Hierarchical preconditioners for high-order FEM

Abstract

The finite element discretization of partial differential equations (PDEs) requires the selection of suitable finite element spaces. While high-order finite elements often lead to solutions of higher accuracy, their associated discrete linear systems of equations are often more difficult to solve (and to set up) compared to those of lower order elements. We will present and compare preconditioners for these types of linear systems of equations. More specifically, we will use hierarchical (H-) matrices to build block H-LU preconditioners. H-matrices provide a powerful technique to compute and store approximations to dense matrices in a data-sparse format. We distinguish between blackbox H-LU preconditioners which factor the entire stiffness matrix and hybrid methods in which only certain subblocks of the matrix are factored after some problem-specific information has been exploited.We conclude with numerical results.