Spectral methods for singular perturbation problems
type: Item
instance of: scholarly article
title: Spectral methods for singular perturbation problems
zbMATH Open document ID: 0812.65100
published in: Applications of Mathematics
publication date: 1995-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/32877
review text: This is a study on spectral discretization methods (where the discretization error tends to zero faster than any power of the grid size) for singular perturbation problems, i.e. homogeneous Dirichlet problems for inhomogeneous second order differential equations with a small parameter at the second-order term in one and two dimensions, e.g. appearing as advection-diffusion equations. First, stabilization of the spectral method is discussed: The introduction of artificial viscosity stabilizes the scheme but makes the method only first-order accurate. To avoid the loss of the high spectral accuracy, the author proposes a simultaneous change of the (smooth) right-hand side, similarly to the second-order accurate ``Mehrstellen''- method due to \textit{H. Yserentant} [Numer. Math. 34, 171-187 (1980; Zbl 0413.65072)]. Numerical examples substantiate the improvement for mesh sizes with sufficiently high spectral accuracy. Second, boundary layer effects are treated by domain decomposition. The interface relaxation of the Dirichlet data follows mainly the pattern of \textit{D. Funaro}, \textit{A. Quarteroni} and \textit{P. Zanolli} [SIAM J. Numer. Anal. 25, No. 6, 1213-1236 (1988; Zbl 0678.65082)], especially in the automatic selection of relaxation parameters. The convergence properties are analyzed for one and two dimensions. Finally, an iterative solver is proposed, getting components for a multigrid method in this way.
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 704269
zbMATH Keywords: Mehrstellen-method
zbMATH Keywords: numerical examples
zbMATH Keywords: singular perturbation
zbMATH Keywords: advection- diffusion equations
zbMATH Keywords: spectral method
zbMATH Keywords: artificial viscosity
zbMATH Keywords: boundary layer
zbMATH Keywords: domain decomposition
zbMATH Keywords: relaxation
zbMATH Keywords: convergence
zbMATH Keywords: multigrid method
author: Wilhelm Heinrichs
reviewed by: E. Lanckau
MaRDI profile type: Publication
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cites work: Spectral multigrid methods for elliptic equations
cites work: Spectral multigrid methods for elliptic equations. II
OpenAlex ID: W3047863071
DOI: 10.21136/AM.1994.134251
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description: scientific article; zbMATH DE number 704269