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On the relationship of local projection stabilization to other stabilized methods for one-dimentional advection-diffusion equations

by Tobiska, Lutz.

Series: 2008-14, Preprints

65L20 Stability and convergence of numerical methods
35B25 Singular perturbations
65L60 Finite elements, ~Rayleigh-Ritz, Galerkin and collocation methods

We consider a singularly perturbed advection-diffusion two-point boundary value problem whose solution exhibits layers. Eliminating the enrichments in the one-level approach of the local projection stabilization we end up with the differentiated residual method (DRM) which coincide for piecewise linears with the streamline upwind Petrov-Galerkin (SUPG) method and for piecwise polynomials of degree r > 2 with the variational multiscale method (VMS). Furthermore, we show that in certain cases the stabilization perameter can be chosen in such a way that the piecwise linear part of the solution becomes nodal exact. In this way, we obtain explicite formulas for the stabilization parameter depending on the local meshsize, the polynomial degree r of the approximation space, and the data of the problem. We discuss the behaviour of different modes of the discrete solution for varying stabilization parameters.

stabilized finite element method, singular perturbation, advection-diffusion equation