Domain decomposition for an advection--diffusion problem with parabolic layers.
type: Item
instance of: scholarly article
title: Domain decomposition for an advection--diffusion problem with parabolic layers.
zbMATH Open document ID: 1037.65129
DOI: 10.1016/S0096-3003(02)00514-3
author: Vic Duoba
author: Igor Boglaev
published in: Applied Mathematics and Computation
publication date: 2003-11-25T00:00:00Z
review text: The authors consider a singularly perturbed convection-diffusion equation in the unit square with the convection from the right to the left. The essential features of the solution are an exponential boundary layer at the left boundary and parabolic boundary layers at the lower and the upper boundary. The equation is discretized by a finite difference upwind scheme using either a Shishkin mesh which is adapted to the exponential layer or a Shishkin mesh which is adapted to the parabolic layers. The numerical solution of the discrete system is performed with domain decomposition methods. In the first method, the domain is decomposed into vertical strips (parallel to the exponential layer) and in the second method into horizontal strips (parallel to the parabolic layers). The main feature of these methods is the use of auxiliary domains which cover the interfaces of the subdomains. In the algorithm, local problems with Dirichlet boundary conditions are solved alternating on the subdomains and on the auxiliary domains. The convergence of this domain decomposition method is proved. The rate of convergence is estimated and it is shown that under certain conditions this rate is independent of the perturbation parameter. Finally, the paper presents numerical tests which confirm the analytical results. In these tests, the number of iterations for the decomposition into horizontal strips was much smaller than for the vertical decomposition.
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N50
zbMATH DE Number: 2009018
zbMATH Keywords: singularly perturbation
zbMATH Keywords: elliptic problem
zbMATH Keywords: advection-diffusion problem
zbMATH Keywords: parabolic layers
zbMATH Keywords: elliptic layer
zbMATH Keywords: domain decomposition method
zbMATH Keywords: parallel computing
zbMATH Keywords: numerical examples
zbMATH Keywords: convection-diffusion equation
zbMATH Keywords: exponential boundary layer
zbMATH Keywords: finite difference upwind scheme
zbMATH Keywords: Shishkin mesh
zbMATH Keywords: algorithm
zbMATH Keywords: convergence
reviewed by: Volker John
MaRDI profile type: Publication
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cites work: A note on the conditioning of upwind schemes on Shishkin meshes
full work available at URL: s0096-3003(02)00514-3
OpenAlex ID: W2077542633
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description: scientific article; zbMATH DE number 2009018