An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition

Abstract

MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O (h(-2)), which can be obtained from other ILU-type preconditioners, to O (h(-1)). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O (h(-1)) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O (h(-1)) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU. (c) 2017 Elsevier Inc. All rights reserved.