The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes.(English)Zbl 1120.65332
Summary: We study the mimetic finite difference discretization of diffusion-type problems on unstructured polyhedral meshes. We demonstrate high accuracy of the approximate solutions for general diffusion tensors, the second-order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes, on non-matching meshes, and even on meshes with irregular-shaped polyhedra with flat faces. We show that in general the meshes with non-flat faces require more than one flux unknown per mesh face to get optimal convergence rates.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
diffusion equation;numerical experiments;finite difference;unstructured polyhedral meshes;convergenceSoftware:
AMG1R5Cite
Full Text:DOI
References:
| [1] | Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci., 6, 405-432 (2002) ·Zbl 1094.76550 |
| [2] | Arbogast, T.; Cowsar, L.; Wheeler, M.; Yotov, I., Mixed finite element methods on non-matching multiblock grids, SIAM J. Numer. Anal., 37, 1295-1315 (2000) ·Zbl 1001.65126 |
| [3] | T. Austin, J. Morel, J. Moulton, M. Shashkov. Mimetic preconditioners for mixed discretizations of the diffusion equation. Technical Report LA-UR-01-807, Los Alamos National Laboratory, 2004. Available from: www.ima.umn.edu/talks/workshops/5-11-15.2004/moulton/moulton.pdf; T. Austin, J. Morel, J. Moulton, M. Shashkov. Mimetic preconditioners for mixed discretizations of the diffusion equation. Technical Report LA-UR-01-807, Los Alamos National Laboratory, 2004. Available from: www.ima.umn.edu/talks/workshops/5-11-15.2004/moulton/moulton.pdf |
| [4] | Brezzi, F.; Lipnikov, K.; Shashkov, M., Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. (2005), (to appear) ·Zbl 1108.65102 |
| [5] | Campbell, J.; Shashkov, M., A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172, 739-765 (2001) ·Zbl 1002.76082 |
| [6] | Grisvard, P., Elliptic Problems in Nonsmooth domains (1985), Pitman: Pitman London ·Zbl 0695.35060 |
| [7] | Hyman, J.; Morel, J.; Shashkov, M.; Steinberg, S., Mimetic finite difference methods for diffusion equations, Comput. Geosci., 6, 3-4, 333-352 (2002) ·Zbl 1023.76033 |
| [8] | Hyman, J.; Shashkov, M., Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion, Progr. Electromagn. Res., 32, 89-121 (2001) |
| [9] | Hyman, J.; Shashkov, M.; Steinberg, S., The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. Comput. Phys., 132, 130-148 (1997) ·Zbl 0881.65093 |
| [10] | Kuznetsov, Y.; Lipnikov, K.; Shashkov, M., Mimetic finite difference method on polygonal meshes for diffusion-type problems, Comput. Geosci., 8, 301-324 (2004) ·Zbl 1088.76046 |
| [11] | Kuznetsov, Y.; Repin, S., New mixed finite element method on polygonal and polyhedral meshes, Russ. J. Numer. Anal. Math. Model., 18, 3, 261-278 (2003) ·Zbl 1048.65113 |
| [12] | Kuznetsov, Y.; Repin, S., Convergence analysis and error estimates for mixed finite element method on distorted meshes, J. Numer. Math., 13, 1, 33-51 (2005) ·Zbl 1069.65114 |
| [13] | Lipnikov, K.; Morel, J.; Shashkov, M., Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes, J. Comput. Phys., 199 (2004) ·Zbl 1057.65071 |
| [14] | Margolin, L.; Shashkov, M.; Smolarkiewicz, P., A discrete operator calculus for finite difference approximations, Comput. Meth. Appl. Mech. Eng., 187, 365-383 (2000) ·Zbl 0978.76063 |
| [15] | Mishev, I., Nonconforming finite volume methods, Comput. Geosci., 6, 253-268 (2002) ·Zbl 1036.76038 |
| [16] | Morel, J.; Roberts, R.; Shashkov, M., A local support-operators diffusion discretization scheme for quadrilateralr-z meshes, J. Comput. Phys., 144, 17-51 (1998) ·Zbl 1395.76052 |
| [17] | P. Raviart, J.-M. Thomas, A mixed finite element method for second order eliptic problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, Berlin, Heidelberg, New York, pp. 292-315, 1977.; P. Raviart, J.-M. Thomas, A mixed finite element method for second order eliptic problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, Berlin, Heidelberg, New York, pp. 292-315, 1977. ·Zbl 0362.65089 |
| [18] | Stüben, K., Algebraic multigrid (AMG): experiences and comparisons, Appl. Math. Comput., 13, 419-452 (1983) ·Zbl 0533.65064 |
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