Domain decomposition methods for large linearly elliptic three-dimensional problems

The idea of solving large problems using domain decomposition technique appears particularly attractive on present day large-scale parallel computers. But the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation. The approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape, and solves for unknown on the interface using the associated trace operator (the Steklov-Poincare operator on the continuous level or the Schur complement matrix after a finite element discretization) and a preconditioned conjugate gradient method. This algorithm involves the solution of Dirichlet and of Neumann problems, defined on each subdomain and which can be solved in parallel. This method has been implemented on a CRAY 2 computer using multitasking and on an INTEL hypercube. It was tested on a large scale, industrial, ill-conditioned, three-dimensional linear elasticity problem, which gives a fair indication of its performance in a real life environment. In such situations, the proposed method appears operational and competitive on both machines: compared to standard techniques, it yields faster results with far less memory requirements.