Numerical Analysis Seminars: Oscillations and their Cure in Discrete Solutions to the Convection-Diffusion Equation
16 March 2001 14:00 in CM105"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 discretization. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this work, 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 discretizations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions that are oscillatory when the mesh Peclet number is large. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Peclet number and boundary conditions of the problem. When streamline upwinding is included in the discretization, we show precisely how the amount of upwinding included in the discrete operator affects solution oscillations and accuracy when boundary layers are present.
Joint work with Alison Ramage of The University of Strathclyde."
Contact David.Bourne@durham.ac.uk for more information