Substructure preconditioners for elliptic saddle point problems. (English) Zbl 0795.65072

Rusten, Torgeir; Winther, Ragnar

Math. Comput. 60, No. 201, 23-48 (1993).

Summary: 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.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems
65F35	Numerical computation of matrix norms, conditioning, scaling
35J25	Boundary value problems for second-order elliptic equations

Keywords:

elliptic saddle point problems; domain decomposition; preconditioners; mixed finite element; second-order elliptic boundary value problems; minimum residual method

References:

[1]	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. · Zbl 0537.65072
[2]	K. Aziz and A. Settari, Petroleum reservoir simulation, Appl. Sci. Publ., London, 1979.
[3]	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. · Zbl 0615.65113 · doi:10.1137/0723075
[4]	J. H. Bramble, R. E. Ewing, J. E. Pasciak, and A. H. Schatz, A preconditioning technique for the efficient solution of problems with local grid refinement, Comput. Methods Appl. Mech. Engrg. 67 (1988), 149-159. · Zbl 0619.76113
[5]	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. , https://doi.org/10.1090/S0025-5718-1988-0917816-8 James H. Bramble and Joseph E. Pasciak, Corrigenda: ”A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems”, Math. Comp. 51 (1988), no. 183, 387 – 388. · Zbl 0643.65017
[6]	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
[7]	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. · Zbl 0595.65111
[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]	Tony F. Chan and Howard C. Elman, Fourier analysis of iterative methods for elliptic problems, SIAM Rev. 31 (1989), no. 1, 20 – 49. · Zbl 0684.65092 · doi:10.1137/1031002
[13]	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). · Zbl 0516.76094
[14]	J. Douglas, Jr. and P. Pietra, A description of some alternating-direction iterative techniques for mixed finite element methods. Mathematical and Computational Methods in Seismic Exploration and Reservoir Modeling , SIAM, Philadelphia, PA, 1986, pp. 37-53.
[15]	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.
[16]	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). · Zbl 0373.65055
[17]	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. · Zbl 0525.65045
[18]	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. · Zbl 0585.65077
[19]	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
[20]	J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. · Zbl 0223.35039
[21]	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. · Zbl 0658.65090
[22]	T. P. Mathew, Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems, Ph.D. thesis, Department of Computer Science, Courant Institute of Mathematical Sciences, 1989.
[23]	J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric \?-matrix, Math. Comp. 31 (1977), no. 137, 148 – 162. · Zbl 0349.65020
[24]	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. · Zbl 0319.65025 · doi:10.1137/0712047
[25]	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.
[26]	T. F. Russel and M. F. Wheeler, Finite element and finite difference methods for continuous flow in porous media, The Mathematics of Reservoir Simulation , SIAM, Philadelphia, PA, 1983.
[27]	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). · Zbl 0760.65033 · doi:10.1137/0613054