An Optimal Streamline Diffusion Finite Element Method for a Singularly Perturbed Problem

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 − uh‖∞ ≤ C infvh∈V h ‖u − vh‖∞, where uh is the SDFEM approximation of the exact solution u and Vh 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.

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