Resumen
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 ...u^I - u^N..._SD, where u^I is the interpolant of the solution u, u^N is the SDFEM solution, and ...·..._SD is the streamline-diffusion norm. This bound implies that ...u - u^N..._L² is of optimal order, thereby settling an open question regarding the L2-accuracy of the SDFEM on rectangular meshes. Furthermore, the bound shows that u^N 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. [ABSTRACT FROM AUTHOR]