A robust smoother for convection-diffusion problems with closed characteristics
type: Item
instance of: scholarly article
title: A robust smoother for convection-diffusion problems with closed characteristics
zbMATH Open document ID: 0976.65093
DOI: 10.1007/PL00010094
author: K. Johannsen
published in: Computing
publication date: 2001-12-19T00:00:00Z
review text: The author focusses attention on the robust property of a block Gauss-Seidel-type smoother. A downwind ordering is used in the inner part of the domain to obtain a lower triangular matrix for the discretization of the convection. The feedback introduced by the cyclic boundary conditions is handled by a block Gauss elimination. For the inversion of the resulting (in general dense) Schur complement, a modification of the frequency filtering iteration is used. It is proved that the resulting smoother is both robust with respect to the strength of the convection and the amount of crosswind diffusion. In a work in preparation, the author shows that the robust smoothing property cannot be complemented by a robust approximation property using a monotone coarse grid discretization. However, a monotone discretization can be found which provides a sufficient approximation to make robust multigrid convergence feasible. The model problem, \(-\varepsilon\Delta u+{\partial u\over\partial x}=b\) in \(]0,1[^2\) and \(u=0\) on the Dirichlet boundary, simulates a convection-diffusion problem with cyclic convection using a linear convection on the unit square with cyclic boundary conditions. The reduced problem has closed characteristics. A full upwind finite difference scheme is used to discretize the problem. Additionally to the strength of the convection, an arbitrary amount of crosswind-diffusion can be added to the discrete level. Numerical results illustrate the convergence and robustness of the proposed method.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1563390
zbMATH Keywords: convection-diffusion problem
zbMATH Keywords: convection-dominated method
zbMATH Keywords: robust multigrid method
zbMATH Keywords: feedback vertex set
zbMATH Keywords: singular perturbation
zbMATH Keywords: block Gauss-Seidel-type smoother
zbMATH Keywords: frequency filtering iteration
zbMATH Keywords: upwind finite difference scheme
zbMATH Keywords: numerical results
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: pl00010094
OpenAlex ID: W2029600401
description: scientific article; zbMATH DE number 1563390