On MGR$[\nu ]$ Multigrid Methods

Abstract

The MGR$[\nu ]$ algorithm of Ries, Trottenberg and Winter with $\nu = 0$, and Algorithm 2.1 of Braess are essentially the same multigrid algorithm for the discrete Poisson equation $ - \Delta _h U = f$. In this report we consider the extension to the general diffusion equation $ - \nabla \cdot p\nabla u = f,p(x,y) \geqq p_0 > 0$. In particular, for the two-grid scheme we re-obtain the basic result: Let $\varepsilon ^0 $ be the error before a single multigrid cycle and let $\varepsilon ^1 $ be the error after this cycle. Then $\|\varepsilon ^1 \|_{Lh} \leqq \frac{1}{2}(1 = Kh)\|\varepsilon ^0 \|_{Lh} $. Computational results indicate that other constant coefficient results carry over as well.