The use of domain decomposition methods in distributed memory parallel environments for solving elliptic partial differential equations with high discontinuity and high anisotropy is the main motivation for this work. In this respect, we propose local algebraic preconditioners for the Schur complement method. We show that these preconditioners are computationally and numerically attractive when used in combination with a probing technique.
We propose and experiment with several coarse space components that are combined with the local preconditioners.
We describe how these preconditioners are efficiently implemented on parallel distributed memory computers using message passing or the virtual shared memory paradigm in combination with efficient linear algebra kernels. Though the algebraic additive Schwarz (AAS) local preconditioner requires communications between neighbouring subdomains and a few more floating-point operations, the cost of one iteration of the preconditioned conjugate gradient method (PCG) when using AAS or block Jacobi is almost the same. Moreover, the number of iterations of the PCG with AAS is reduced by 40% for highly anisotropic problems.
We experiment with the new preconditioners on the linear systems that arise from a device simulation code that we have parallelised. Although the preconditioners are not optimal, as their convergence depends on both H and H/h, the experiments show that Schur complement domain decomposition methods, using those preconditioners, solve efficiently the proposed device problems on parallel distributed computers.