A Hierarchical Low Rank Schur Complement Preconditioner for Indefinite Linear Systems

Abstract

Nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods. This paper combines ideas from the multilevel Schur low rank preconditioner developed by Y. Xi, R. Li, and Y. Saad [SIAM J. Matrix Anal., 37 (2016), pp. 235--259] with classic block preconditioning strategies in order to handle this case. The method to be described generates a tree structure $\mathcal{T}$ that represents a hierarchical decomposition of the original matrix. This decomposition gives rise to a block structured matrix at each level of $\mathcal{T}$. An approximate inverse of the original matrix based on its block $LU$ factorization is computed at each level via a low rank property that characterizes the difference between the inverses of the Schur complement and another block of the reordered matrix. The low rank correction matrix is computed by several steps of the Arnoldi process. Numerical results illustrate the robustness of the proposed preconditioner with respect to indefiniteness for a few discretized partial differential equations and publicly available test problems.