability of saddle-points in finite dimensions. In: Blowey J.F., Craig A.W., and Shardlow T. (Eds). Frontiers in Numerical Analysis, Universitext (2003), Springer, Berlin 17-62
14. Ciarlet P.G. The Finite Element Method for Elliptic Problems (1980), North-Holland Publishing Company, Amsterdam, Holland
15. Coudière Y., Vila J.-P., and Villedieu P. Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. Math. Model. Numer. Anal. 33 3 (1999) 493-516
16. Coudière Y., and Villedieu P. Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. M2AN Math. Model. Numer. Anal. 34 6 (1999) 1123-1149 ISSN 0764-583X
17. Domelevo K., and Omnes P. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN 39 6 (2005) 1203-1249
18. Eymard E., Gallouët T., and Herbin R. A finite volume scheme for anisotropic diffusion problem. Comptes Rendus Acad. Sci., Mathématiques 339 4 (2004) 299-302
19. Eymard R., Gallouët T., and Herbin R. In: Ciarlet Ph., and Lions J.L. (Eds). Handbook for Numerical Analysis (2000), North-Holland, Amsterdam 715-1022 (Chapter: The finite volume method)
20. Havu V. An analysis of asymptotic consistency error in a parameter dependent model problem. Calcolo 40 2 (2003) 121-130 ISSN 0008-0624
21. Havu V., and Pitkäranta J. An analysis of finite element locking in a parameter dependent model problem. Numer. Math. 89 4 (2001) 691-714 ISSN 0029-599X
22. Hermeline F. A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481-499
23. Hyman J., and Shashkov M. Solving diffusion equations with rough coefficients in rough grids. J. Comput. Phys. 129 (1997) 383-405
24. Hyman J., Shashkov M., and Steinberg S. The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130-148
25. Jayantha P.A., and Turner I.W. On the use of surface interpolation techniques in generalised finite volume strategies for simulating transport in highly anisotropic porous media. J. Comput. Appl. Math. 152 1-2 (2003) 199-216 ISSN 0377-0427
26. Jayantha P.A., and Turner I.W. A second order finite volume technique for simulating transport in anisotropic media. Int. J. Numer. Methods Heat Fluid Flow 13 1 (2003) 31-56
27. Jayantha P.A., and Turner I.W. A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media. J. Comput. Math. 23 1 (2005) 1-16 ISSN 0254-9409
28. Křižek M., and Liu L. Finite element approximation of a nonlinear heat conduction problem in anisotropic media. Comput. Methods Appl. Mech. Eng. 157 (1998) 387-397
29. Manzini G., and Ferraris S. Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation. Adv. Water Resour. 27 12 (2004) 1199-1215
30. Ollivier-Gooch C., and Van Altena M. A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation. J. Comput. Phys. 181 2 (2002) 729-752 ISSN 0021-9991
31. Ollivier-Gooch C.F. Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction. J. Comput. Phys. 133 (1997) 6-17
32. A. Paster, Modeling the salt-water body in the northern Yarkon-Taninim aquifer, Israel. M.Sc. Thesis, Tel Aviv University, 2004.
33. Putti M., and Cordes C. Finite element approximation of the diffusion operator on tetrahedra. SIAM, J. Sci. Comput. 19 4 (1998) 1154-1168
34. Quarteroni A., and Valli A. Numerical Approximation of Partial Differential Equations (1994), Springer, Berlin. 3-540-57111-6