×
Cowsar, Lawrence C.; Mandel, Jan; Wheeler, Mary F.

Balancing domain decomposition for mixed finite elements. (English) Zbl 0828.65135

Math. Comput. 64, No. 211, 989-1015 (1995).
The convergence rate of the balancing domain decomposition method applied to mixed finite element discretizations of second-order elliptic equations is analyzed.
The balancing domain decomposition method was introduced by J. Mandel. It is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet data, a local problem with Neumann data, and a “coarse grid” problem to propagate information globally and to insure the consistency of the Neumann problem. It is shown that the condition number grows at work like the logarithm squared of the ratio of the subdomain size to the element size, both in two and three dimensions. Computational results on an INTEL-Delta machine demonstrate the scalability properties of the method.
Reviewer: W.Heinrichs (Düsseldorf)

Cited in 1 Review
Cited in 45 Documents

MSC:

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
65Y05 Parallel numerical computation
35J25 Boundary value problems for second-order elliptic equations

Keywords:

parallel computing; computational results; balancing domain decomposition; mixed finite element; second-order elliptic equations; substructuring method; condition number

Software:

PICL
Cite Review PDF
Full Text: DOI

References:

[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7 – 32 (English, with French summary). · Zbl 0567.65078
[3] Steven F. Ashby, Thomas A. Manteuffel, and Paul E. Saylor, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal. 27 (1990), no. 6, 1542 – 1568. · Zbl 0723.65018 · doi:10.1137/0727091
[4] J.-F. Bourgat, Roland Glowinski, Patrick Le Tallec, and Marina Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, Domain decomposition methods (Los Angeles, CA, 1988) SIAM, Philadelphia, PA, 1989, pp. 3 – 16. · Zbl 0684.65094
[5] J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103 – 134. · Zbl 0615.65112
[6] James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1 – 24. · Zbl 0668.65082
[7] Susanne C. Brenner, A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method, SIAM J. Numer. Anal. 29 (1992), no. 3, 647 – 678. · Zbl 0759.65080 · doi:10.1137/0729042
[8] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129 – 151 (English, with loose French summary). · Zbl 0338.90047
[9] Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237 – 250. · Zbl 0631.65107 · doi:10.1007/BF01396752
[10] Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 581 – 604 (English, with French summary). · Zbl 0689.65065
[11] Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217 – 235. · Zbl 0599.65072 · doi:10.1007/BF01389710
[12] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[13] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[14] L. C. Cowsar, Domain decomposition methods for nonconforming finite element spaces of Lagrange-type, Sixth Copper Mountain Conference on Multigrid Methods , NASA CP 3224, Hampton, VA, 1993, pp. 93-109.
[15] -, Dual-variable Schwarz methods for mixed finite elements, Numer. Math., submitted.
[16] Lawrence C. Cowsar and Mary F. Wheeler, Parallel domain decomposition method for mixed finite elements for elliptic partial differential equations, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990) SIAM, Philadelphia, PA, 1991, pp. 358 – 372. · Zbl 0770.65080
[17] Yann-Hervé De Roeck and Patrick Le Tallec, Analysis and test of a local domain-decomposition preconditioner, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990) SIAM, Philadelphia, PA, 1991, pp. 112 – 128. · Zbl 0770.65082
[18] L. J. Durlofsky and M. C. H. Chien, Development of a mixed finite-element based compositional reservoir simulator, Proceedings 12th SPE Symposium on Reservoir Simulation, SPE, Inc., 1993, Society of Petroleum Engineers, pp. 221-231.
[19] G. A. Geist, M. T. Heath, B. W. Peyton, and P. H. Worley, A users’ guide to PICL: A portable instrumented communication library, Tech. Rep. ORNL/TM-11616, Oak Ridge National Laboratory, Aug. 1990.
[20] Roland Glowinski, Gene H. Golub, Gérard A. Meurant, and Jacques Périaux , First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. · Zbl 0649.00019
[21] Roland Glowinski, Yuri A. Kuznetsov, Gérard Meurant, Jacques Périaux, and Olof B. Widlund , Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1991. · Zbl 0758.00010
[22] Roland Glowinski and Mary Fanett Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 144 – 172. · Zbl 0661.65105
[23] J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag, Berlin and New York, 1972. · Zbl 0223.35039
[24] Jan Mandel, Balancing domain decomposition, Comm. Numer. Methods Engrg. 9 (1993), no. 3, 233 – 241. · Zbl 0796.65126 · doi:10.1002/cnm.1640090307
[25] J. Mandel and M. Brezina, Balancing domain decomposition: theory and performance in two and three dimensions, Report No. 2, Center for Computational Mathematics, University of Colorado at Denver, 1993.
[26] J. Necas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967.
[27] J.-C. Nédélec, Mixed finite elements in \?³, Numer. Math. 35 (1980), no. 3, 315 – 341. · Zbl 0419.65069 · doi:10.1007/BF01396415
[28] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292 – 315. Lecture Notes in Math., Vol. 606.
[29] T. F. Russell and M. F. Wheeler, Finite element and finite difference methods for continuous flows in porous media, Mathematics of Reservoir Simulation , SIAM, Philadelphia, PA, 1983, ch. II, pp. 35-106.
[30] J. M. Thomas, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes, thèse d’état, Université Pierre et Marie Curie, Paris, 1977.
[31] Alan Weiser and Mary Fanett Wheeler, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal. 25 (1988), no. 2, 351 – 375. · Zbl 0644.65062 · doi:10.1137/0725025
[32] O. B. Widlund, An extension theorem for finite element spaces with three applications, Numerical Techniques in Continuum Mechanics , GAMM, 1987, pp. 110-122.
[33] -, Iterative substructuring methods: Algorithms and theory for problems in the plane, in Glowinski et al. [20], pp. 113-128.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.