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Iterative methods for elliptic problems on regions partitioned into substructures and the biharmonic Dirichlet problem. (English) Zbl 0569.65081

Computing methods in applied sciences and engineering VI, Proc. 6th Int. Symp., Versailles 1983, 33-45 (1984).
[For the entire collection see Zbl 0547.00045.]
The author reviews several methods which combine preconditioned conjugate gradient iterations (concerning the unknown function values on lines connecting the subregions) with direct methods (solving the discrete selfadjoint elliptic problems posed on the subregions). Numerical examples show that reasonable accuracy is obtained in less than 10 iterations. For the convergence theory, properties of Schur complements and regularity results for finite elements are needed. To the biharmonic equation, similar techniques (including splitting of the forth order into two second order equations and Dirichlet to Neumann data mappings) are briefly surveyed, too.
Reviewer: G.Stoyan

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J40 Boundary value problems for higher-order elliptic equations
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions

Citations:

Zbl 0547.00045