Robust Multi-grid with 7-point ILU Smoothing

Abstract

This paper deals with MG applied to discretized anisotropic boundary value problems. A model problem is given by $$\left\{ \begin{gathered} - ( \in \partial _1^2 {\text{ + }}\partial )u{\text{ = }}f on \Omega = \left( {0,1} \right)^2 , \hfill \\ u = f on \partial \Omega \hfill \\ \end{gathered} \right. $$ discretized using linear finite elements corresponding to a regular triangulation of Ω Besides this problem, we consider the problems that arise when the unit square is replaced by a domain having a re-entrant corner (less-regular problems) or when the differential operator is replaced by a rotated operator so that possibly the grid is no longer oriented with the direction of the anisotropy. We are interested in the question of whether the MGM is robust, that is, whether it converges uniformly not only in the “level” l, but also in ε. We concentrate on 7-point ILU smoothers which are known to give often good results in practice.KeywordsMultigrid MethodContraction NumberLanczos MethodLinear Finite ElementRegular TriangulationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.