Parallel preconditioning based on h-hierarchical finite elements with application to acoustics

Abstract

In this paper, we consider the hierarchical basis preconditioning approach which is closely related to the h-version of the hierarchical finite element method. In general, finite element formulations that employ hierarchical shape functions yield better conditioned matrix problems than those based on the Lagrange nodal basis. These matrix problems are also better-suited to a faster rate of convergence with Krylov-subspace iterative methods. We consider a purely algebraic approach to describe projections between the nodal and the h-hierarchical bases functions, which are then used to construct the preconditioning operator. Implementation details of the preconditioner are provided for solving finite element problems on unstructured grids. We employ the preconditioning approach in the iterative solution of linear systems that arise from finite element discretization of the exterior Helmholtz problem. Numerical results are presented to examine convergence rates for practical discretizations in acoustics, and to illustrate the computational performance of the preconditioning algorithm on serial and data-parallel computers.