Domain decomposition based preconditioner combined local low-rank approximation with global corrections

To solve general sparse linear systems, this paper presents a domain decomposition based parallel preconditioner. Vertex-based partitioning is utilized to reorder the original coefficient matrix, resulting in a s x s block structure. Here, s is the number of subdomains used in the partition. Variables corresponding to the interface nodes are obtained by solving a linear system with coefficient matrix being the Schur complement S of the reordered matrix. Combining local low-rank correction approximation with a global low-rank correction technique to approximate the inverse of S, the method presented in this paper is different with previous Schur complement based preconditioners. The global low-rank correction terms are obtained by using the information comes from the local low-rank correction terms. In addition, variables corresponding to the interior nodes are computed by solving s small linear systems in parallel. Some numerical tests are presented to show the efficiency and robustness of the proposed preconditioner.