The hp Streamline Diffusion Finite Element Method for Convection Dominated Problems in one Space Dimension

Abstract

We analyze the hp Streamline Diffusion Finite Element Method (SDFEM) and the standard Galerkin FEM for one dimensional stationary convection-diffusion problems. Under the assumption of analyticity of the input data, a mesh is exhibited on which approximation with continuous piecewise polynomials of degree p allows for resolution of the boundary layer. On such meshes, both the SDFEM and the Galerkin FEM lead to robust exponential convergence in the "energy norm" and in the $L^\infty$ norm. Next, we show that even in the case that the boundary layers are not resolved, robust exponential convergence on compact subsets "upstream" of the layer can be achieved with the hp-SDFEM. This is possible on sequences of meshes that would typically be generated by an hp-adaptive scheme. Detailed numerical experiments confirm our convergence estimates.