The construction of preconditioners for elliptic problems by substructuring. II

Bramble, J. H.; Pasciak, J. E.; Schatz, A. H.

doi:10.1090/S0025-5718-1987-0890250-4

The construction of preconditioners for elliptic problems by substructuring. II
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by J. H. Bramble, J. E. Pasciak and A. H. Schatz PDF

Math. Comp. 49 (1987), 1-16 Request permission

Abstract:

We give a method for constructing preconditioners for the discrete systems arising in the approximation of solutions of elliptic boundary value problems. These preconditioners are based on domain decomposition techniques and lead to algorithms which are well suited for parallel computing environments. The method presented in this paper leads to a preconditioned system with condition number proportional to $d/h$ where d is the subdomain size and h is the mesh size. These techniques are applied to singularly perturbed problems and problems in three dimensions. The results of numerical experiments illustrating the performance of the method on problems in two and three dimensions are given.

References

Petter E. Bjørstad and Olof B. Widlund, Solving elliptic problems on regions partitioned into substructures, Elliptic problem solvers, II (Monterey, Calif., 1983) Academic Press, Orlando, FL, 1984, pp. 245–255. MR 764237
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, 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
B. L. Buzbee and Fred W. Dorr, The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions, SIAM J. Numer. Anal. 11 (1974), 753–763. MR 362944, DOI 10.1137/0711061
B. L. Buzbee, F. W. Dorr, J. A. George, and G. H. Golub, The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal. 8 (1971), 722–736. MR 292316, DOI 10.1137/0708066

Elliptic Problem Solvers II

Proc. 6th Internat. Conf. Comput. Methods in Sci. and Engng.

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