On the convergence of multigrid methods for flow problems
type: Item
instance of: scholarly article
title: On the convergence of multigrid methods for flow problems
zbMATH Open document ID: 0937.76036
author: Ingemar Persson
author: Klas Samuelsson
author: Anders Szepessy
publication date: 1999-06-21T00:00:00Z
full work available at URL: https://eudml.org/doc/120310
full work available at URL: pp46-87.html
review text: We prove two theorems on the residual damping in multigrid methods when solving convection dominated diffusion equations and shock wave problems, discretized by the streamline diffusion finite element method. The first theorem shows that a \(V\)-cycle, including sufficiently many pre and post smoothing steps, damps the residual in \(L^{\text{loc}}_1\) for a constant coefficient convection problem with small diffusion in two space dimensions, without the assumption that the coarse grid is sufficiently fine. The proof is based on discrete Green's functions for the smoothing and correction operators on a uniform unbounded mesh aligned with the characteristic. The second theorem proves a similar result for a certain continuous version of a two-grid method, with isotropic artificial diffusion, applied to a two-dimensional Burgers shock wave problem.
Mathematics Subject Classification ID: 76M10
Mathematics Subject Classification ID: 76R50
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 1306004
zbMATH Keywords: multigrid methods
zbMATH Keywords: convergence
zbMATH Keywords: conservation laws
zbMATH Keywords: residual damping
zbMATH Keywords: convection dominated diffusion equations
zbMATH Keywords: streamline diffusion finite element method
zbMATH Keywords: discrete Green's functions
zbMATH Keywords: smoothing
zbMATH Keywords: two-grid method
zbMATH Keywords: isotropic artificial diffusion
zbMATH Keywords: two-dimensional Burgers shock wave problem
MaRDI profile type: Publication
published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 1306004