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

Howard C. Elman, Alison Ramage

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)263-288
Number of pages26
JournalMathematics of Computation
Issue number241
Early online date4 Dec 2001
Publication statusPublished - 1 Jan 2003


  • computation mathematics
  • convection-diffusion equations
  • oscillations


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