An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Journal of Computational Physics

Volumn 229, Issue 21, 2010, Pages 8072-8094

  Li, Xianping ^a   Huang, Weizhang ^a  

^a UNIVERSITY OF KANSAS   (United States)

Author keywords

Anisotropic coefficient; Anisotropic diffusion; Anisotropic mesh adaptation; Anisotropic mesh generation; Discrete maximum principle; Finite element; Mesh adaptation

Indexed keywords

DIFFUSION; DIHEDRAL ANGLE; FINITE ELEMENT METHOD; IMAGE SEGMENTATION; MAXIMUM PRINCIPLE; MESH GENERATION; PETROLEUM ENGINEERING; PROBLEM SOLVING; SOLVENTS; TENSORS;

ANISOTROPIC COEFFICIENTS; ANISOTROPIC DIFFUSION; ANISOTROPIC MESH ADAPTATION; ANISOTROPIC MESH GENERATION; CONDITION; DIFFUSION PROBLEMS; DISCRETE MAXIMUM PRINCIPLES; FINITE ELEMENT; FINITE ELEMENT SOLUTION; MESH ADAPTATION;

NUMERICAL METHODS;

EID: 77955924295     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.07.009     Document Type: Article

References (72)

1. Aavatsmark I., Barkve T., Bøe Ø., Mannseth T. Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 1998, 19:1700-1716.
2. Aavatsmark I., Barkve T., Bøe Ø., Mannseth T. Discretization on unstructured grids for inhomogeneous, anisotropic media. II. Discussion and numerical results. SIAM J. Sci. Comput. 1998, 19:1717-1736.
3. Ait-Ali-Yahia D., Baruzzi G., Habashi W.G., Fortin M., Dompierre J., Vallet M.-G. Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part II: structured grids. Int. J. Numer. Meth. Fluids 2002, 39:657-673.
4. Borouchaki H., George P.L., Hecht P., Laug P., Saletl E. Delaunay mesh generation governed by metric specification. Part I: algorithms. Fin. Elem. Anal. Des. 1997, 25:61-83.
5. Borouchaki H., George P.L., Mohammadi B. Delaunay mesh generation governed by metric specification. Part II: applications. Fin. Elem. Anal. Des. 1997, 25:85-109.
6. F.J. Bossen, P.S. Heckbert, A pliant method for anisotropic mesh generation, in: Proceedings of the Fifth International Meshing Roundtable, Sandia National Laboratories, Albuquerque, NM, 1996, Sandia Report 96-2301, pp. 63-74.
7. Brackbill J.U., Saltzman J.S. Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 1982, 46:342-368.
8. Brandts J., Korotov S., Křížek M. Dissection of the path-simplex in Rn into n path-subsimplices. Linear Algebra Appl. 2007, 421:382-393.
9. Brandts J., Korotov S., Křížek M. The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem. Linear Algebra Appl. 2008, 429:2344-2357.
10. Burman E., Ern A. Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. C.R. Acad. Sci. Paris, Ser. I 2004, 338:641-646.
11. Castro-Díaz M.J., Hecht F., Mohammadi B., Pironneau O. Anisotropic unstructured mesh adaption for flow simulations. Int. J. Numer. Meth. Fluids 1997, 25:475-491.
12. Chan T.F., Shen J. Non-texture inpainting by curvature driven diffusions (CDD). J. Vis. Commun. Image Rep. 2000, 12:436-449.
13. Chan T.F., Shen J., Vese L. Variational PDE models in image processing. Not. AMS J. 2003, 50:14-26.
14. Ciarlet P.G. Discrete maximum principle for finite difference operators. Aequationes Math. 1970, 4:338-352.
15. Ciarlet P.G., Raviart P.-A. Maximum principle and uniform convergence for the finite element method. Comput. Meth. Appl. Mech. Eng. 1973, 2:17-31.
16. Crumpton P.I., Shaw G.J., Ware A.F. Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. J. Comput. Phys. 1995, 116:343-358.
17. Dompierre J., Vallet M.-G., Bourgault Y., Fortin M., Habashi W.G. Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part III: unstructured meshes. Int. J. Numer. Meth. Fluids 2002, 39:675-702.
18. Drǎgǎnescu A., Dupont T.F., Scott L.R. Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comput. 2004, 74:1-23.
