MPI parallel implementation of CBF preconditioning for 3D elasticity problems

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second-order elliptic partial differential equations. This system is discretized by trilinear brick finite elements. The PCG iterative method is used for solving the large-scale linear algebraic systems arising after the FEM discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. New construction of a parallel algorithm for the discussed preconditioning method is proposed. The theoretical part of this study includes analysis of the execution time on various parallel architectures and asymptotic estimates of the parallel speedup and the parallel efficiency. The parallel performance estimates indicate that the proposed algorithm will be especially efficient on coarse-grain parallel systems, which is also confirmed by the numerical experiments. A portable MPI parallel code is developed. Numerical tests on three symmetric multiprocessor systems: SUN Enterprise 3000, SUN SPARCstation 10 and Origin 2000 are presented. The reported speedup and parallel efficiency illustrate well the features of the proposed method and its implementation.