Convection-diffusion problems, SDFEM/SUPG and a priori meshes

This paper aims to give the reader a summary of current understanding of the streamlinediffusion finite element method (SDFEM), as applied to linear steady-state convection-diffusion problems. Towards this end, we begin with a brief description of the nature of convectiondiffusion problems: the structure of their solutions will be examined, with special emphasis on the main phenomena of exponential and characteristic/parabolic layers. See [34] for a more leisurely and detailed exposition of this material. Next, Shishkin meshes will be presented and discussed. These piecewise-uniform meshes are suited to the numerical solution of convection-diffusion problems with boundary layers. Further information on them appears in [7, 27, 30, 31, 34]. Finally, we come to the Streamline Diffusion Finite Element Method (SDFEM), which is also known as the Streamline-Upwinded Petrov-Galerkin method (SUPG). Since its inception [12] in 1979, this method has been the subject of a huge number of theoretical analyses and numerical investigations that continue to this day; see the references in [30, 31, 34]. We shall give a comprehensive survey of the application of the method to convection-diffusion problems, including discussions of its strengths and weaknesses, and present recent theoretical results.