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Analysis of a new stabilized higher order finite element method for advection-diffusion equations

by Tobiska, L..

Series: 2005-36, Preprints

65N12 Stability and convergence of numerical methods
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

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.

stabilized finite element method, singular perturbation, advection-diffusion, Shishkin mesh, superconvergence, postprocessing