The SDFEM for a convection-diffusion problem with a boundary layer: Optimal error analysis and enhancement of accuracy

摘要

The streamline-diffusion finite element method (SDFEM) is applied to a convection-diffusion problem posed on the unit square, using a Shishkin rectangular mesh with piecewise bilinear trial functions. The hypotheses of the problem exclude interior layers but allow exponential boundary layers. An error bound is proved for parallel tou(I) - u(N)parallel to(SD), where u(I) is the interpolant of the solution u, u(N) is the SDFEM solution, and parallel to . parallel to(SD) is the streamline-diffusion norm. This bound implies that parallel tou - u(N)parallel to(L2) is of optimal order, thereby settling an open question regarding the L-2-accuracy of the SDFEM on rectangular meshes. Furthermore, the bound shows that uN is superclose to u(I), which allows the construction of a simple postprocessing that yields a more accurate solution. Enhancement of the rate of convergence by using a discrete streamline-diffusion norm is also discussed. Finally, the verification of these rates of convergence by numerical experiments is examined, and it is shown that this practice is less reliable than was previously believed. [References: 25]