Domain decomposition based $${\mathcal H}$$ -LU preconditioning

Abstract

Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an $${{\mathcal H}}$$ -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based $${{\mathcal H}}$$ -matrices, this new approach yields $${\mathcal H}$$ -LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an $${\mathcal H}$$ -matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based $${{\mathcal H}}$$ -LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.