Multigrid algorithm for the cell-centered finite difference method II: Discontinuous coefficient case

Abstract

We consider a multigrid algorithm for the cell centered finite difference scheme with a prolongation operator depending on the diffusion coefficient. This prolongation operator is designed mainly for solving diffusion equations with strong varying or discontinuous coefficient and it reduces to the usual bilinear interpolation for Laplace equation. For simple interface problem, we show that the energy norm of this operator is uniformly bounded by 11/8, no matter how large the jump is, from which one can prove that W-cycle with one smoothing converges with reduction factor independent of the size of jump using the theory developed by Bramble et al. (Math Comp 56 (1991), 1-34). For general interface problem, we show that the energy norm is bounded by some constant C-* (independent of the jumps of the coefficient). In this case, we can conclude W-cycle converges with sufficiently many smoothings. Numerical experiment shows that even V-cycle multigrid algorithm with our prolongation works well for various interface problems. (C) 2004 Wiley Periodicals, Inc.