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Block incremental unknowns for anisotropic elliptic equations. (English) Zbl 1019.65081

Authors’ abstract: We define the notion of block incremental unknowns for anisotropic elliptic equations. Written in a block structure, they are nothing more than the classical second order incremental unknowns in one space dimension. In particular, we obtain a priori estimates and we observe a significant gain in the condition number for stiff problems when compared with the classical two-dimensional second-order incremental unknowns.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] Bank, R. E.; Dupont, T. F.; Yserentant, H., The hierarchical basis multigrid method, Numer. Math., 52, 427-458 (1988) · Zbl 0645.65074
[2] Chehab, J. P., A nonlinear adaptive multiresolution method in finite differences with incremental unknowns, RAIRO Modél Math. Anal. Numér., 29, 4, 451-475 (1995) · Zbl 0836.65114
[3] Chehab, J. P., Solution of generalized Stokes problems using hierarchical methods and incremental unknowns, Appl. Numer. Math., 21, 9-42 (1996) · Zbl 0853.76044
[4] J.P. Chehab, A. Miranville, Induced hierarchical preconditioners: the finite difference case, Publication ANO-371, Laboratoire ANO, Lille, 1997; J.P. Chehab, A. Miranville, Induced hierarchical preconditioners: the finite difference case, Publication ANO-371, Laboratoire ANO, Lille, 1997
[5] Chehab, J. P.; Miranville, A., Incremental unknowns on nonuniform meshes, RAIRO Modél Math. Anal. Numér., 32, 5, 539-577 (1998) · Zbl 0913.65088
[6] Chehab, J. P.; Temam, R., Incremental unknowns for solving nonlinear eigenvalue problems. New multiresolution methods, Numer. Methods Partial Differential Equations, 11, 199-228 (1995) · Zbl 0828.65124
[7] Chen, M.; Temam, R., Incremental unknowns for solving partial differential equations, Numer. Math., 59, 255-271 (1991) · Zbl 0712.65103
[8] Chen, M.; Temam, R., Incremental unknowns in finite differences: Condition number of the matrix, SIAM J. Matrix Anal. Appl., 14, 2, 432-455 (1993) · Zbl 0773.65080
[9] Chen, M.; Temam, R., Incremental unknowns for solving convection diffusion equations, Appl. Numer. Math., 11, 365-383 (1993) · Zbl 0774.65074
[10] Chen, M.; Temam, R., Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns, Numer. Math., 64, 271-294 (1993) · Zbl 0798.65093
[11] Chen, M.; Miranville, A.; Temam, R., Incremental unknowns in finite differences in space dimension 3, Comp. Appl. Math., 14, 3, 219-252 (1995) · Zbl 0841.65089
[12] Elman, H. C.; Zhang, X., Algebraic analysis of the hierarchical basis preconditioner, SIAM J. Matrix Anal. Appl., 16, 1, 192-206 (1995) · Zbl 0828.65032
[13] P. Poullet, Thèse, Université Paris-XI, 1994; P. Poullet, Thèse, Université Paris-XI, 1994
[14] T. Tachim-Medjo, Thèse, Université Paris-XI, 1995; T. Tachim-Medjo, Thèse, Université Paris-XI, 1995
[15] M. Marion, R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal. 26, 1139-1157; M. Marion, R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal. 26, 1139-1157 · Zbl 0683.65083
[16] Temam, R., Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21, 154-178 (1990) · Zbl 0715.35039
[17] Yserentant, H., On multilevel splitting of finite element spaces, Numer. Math., 49, 379-412 (1986) · Zbl 0608.65065
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