C3

Long Chen and Jinchao Xu. An Optimal Streamline Diffusion Finite Element Method for a Singularly Perturbed Problem. AMS Contemporary Mathematics Series: Recent Advances in Adaptive Computation, 383:236--246, 2005.

Chen.L%3BXu.J2005.pdf

ABSTRACT: The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a special choice of the stabilization quadratic bubble function, the SDFEM is shown to have an optimal second order in the sense that $\|u-u_{h}\|_{\infty}\leq C\inf_{v_{h}\in V^{h}}\|u-v_{h}\|_{\infty},$ where $u_{h}$ is the SDFEM approximation of the exact solution $u$ and $V_{h}$ is the linear finite element space. With the quasi-optimal interpolation error estimate, quasi-optimal convergence results for the SDFEM are obtained. As a consequence, an open question about the optimal choice of the monitor function for a second order scheme in the moving mesh method is answered.