06-08
Using rectangular $Q_p$ elements in the SDFEM for a convection-diffusion problem with a boundary layer
Preprint series: 06-08, Preprints
- MSC:
- 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
- 65N15 Error bounds
Abstract: The streamline diffusion finite element method (SDFEM; the method is also known as SUPG) is applied to a convection-diffusion problem posed on the unit square whose solution has exponential boundary layers. A rectangular Shishkin mesh is used. The trial functions in the SDFEM are piecewise polynomials that lie in the space $Q_p$, i.e., are tensor products of polynomials of degree $p$ in one variable, where $p>1$. The error bound $\|I_N u-u^N\|_{SD}\le C N^{-(p+1/2)}$ is proved; here $u^N$ is the computed SDFEM solution, $I_N u$ is chosen in the finite element space to be a special approximant of the true solution $u$, and $\|\cdot\|_{SD}$ is the streamline-diffusion norm. This result is compared with previously known results for the case $p=1$. The error bound is a superclose result; $u^N$ can be enhanced using local postprocessing to yield a modified solution $\tilde u^N$ for which $\|u-\tilde u^N\|_{SD}\le C N^{-(p+1/2)}$.
Keywords: convection-diffusion problems, streamline-diffusion method, finite elements, error estimates
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