Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations
type: Item
instance of: scholarly article
title: Finite element analysis of an exponentially fitted non-lumped scheme for advection-diffusion equations
zbMATH Open document ID: 0826.65089
DOI: 10.1016/0168-9274(94)00050-6
author: Wen Guo
author: Martin Stynes
published in: Applied Numerical Mathematics
publication date: 1995-11-26T00:00:00Z
review text: The analysis of a finite element method for the numerical solution of singulary perturbed problems which occur in fluid mechanics, gas dynamics and wave propagation is considered. Problems of this form are of mixed hyperbolic-parabolic type. Finite element techniques to examine a difference scheme generated by the Petrov-Galerkin method for advection- diffusion equations in one space variable are used. The scheme is exponentially fitted in the \(x\)-direction. The present paper is devoted to the analysis of a non-lumped method. The authors show that for constant coefficients, when the Courant number is equal to one and the mesh is coarse, the non-lumped scheme is locally higher-order convergent, whereas the lumped scheme is locally only first- order convergent. Under reasonable global assumptions on the behaviour of the exact solution and a certain stability constant, the scheme is shown to be convergent, uniformly in the diffusion parameter, in global energy and \(L^2\) norms and a local discrete \(L^\infty\) norm. These local estimates show that the method does not smear interior and boundary layers excessively and that it is first-order accurate, away from layers, on arbitrary equidistant meshes. Numerical results support these conclusions.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35M10
zbMATH DE Number: 699086
zbMATH Keywords: singular perturbation
zbMATH Keywords: convergence
zbMATH Keywords: numerical results
zbMATH Keywords: finite element method
zbMATH Keywords: mixed hyperbolic-parabolic type
zbMATH Keywords: difference scheme
zbMATH Keywords: Petrov- Galerkin method
zbMATH Keywords: advection-diffusion equations
zbMATH Keywords: non-lumped method
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: 0168-9274(94)00050-6
OpenAlex ID: W1994630953
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description: scientific article; zbMATH DE number 699086