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 | References | Similar Articles | Additional Information

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.

**1.**H.C. Elman and A. Ramage,*An analysis of smoothing effects of upwinding strategies for the convection-diffusion equation*, Tech. Report UMCP-CSD:CS-TR-4160, University of Maryland, College Park MD 20742, 2000.**2.**P.M. Gresho and R.L. Sani,*Incompressible flow and the finite element method*, John Wiley and Sons, Chichester, 1999.**3.**Claes Johnson,*Numerical solution of partial differential equations by the finite element method*, Cambridge University Press, Cambridge, 1987. MR**925005****4.**K. W. Morton,*Numerical solution of convection-diffusion problems*, Applied Mathematics and Mathematical Computation, vol. 12, Chapman & Hall, London, 1996. MR**1445295****5.**H.-G. Roos, M. Stynes, and L. Tobiska,*Numerical methods for singularly perturbed differential equations*, Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems. MR**1477665****6.**Bill Semper,*Numerical crosswind smear in the streamline diffusion method*, Comput. Methods Appl. Mech. Engrg.**113**(1994), no. 1-2, 99–108. MR**1266924**, 10.1016/0045-7825(94)90213-5**7.**M.R. Spiegel,*Mathematical handbook of formulas and tables*, Schaum's outline series, McGraw-Hill, New York, 1990.**8.**Paul N. Swarztrauber,*The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle*, SIAM Rev.**19**(1977), no. 3, 490–501. MR**0438732**

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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

Email:
elman@cs.umd.edu

**Alison Ramage**

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

Email:
alison@maths.strath.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-01-01392-8

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