19. Dvinsky A.S. Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 1991, 95:450-476.
20. Eigestad G.T., Aavatsmark I., Espedal M. Symmetry and M-matrix issues for the O-method on an unstructured grid. Comput. Geosci. 2002, 6:381-404.
21. Ern A., Guermond J.L. Theory and Practice of Finite Elements 2004, Springer, New York.
22. T. Ertekin, J.H. Abou-Kassem, G.R. King, Basic applied reservoir simulation, in: SPE Textbook Series, vol. 7, Richardson, Texas, 2001.
23. Frey P., George P.L. Mesh Generation: Application to Finite Elements 2000, Hermes Science, Oxford and Paris.
24. R.V. Garimella, M.S. Shephard, Boundary layer meshing for viscous flows in complex domain, in: Proceedings of the Seventh International Meshing Roundtable, Sandia National Laboratories, Albuquerque, NM, 1998, pp. 107-118.
25. Gu{double acute}nter S., Lackner K. A mixed implicit-explicit finite difference scheme for heat transport in magnetised plasmas. J. Comput. Phys. 2009, 228:282-293.
26. Gu{double acute}nter S., Lackner K., Tichmann C. Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas. J. Comput. Phys. 2007, 226:2306-2316.
27. Gu{double acute}nter S., Yu Q., Kruger J., Lackner K. Modelling of heat transport in magnetised plasmas using non-aligned coordinates. J. Comput. Phys. 2005, 209:354-370.
28. Habashi W.G., Dompierre J., Bourgault Y., Ait-Ali-Yahia D., Fortin M., Vallet M.-G. Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part I: general principles. Int. J. Numer. Meth. Fluids 2000, 32:725-744.
29. F. Hecht, Bidimensional anisotropic mesh generator. Technical report, INRIA, Rocquencourt, 1997. Source code available from: http://www.ann.jussieu.fr/hecht/ftp/bamg/.
30. Huang W. Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 2001, 174:903-924.
31. Huang W. Metric tensors for anisotropic mesh generation. J. Comput. Phys. 2005, 204:633-665.
32. Huang W. Mathematical principles of anisotropic mesh adaptation. Comm. Comput. Phys. 2006, 1:276-310.
33. Huang W., Li X.P. An anisotropic mesh adaptation method for the finite element solution of variational problems. Fin. Elem. Anal. Des. 2010, 46:61-73.
34. Huang W., Kamenski L., Lang J. A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates. J. Comput. Phys. 2010, 229:2179-2198.
35. Jacquotte Olivier-P. A mechanical model for a new grid generation method in computational fluid dynamics. Comput. Meth. Appl. Mech. Eng. 1988, 66:323-338.
36. Karátson J., Korotov S. Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 2005, 99:669-698.
37. Karátson J., Korotov S. Discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures. J. Comput. Appl. Math. 2006, 192:75-88.
38. Karátson J., Korotov S., Křížek M. On discrete maximum principles for nonlinear elliptic problems. Math. Comput. Simulat. 2007, 76:99-108.
39. Karras D.A., Mertzios G.B. New PDE-based methods for image enhancement using SOM and Bayesian inference in various discretization schemes. Meas. Sci. Technol. 2009, 20:104012.
40. Knupp P., Margolin L., Shashkov M. Reference jacobian optimization-based rezone strategies for arbitrary lagrangian eulerian methods. J. Comput. Phys. 2002, 176:93-128.
41. Knupp P.M. Jacobian-weighted elliptic grid generation. SIAM J. Sci. Comput. 1996, 17:1475-1490.
42. Kornhuber R., Roitzsch R. On adaptive grid refinement in the presence of internal or boundary layers. IMPACT Comput. Sci. Eng. 1990, 2:40-72.
43. Kuzmin D., Shashkov M.J., Svyatskiy D. A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. J. Comput. Phys. 2009, 228:3448-3463.
44. Křížek M., Lin Q. On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 1995, 3:59-69.
