On an Estimate for the Three-Grid MGR Multigrid Method

Abstract

The ${\operatorname{MGR}}[\nu ]$ algorithm of Ries, Trottenberg and Winter, Algorithm 2.1 of Braess and Algorithm 4.1 of Verfi rth are all algorithms for the numerical solution of the discrete Poisson equation based on red-black Gauss-Seidel smoothing iterations. In this work we consider the extension of the ${\operatorname{MGR}}[0]$ method to the general diffusion equation $ - \nabla \cdot p\nabla u = f$. In particular, for the three-grid scheme we extend an interesting and important result of Ries, Trottenberg and Winter, whose results are based on Fourier analysis and hence intrinsically limited to the case where $\Omega $ is a rectangle. Let $\Omega $ be a general polygonal domain whose sides have slope $ \pm 1$, 0 and $\infty $. Let $\varepsilon ^0 $ be the error before a single multigrid cycle and let $\varepsilon ^1 $ be the error after this cycle. Then $\|\varepsilon ^1 \|_{Ln} \leq \frac{1}{2}(1 + Kh)||\varepsilon ^0 ||_{L_h } $, where denotes the energy or operator norm. When $p(x,y)\equiv $ constant, then $K \equiv 0$.