The construction of preconditioners for elliptic problems by substructuring. IV

Bramble, James H.; Pasciak, Joseph E.; Schatz, Alfred H.

doi:10.1090/S0025-5718-1989-0970699-3

The construction of preconditioners for elliptic problems by substructuring. IV
HTML articles powered by AMS MathViewer

by James H. Bramble, Joseph E. Pasciak and Alfred H. Schatz PDF

Math. Comp. 53 (1989), 1-24 Request permission

Abstract:

We consider the problem of solving the algebraic system of equations which result from the discretization of elliptic boundary value problems defined on three-dimensional Euclidean space. We develop preconditioners for such systems based on substructuring (also known as domain decomposition). The resulting algorithms are well suited to emerging parallel computing architectures. We describe two techniques for developing these preconditioners. A theory for the analysis of the condition number for the resulting preconditioned system is given and the results of supporting numerical experiments are presented.

References

G. P. Astrahancev, The method of fictitious domains for a second order elliptic equation with natural boundary conditions, Ž. Vyčisl. Mat i Mat. Fiz. 18 (1978), no. 1, 118–125, 269 (Russian). MR 468228
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
J. H. Bramble, A second order finite difference analog of the first biharmonic boundary value problem, Numer. Math. 9 (1966), 236–249. MR 205478, DOI 10.1007/BF02162087

Comput. Methods Appl. Mech. Engrg.

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
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, The construction of preconditioners for elliptic problems by substructuring. II, Math. Comp. 49 (1987), no. 179, 1–16. MR 890250, DOI 10.1090/S0025-5718-1987-0890250-4
James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. III, Math. Comp. 51 (1988), no. 184, 415–430. MR 935071, DOI 10.1090/S0025-5718-1988-0935071-X

Elliptic Problem Solvers II

The Use of Preconditioning Over Irregular Regions

A Comparison on Domain Decomposition Techniques for Elliptic Partial Differential Equations and the Parallel Implementation

Inequalities

S. G. Kreĭn and Ju. I. Petunin, Scales of Banach spaces, Uspehi Mat. Nauk 21 (1966), no. 2 (128), 89–168 (Russian). MR 0193499
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
S. McCormick and J. Thomas, The fast adaptive composite grid (FAC) method for elliptic equations, Math. Comp. 46 (1986), no. 174, 439–456. MR 829618, DOI 10.1090/S0025-5718-1986-0829618-X

Les Méthodes Directes en Théorie des Équations Elliptiques

Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502