Optimal Order $L^2$ Error Estimate of SDFEM on Shishkin Triangular Meshes for Singularly Perturbed Convection-Diffusion Equations

Abstract

In this paper, we provide a convergence analysis for a streamline diffusion finite element method (SDFEM) for the singularly perturbed convection-diffusion equation on a Shishkin triangular mesh. The main result is to show that the SDFEM solution on the triangular mesh has the optimal order $L^2$ accuracy. The argument relies on a series of novel integral inequalities, which give the delicate estimations for the error terms related to the convection. Numerical experiments illustrate the proved results.