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The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. (English) Zbl 0881.65093
The paper is the third of a series on the support-operators method [see M. Shashkov and S. Steinberg, J. Comput. Phys. 118, No. 1, 131-151 (1995; Zbl 0824.65101) and ibid. 129, No. 2, 383-40, (1996; Zbl 0874.65062)].
The authors describe and investigate a new second-order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non-isotropic media.
At first, the boundary value problem is formulated as a system of first-order equations. This formulation is given in operator form to illuminate the properties of the operators that should have analogs in the discrete case. Then, the construction of logically rectangular grids and the discretization of scalar and vector functions are described. Here, cell-centered discretizations of scalar functions, and both nodal and face-centered discretizations of vector functions (e.g. the flux) are used. Following the support-operators method, approximations for \(\text{div}\) and \(K \text{grad}\) (\(K\) is the conductivity matrix) are derived in the cases of the nodal and surface discretizations. Based on these discrete operators the finite difference scheme for \(\text{div} K \text{grad}\) is constructed.
Furthermore, iterative methods for solving the discrete problems are discussed.
Finally, the presented algorithms are compared on five examples. The experiments show that the surface discretization approach performs reliably on all examples, and the nodal discretization gives reasonable results.
Reviewer: M.Jung (Chemnitz)

65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] Favorskii, A.; Samarskii, A.; Shashkov, M.; Tishkin, V., Operational finite-difference schemes, Differential equations, 17, 854, (1981)
[2] Shashkov, M., Conservative finite-difference methods on general grids, (1995), CRC Press Boca Raton · Zbl 0844.65067
[3] Shashkov, M.; Steinberg, S., Support-operator finite-difference algorithms for general elliptic problems, J. comput. phys., 118, 131, (1995) · Zbl 0824.65101
[4] M. Shashkov, S. Steinberg, 1996, Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys. 129, 383, 405 · Zbl 0874.65062
[5] Durlofsky, L.J., A triangle based mixed finite element-finite volume technique for modeling two phase flow through porous media, J. comput. phys., 105, 252, (1993) · Zbl 0768.76046
[6] M. F. Wheeler, R. Gonzalez, Mixed finite element methods for petroleum reservoir engineering problems, Computing Methods in Applied Science and Engineering, VI, 639, 657 · Zbl 0598.76102
[7] Knupp, P.M.; Steinberg, S., The fundamentals of grid generation, (1993), CRC Press Boca Raton
[8] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16, (1992) · Zbl 0759.65006
[9] Carpetner, M.H.; Gottlieb, D.; Abarbanel, S., Time stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. comput. phys., 111, 220, (1994) · Zbl 0832.65098
[10] Leventhal, S.H., An operator compact implicit method of exponential type, J. comput. phys., 46, 138, (1982) · Zbl 0514.76086
[11] Hyman, J.M.; Shashkov, M., Natural discretizations for the divergence, gradient and curl on logically rectangular grid, Int. J. comput. math. appl., 33, 81, (1997) · Zbl 0868.65006
[12] J. M. Hyman, M. Shashkov, S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, Los Alamos National Laboratory, Los Alamos, NM · Zbl 0881.65093
[13] L. G. Margolin, J. J. Pyun, A method for treating hourglass pattern, Los Alamos National Laboratory, Los Alamos, NM
[14] Alcouffe, R.E.; Brandt, A.; Dendy, J.E.; Painter, J.W., The multi-grid method for the diffusion equation with strongly discontinuous coefficients, SIAM J. sci. statist. comp., 2, 430, (1981) · Zbl 0474.76082
[15] Dendy, J.E., Black box multigrid, J. comput. phys., 48, 366, (1982) · Zbl 0495.65047
[16] de Zeeuw, P.M., Matrix-dependent prolongation and restriction in the blackbox multi-grid solver, J. comput. appl. math., 3, 1, (1990) · Zbl 0717.65099
[17] Crumpton, P.I.; Shaw, G.I.; Ware, A.F., Discretization and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients, J. comput. phys., 116, 343, (1995) · Zbl 0818.65113
[18] Kershaw, D.S., The incomplete choleski-conjugate gradient methods for the iterative solution of systems of linear equations, J. comput. phys., 26, 43, (1978) · Zbl 0367.65018
[19] Morel, J.M.; Dendy, J.E.; Hall, M.L.; White, S.W., A cell-centered Lagrangian-mesh diffusion differencing scheme, J. comput. phys., 103, 286, (1992) · Zbl 0763.76052
[20] Durlofsky, L.J., Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities, Water resources res., 30, 965, (1994)
[21] Das, B.; Schaffer, S.; Steinberg, S.; Weber, S., Finite difference methods for modeling porous media flows, Transport porous media, 17, 171, (1994)
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