MULTIPLE GALERKIN ADAPTIVE ALGEBRAIC MULTIGRID ALGORITHM FOR THE HELMHOLTZ EQUATIONS

摘要

Introduced in this paper is a multiple Galerkin adaptive algebraic multigrid algorithm for solving linear discrete equations arising from discretization of partial differential operators with a rich and diverse near-kernel. Standard multigrid methods struggle with solving such equations due to their inability to produce accurate coarse correction to all near-kernel components. The motivating model problem here is the indefinite Helmholtz operators for which the character of near-kernel components is, at least approximately, known. In the algorithm, some of these components are used to create multiple prolongation operators and consecutive coarse descriptions which collectively approximate the entire near-kernel. The algorithm consists of two parts, with multiple Galerkin corrections accelerating a standard multigrid V-cycle. Both parts largely employ standard multigrid techniques. The resulting algorithm is then used to precondition the GMRES. Numerical experiments in one and two dimensions are presented, demonstrating the applicability of the method for problems with different types of wave numbers. Further leads in improving and extending the proposed approach conclude the paper.