zbMATH — the first resource for mathematics

Analysis of multilevel decomposition iterative methods for mixed finite element methods. (English) Zbl 0823.65035
Multilevel extensions of the two-level Schwarz algorithm [see R. E. Ewing and J. Wang, ibid. 26, No. 6, 739-756 (1992; Zbl 0765.65104)] are considered for second-order elliptic equations. The authors show that, without any regularity assumptions, the algorithms converge with rate bounded by \(1 - w(2-w)/ (CJ)\) for some constant \(C\) where \(J\) is the number of levels and \(0 < w < 2\) is a relaxation parameter. Furthermore, if the full \(H^ 2\) regularity is satisfied for the operator \(- \nabla \cdot (c\nabla)\) with homogeneous Dirichlet boundary condition, then uniform convergence is possible. Numerical results are presented to illustrate the efficiency of the proposed algorithms.
Reviewer: L.S.Ioffe (Haifa)

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
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI EuDML
[1] I. BABUŠKA, 1973, The finite element method with Lagrangian multipliers, Numer, Math., 20, 179-192. Zbl0258.65108 MR359352 · Zbl 0258.65108 · doi:10.1007/BF01436561 · eudml:132183
[2] R. E. BANK and T. F. DUPONT, 1981, An optimal order process for solving elliptic finite element equations, Math. Comp., 36, 35-51. Zbl0466.65059 MR595040 · Zbl 0466.65059 · doi:10.2307/2007724
[3] R. E. BANK, T. F. DUPONT and H. YSERENTANT, 1988, The Hierarchical basis multigrid method, Numer. Math., 52, 427-458. Zbl0645.65074 MR932709 · Zbl 0645.65074 · doi:10.1007/BF01462238 · eudml:133245
[4] J. H. BRAMBLE, R. E. EWING, J. E. PASCIAK and A. H. SCHATZ, 1988, A preconditioning technique for the efficient solution of problems with local grid refinement, Compt. Meth. Appl. Mech. Eng., 67, 149-159. Zbl0619.76113 · Zbl 0619.76113 · doi:10.1016/0045-7825(88)90122-3
[5] J. H. BRAMBLE, J. E. PASCIAK, J. WANG and J. XU, 1991, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., 57, 1-22. Zbl0754.65085 MR1090464 · Zbl 0754.65085 · doi:10.2307/2938660
[6] J. H. BRAMBLE, J. E. PASCIAK, J. WANG and J. XU, 1911, Convergence estimate for multigrid algorithms without regularity assumptions, Math. Comp., 57, 23-45. Zbl0727.65101 MR1079008 · Zbl 0727.65101 · doi:10.2307/2938661
[7] F. BREZZI, 1974, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O., Anal. Numér., 2, 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047 · eudml:193255
[8] F. BREZZI, J. Jr. DOUGLAS, M. FORTIN and L. D. MARINI, 1987, Efficient rectangular mixed finite elements in two and three space variables, R.A.I.R.O., Anal. Numér., 21, 581-604. Zbl0689.65065 MR921828 · Zbl 0689.65065 · eudml:193515
[9] F. BREZZI, J. Jr. DOUGLAS and L. D. MARINI, 1985, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235. Zbl0599.65072 MR799685 · Zbl 0599.65072 · doi:10.1007/BF01389710 · eudml:133032
[10] J. Jr. DOUGLAS and J. E. ROBERTS, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 45, 39-52. Zbl0624.65109 MR771029 · Zbl 0624.65109 · doi:10.2307/2007791
[11] R. E. EWING and J. WANG, 1992, Analysis of the Schwarz algorithm for mixed finite element methods, R.A.I.R.O. M2AN, 26, 739-756. Zbl0765.65104 MR1183415 · Zbl 0765.65104 · eudml:193683
[12] M. DRYJA and O. WIDLUND, 1987, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report, Courant Institute of Mathematical Sciences, 339.
[13] M. DRYJA and O WIDLUND, 1989, Some domain decomposition algorithms for elliptic problems, Technical Report, Courant Institute of Mathematical Sciences, 438. MR1038100 · Zbl 0668.65084
[14] R. FALK and J. OSBORN, 1980, Error estimates for mixed methods, R.A.I.R.O., Anal. Numér., 14, 249-277. Zbl0467.65062 MR592753 · Zbl 0467.65062 · eudml:193361
[15] M. FORTIN, An analysis of the convergence of mixed finite element methods, R.A.I.R.O., Anal. Numér., 11, 341-354. Zbl0373.65055 MR464543 · Zbl 0373.65055 · eudml:193306
[16] R. GLOWINSKI and M. F. WHEELER, 1988, Domain decomposition and mixed finite element methods for elliptic problems, Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Dijferential Equations (R. Glowinski, G. H. Golub, G. A. Meurant and J. Périaux, eds.). Zbl0661.65105 MR972509 · Zbl 0661.65105
[17] W. HACKBUSCH, 1985, Multi-Grid Methods and Applications, Springer-Verlag, New York. Zbl0595.65106 · Zbl 0595.65106
[18] P. L. LIONS, 1988, On the Schwarz alternating method, Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations (R. Glowinski, G. H. Golub, G. A. Meurant and J. Périaux, eds.). Zbl0658.65090 MR972509 · Zbl 0658.65090
[19] T. P. MATHEW, 1989, Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretizations of Elliptic Problems, Ph. D. Thesis, New York University.
[20] P.-A. RAVIART and J.-M. THOMAS, 1977, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics (606), Springer-Verlag, Berlin and New York, 292-315. Zbl0362.65089 MR483555 · Zbl 0362.65089
[21] H. A. SCHWARZ, 1869, Über einige Abbildungsaufgaben, Ges. Math. Abh., 11, 65-83.
[22] J. WANG, 1992, Convergence analysis without regularity assumptions for multigrid algorithms based on SOR smoothing, SIAM J. Numer. Anal., 29, 987-1001. Zbl0753.65093 MR1173181 · Zbl 0753.65093 · doi:10.1137/0729060
[23] J. WANG, 1992, Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods I : selfadjoint and positive definite elliptic problems, in Iterative Methods in Linear Algebra, R. Beauwens and P. de Groen (eds.), North-Holland, Amsterdam. Zbl0785.65115 MR1159720 · Zbl 0785.65115
[24] J. WANG, 1993, Convergence analysis of the Schwarz algorithm and multilevel decomposition iterative methods II : non-selfadjoint and indefinite elliptic problems, SIAM J. Numer. Anal., 30, 953-970. Zbl0777.65066 MR1231322 · Zbl 0777.65066 · doi:10.1137/0730050
[25] H. YSERENTANT, 1986, On the multi-level splitting of finite element spaces, Numer. Math., 49, 379-412. Zbl0608.65065 MR853662 · Zbl 0608.65065 · doi:10.1007/BF01389538 · eudml:133120
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.