Substructure preconditioners for elliptic saddle point problems

Rusten, Torgeir; Winther, Ragnar

doi:10.1090/S0025-5718-1993-1149293-0

Substructure preconditioners for elliptic saddle point problems
HTML articles powered by AMS MathViewer

by Torgeir Rusten and Ragnar Winther PDF

Math. Comp. 60 (1993), 23-48 Request permission

Abstract:

Domain decomposition preconditioners for the linear systems arising from mixed finite element discretizations of second-order elliptic boundary value problems are proposed. The preconditioners are based on subproblems with either Neumann or Dirichlet boundary conditions on the interior boundary. The preconditioned systems have the same structure as the nonpreconditioned systems. In particular, we shall derive a preconditioned system with conditioning independent of the mesh parameter h. The application of the minimum residual method to the preconditioned systems is also discussed.

References

O. Axelsson and V. A. Barker, Finite element solution of boundary value problems, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. Theory and computation. MR 758437

Petroleum reservoir simulation

Sci. Publ.

Petter E. Bjørstad and Olof B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal. 23 (1986), no. 6, 1097–1120. MR 865945, DOI 10.1137/0723075

A preconditioning technique for the efficient solution of problems with local grid refinement

67

James H. Bramble and Joseph E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), no. 181, 1–17. MR 917816, DOI 10.1090/S0025-5718-1988-0917816-8
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. MR 842125, DOI 10.1090/S0025-5718-1986-0842125-3
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), no. 174, 361–369. MR 829613, DOI 10.1090/S0025-5718-1986-0829613-0
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 French summary). MR 365287
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. MR 890035, DOI 10.1007/BF01396752
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). MR 921828, DOI 10.1051/m2an/1987210405811
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. MR 799685, DOI 10.1007/BF01389710
Tony F. Chan and Howard C. Elman, Fourier analysis of iterative methods for elliptic problems, SIAM Rev. 31 (1989), no. 1, 20–49. MR 986481, DOI 10.1137/1031002
Jim Douglas Jr., Richard E. Ewing, and Mary Fanett Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numér. 17 (1983), no. 1, 17–33 (English, with French summary). MR 695450

A description of some alternating-direction iterative techniques for mixed finite element methods

Richard E. Ewing and Mary Fanett Wheeler, Computational aspects of mixed finite element methods, Scientific computing (Montreal, Que., 1982) IMACS Trans. Sci. Comput., I, IMACS, New Brunswick, NJ, 1983, pp. 163–172. MR 751617
Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341–354, iii (English, with French summary). MR 464543, DOI 10.1051/m2an/1977110403411
Michel Fortin and Roland Glowinski, Augmented Lagrangian methods, Studies in Mathematics and its Applications, vol. 15, North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
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. MR 972516
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR 972510

Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems

J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric $M$-matrix, Math. Comp. 31 (1977), no. 137, 148–162. MR 438681, DOI 10.1090/S0025-5718-1977-0438681-4
C. C. Paige and M. A. Saunders, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 617–629. MR 383715, DOI 10.1137/0712047
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) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555

Finite element and finite difference methods for continuous flow in porous media

Torgeir Rusten and Ragnar Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), no. 3, 887–904. Iterative methods in numerical linear algebra (Copper Mountain, CO, 1990). MR 1168084, DOI 10.1137/0613054