Finite Element Approximate Inverse Preconditioning for solving 3D Biharmonic Problems on Shared Memory Systems

摘要

In this paper we present parallel explicit approximate inverse matrix techniques for solving sparse linear systems on shared memory systems, which are derived using the finite element method for biharmonic equations in three space variables. Our approach for solving such equations is by considering the biharmonic equation as a "coupled equation approach" (pair of Poisson equation), using a FE approximation scheme, yielding an "inner-outer" iteration method. Additionally, parallel approximate inverse matrix algorithms are introduced for the efficient solution of sparse linear systems, based on an anti-diagonal computational approach that eliminates the data dependencies. Parallel explicit preconditioned conjugate gradient-type schemes in conjunction with parallel approximate inverse matrix algorithms are presented for the efficient solution of sparse linear systems. Theoretical estimates on computational complexity of the parallel explicit preconditioned conjugate gradient method along with theoretical speedups and efficiency are also presented. Applications of the proposed methods on characteristic biharmonic problems are discussed and numerical results are given.