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Multigrid preconditioned conjugate-gradient solver for mixed finite-element method. (English) Zbl 1198.65064

The mixed finite-element approximation of a porous media flow by the lowest order Raviart-Thomas elements is transformed by elimination of the fluxes into a positive definite Schur complement system for the pressures, which is solved by conjugate gradients preconditioned by a multigrid method for related cell-centered finite differences. The matrix of the fluxes can be inverted cheaply for rectangular grids, but for distorted meshes, the elimination is done by inner iterations with incomplete Cholesky preconditioning.

MSC:

65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65N06 Finite difference methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76M10 Finite element methods applied to problems in fluid mechanics
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[1] Akin, J.E.: Application and Implementation of Finite Element Methods, pp. 153–158. Academic, London (1982) · Zbl 0535.73063
[2] Allen, M.B., Ewing, R.E., Lu, P.: Well-conditioned iterative schemes for mixed finite-element models of porous-media flows. SIAM J. Sci. Statist. Comput. 13(3), 794–814 (1992) · Zbl 0777.76047
[3] Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, New York (1988) · Zbl 0681.68025
[4] Bank, R., Welfert, B., Yserentant, H.: A class of iterative methods for solving saddle point problems. Numer. Math. 55, 645–666 (1990) · Zbl 0684.65031
[5] Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., der Vorst, H.V.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994). www.netlib.org/linalg/html_templates/report.html · Zbl 0814.65030
[6] Bramble, J., Ewing, R., Pasciak, J., Shen, J.: The analysis of multigrid algorithms for cell centered finite difference methods. Adv. Comput. Math. 5(1), 15–29 (1996) · Zbl 0848.65082
[7] Bramble, J., Pasciak, J., Apostol, T.: Analysis of the inexact uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34(3), 1072–1092 (1987) · Zbl 0873.65031
[8] Bramble, J., Pasciak, J., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comput. 56(193), 1–34 (1991) · Zbl 0718.65081
[9] Brenner, S.C.: A mutligrid algorithm for the lowest-order Raviart–Thomas mixed triangular finite element method. SIAM J. Numer. Anal. 29(3), 647–678 (1992) · Zbl 0759.65080
[10] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991) · Zbl 0788.73002
[11] Briggs, W.L.: A Multigrid Tutorial. Siam, Philadelphia (1987)
[12] Cai, Z., Jones, J., McCormick, S., Russell, T.: Control-volume mixed finite elements methods. Comput. Geosci. 1, 289–315 (1997) · Zbl 0941.76050
[13] Chou, S., Kwak, D.Y., Kim, K.Y.: Flux recovery from primal hybrid finite element methods. SIAM J. Numer. Anal. 40(2), 403–415 (2003) · Zbl 1021.65057
[14] Dougherty, D.: PCG solutions of flow problems in random porous media using mixed finite elements. Adv. Water Resour. 13(1) (1990)
[15] Ewing, R., Shen, J.: A multigrid algorithm for the cell-centered finite difference scheme. In: The Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods. NASA Conference Publication 3224 (1993)
[16] Ewing, R., Wheeler, M.: Computational aspects of mixed finite element methods. In: Numerical Methods for Scientific Computing, pp. 163–172 (1983)
[17] Golub, G.H., Van Loan, C.F.: Matrix Computations. John Hopkins University Press, Baltimore (1989)
[18] Harbaugh, A.W., Banta, E.R., Hill, M.C., McDonald, M.G.: Modflow-2000, the U.S. geological survey modular ground-water model user guide to modularization concepts and the ground-water flow process. Tech. rep., U.S. Geological Survey. Open-File Report 00-92 (2000)
[19] Howard, C.E., Golub, G.H.: Inexact and preconditioned uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645–1661 (1994) · Zbl 0815.65041
[20] Hughes, T.: The Finite Element Method, pp. 123–125. Prentice-Hall, Englewood Cliffs (1987)
[21] Johnson, C.: Numerical Solutions of Partial Differential Equations by the Finite Element Method, pp. 141–144. Cambridge University Press, Cambridge (1987)
[22] Naff, R., Russell, T., Wilson, J.: Test functions for three-dimensional control-volume finite-element methods on irregular grids. In: Computational Methods in Water Resources, vol. 2, pp. 677–684 (2000). http://wwwbrr.cr.usgs.gov/projects/GW_stoch/index.html
[23] Naff, R., Russell, T., Wilson, J.: Shape functions for three-dimensional control-volume mixed finite-element methods on irregular grids. In: Computational Methods in Water Resources, pp. 359–366 (2002). http://wwwbrr.cr.usgs.gov/projects/GW_stoch/index.html
[24] Naff, R., Russell, T.R., Wilson, J.: Shape functions for velocity interpolation in general hexahedral cells. Comput. Geosci. 6, 285–314 (2002). http://wwwbrr.cr.usgs.gov/projects/GW_stoch/index.html · Zbl 1094.76542
[25] Raviart, P., Thomas, J.: A mixed finite element method for 2nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, pp. 292–315. Springer, New York (1977) · Zbl 0362.65089
[26] Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E. (ed.) The Mathematics of Reservoir Simulation, pp. 35–106. Society of Industrial and Applied Mathematics, Philadelphia (1983) · Zbl 0572.76089
[27] Sameh, A., Baggag, A.: Nested iterative schemes for indefinite linear systems. In: Mang, H.A., Rammerstorfer, F.G., Eberhardsteiner, J. (eds.) Fifth World Congress on Computational Mechanics (2002) · Zbl 1067.76559
[28] Tong, Z., Sameh, A.: On an iterative method for saddle point problems. Numer. Math. 79, 643–646 (1998) · Zbl 0906.65036
[29] Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic, London (2001)
[30] Wilson, J., Naff, R.: Modflow-2000, the U.S. Geological Survey modular ground-water model – GMG linear equation solver package documentation. Tech. rep., U.S. Geological Survey (2004). http://pubs.water.usgs.gov/ofr2004-1261/
[31] Xu, J.: Iterative methods by space decomposition and subspace corrections. SIAM Rev. 34(4), 581–613 (1992) · Zbl 0788.65037
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