Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. The Schur complements are based on a two-by-two block decomposition of the matrix, and their approximations are computed by assembly of local (macroelement) Schur complements. The block partitioning is done by imposing a particular node ordering following the grid refinement hierarchy in the discretization mesh. For the theoretical derivation of condition number bounds, but not for the actual application of the method, we assume that the corresponding differential operator is self-adjoint and positive definite. The numerical efficiency of the proposed Schur complement approximation is illustrated in the framework of block incomplete factorization preconditioners of multilevel type, which require approximations of a sequence of arising Schur complement matrices. The behavior of the proposed approximation is compared with that of the coarse mesh finite element matrix, commonly used as an approximation of the Schur complement in the context of the above preconditioning methods. Moreover, the influence of refining a coarse mesh using a higher refinement number ($m$) than the customary $m=2$ is analyzed and its efficiency is also illustrated by numerical tests.