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A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

Authors: Howard C. Elman and Alison Ramage
Journal: Math. Comp. 72 (2003), 263-288
MSC (2000): Primary 65N22, 65N30, 65Q05, 35J25
Published electronically: December 4, 2001
MathSciNet review: 1933821
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Abstract: It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

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Additional Information

Howard C. Elman
Affiliation: Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742

Alison Ramage
Affiliation: Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, Scotland

Keywords: Convection-diffusion equation, oscillations, Galerkin finite element method
Received by editor(s): March 27, 2000
Received by editor(s) in revised form: February 22, 2001
Published electronically: December 4, 2001
Additional Notes: The work of the first author was supported by National Science Foundation grant DMS9972490.
The work of the second author was supported by the Leverhulme Trust.
Article copyright: © Copyright 2001 American Mathematical Society