ℋ︁‐matrix preconditioners for symmetric saddle‐point systems from meshfree discretization

Abstract

AbstractMeshfree methods are suitable for solving problems on irregular domains, avoiding the use of a mesh. To deal with the boundary conditions, we can use Lagrange multipliers and obtain a sparse, symmetric and indefinite system of saddle‐point type. Many methods have been developed to solve the indefinite system. Previously, we presented an algebraic method to construct an LU‐based preconditioner for the saddle‐point system obtained by meshfree methods, which combines the multilevel clustering method with the ℋ︁‐matrix arithmetic. The corresponding preconditioner has both ℋ︁‐matrix and sparse matrix subblocks. In this paper we refine the above method and propose a way to construct a pure ℋ︁‐matrix preconditioner. We compare the new method with the old method, JOR and smoothed algebraic multigrid methods. The numerical results show that the new preconditioner outperforms the preconditioners based on the other methods. Copyright © 2008 John Wiley & Sons, Ltd.