Abstract
We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L ∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter τ0 specifying the mesh. When τ0 is too small, the error becomes O(1), but for τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results.
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Stynes, M., Tobiska, L. A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numerical Algorithms 18, 337–360 (1998). https://doi.org/10.1023/A:1019185802623
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DOI: https://doi.org/10.1023/A:1019185802623