Report 1998-10

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

J.M. Melenk and C. Schwab

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.

Paper: Available as PostScript (708 KB) or as hardcopy to order reports@sam.math.ethz.ch.

Publishing information: East-West J. Numer. Anal., 7 (1999), 31-60