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

Elman, Howard; Ramage, Alison

doi:10.1090/S0025-5718-01-01392-8

A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation
HTML articles powered by AMS MathViewer

by Howard C. Elman and Alison Ramage PDF

Math. Comp. 72 (2003), 263-288 Request permission

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.

References

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.
P.M. Gresho and R.L. Sani, Incompressible flow and the finite element method, John Wiley and Sons, Chichester, 1999.
Claes Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1987. MR 925005
K. W. Morton, Numerical solution of convection-diffusion problems, Applied Mathematics and Mathematical Computation, vol. 12, Chapman & Hall, London, 1996. MR 1445295
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, DOI 10.1007/978-3-662-03206-0
Bill Semper, Numerical crosswind smear in the streamline diffusion method, Comput. Methods Appl. Mech. Engrg. 113 (1994), no. 1-2, 99–108. MR 1266924, DOI 10.1016/0045-7825(94)90213-5
M.R. Spiegel, Mathematical handbook of formulas and tables, Schaum’s outline series, McGraw-Hill, New York, 1990.
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 438732, DOI 10.1137/1019071