45. J. Lang, An adaptive finite element method for convection-diffusion problems by interpolation techniques. Technical Report TR 91-4, Konrad-Zuse-Zentrum Berlin, 1991.
46. Le Potier C. Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non-structurés. C.R. Math. Acad. Sci. Paris 2005, 341:787-792.
47. Le Potier C. A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. Int. J. Finite 2009, 6:20.
48. Le Potier C. Un schéma linéaire vérifiant le principe du maximum pour des opérateurs de diffusion très anisotropes sur des maillages déformés. C.R. Math. Acad. Sci. Paris 2009, 347:105-110.
49. Letniowski F.W. Three-dimensional delaunay triangulations for finite element approximations to a second-order diffusion operator. SIAM J. Sci. Stat. Comput. 1992, 13:765-770.
50. X.P. Li, D. Svyatskiy, M. Shashkov, Mesh adaptation and discrete maximum principle for 2D anisotropic diffusion problems, LANL Technical report, LA-UR 10-01227, 2007.
51. Lipnikov K., Shashkov M., Svyatskiy D., Vassilevski Yu. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 2007, 227:492-512.
52. Liska R., Shashkov M. Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems. Commun. Comput. Phys. 2008, 3:852-877.
53. Mlacnik M.J., Durlofsky L.J. Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. J. Comput. Phys. 2006, 216:337-361.
54. Mumford D., Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math 1989, 42:577-685.
55. Nishikawa K., Wakatani M. Plasma Physics 2000, Springer, Berlin.
56. Peraire J., Vahdati M., Morgan K., Zienkiewicz O.C. Adaptive remeshing for compressible flow computations. J. Comput. Phys. 1987, 72:449-466.
57. Perona P., Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intel. 1990, 12:629-639.
58. Rachowicz W. An anisotropic h-type mesh refinement strategy. Comput. Meth. Appl. Mech. Eng. 1993, 109:169-181.
59. Rachowicz W. An anisotropic h-adaptive finite element method for compressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 1997, 146:231-252.
60. Remacle J., Li X., Shephard M.S., Flaherty J.E. Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods. Int. J. Numer. Meth. Eng. 2003, 62:899-923.
61. Sharma P., Hammett G.W. Preserving monotonicity in anisotropic diffusion. J. Comput. Phys. 2007, 227:123-142.
62. Sommerville D.M.Y. An Introduction to the Geometry of n Dimensions 1929, Methuen & Co. Ltd., London.
63. Stix T.H. Waves in Plasmas 1992, American Institute of Physics, New York.
64. Stoyan G. On a maximum principle for matrices, and on conservation of monotonicity. With applications to discretization methods. Z. Angew. Math. Mech. 1982, 62:375-381.
65. Stoyan G. On maximum principles for monotone matrices. Linear Algebra Appl. 1986, 78:147-161.
66. Strang G., Fix G.J. An Analysis of the Finite Element Method 1973, Prentice-Hall, Englewood Cliffs, NJ.
67. Varga R.S. Matrix Iterative Analysis 1962, Prentice-Hall, Englewood Cliffs, NJ.
68. Varga R.S. On a discrete maximum principle. SIAM J. Numer. Anal. 1966, 3:355-359.
69. Weickert J. Anisotropic Diffusion in Image Processing 1998, Teubner-Verlag, Stuttgart, Germany.
70. A.M. Winslow, Adaptive mesh zoning by the equipotential method. Technical Report UCID-19062, Lawrence Livemore Laboratory, 1981 (unpublished).
71. Xu J., Zikatanov L. A monotone finite element scheme for convection-diffusion equations. Math. Comput. 1999, 69:1429-1446.
72. S. Yamakawa, K. Shimada, High quality anisotropic tetrahedral mesh generation via ellipsoidal bubble packing, in: Proceedings of the Ninth International Meshing Roundtable, Sandia National Laboratories, Albuquerque, NM, 2000, Sandia Report 2000-2207, pp. 263-273.