Geometric multigrid methods based on Generic Approximate Sparse Inverses

Abstract

Multigrid methods are considered amongst the most efficient methods for the solution of large sparse linear systems derived from the discretization of Partial Differential Equations (PDEs). In order to accelerate multigrid convergence a new class of Generic Approximate Inverses utilized, in conjunction with the Richardson's method, as smoothers. The Generic Approximate Inverses in conjunction with approximate inverse sparsity patterns, based on powers of sparsified matrices, are computed from the Incomplete LU factorization of the coefficient matrices. The proposed schemes are used to solve large linear systems derived from the discretization of a PDE with the Mehrstellen scheme which leads to solutions with fourth order accuracy by increasing slightly the computation effort. Moreover, the proposed schemes are used to solve boundary value problems in three dimensions. Furthermore, numerical results are presented for the proposed schemes to outline the applicability and effectiveness of the parametric design of the Approximate Inverse.