05-36
Analysis of a new stabilized higher order finite element method for advection-diffusion equations
by Tobiska, L.
Preprint series: 05-36, Preprints
- MSC:
- 65N12 Stability and convergence of numerical methods
- 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Abstract: We consider a singularly perturbed advection-diffusion two-point boundary value problem whose solution has a single boundary layer. Based on piecewise polynomial approximations of degree $k\ge 1$, a new stabilized finite element method is derived in the framework of a variational multiscale approach. The method coincides with the SUPG method for $k=1$ but differs from it for $k\ge 2$. Estimates for the error to an appropriate interpolant are given in several norms on different types of meshes. For $k=1$ enhanced accuracy is achieved by superconvergence. Postprocessing guarantees the same estimates for the error to the solution itself.
Keywords: stabilized finite element method, singular perturbation, advection-diffusion, Shishkin mesh, superconvergence, postprocessing
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