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A local projection type stabilization with exponential enrichments applied to one-dimensional advection-diffusion equations

by Juhnke, D., Tobiska, L..

Series: 2011-01, Preprints

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

We consider the local projection stabilization (LPS) for solving a singularly
perturbed advection-diffusion two-point boundary value problem.
In its classical one-level variant, the LPS uses polynomial
bubble functions to enrich the standard finite element spaces of continuous,
piecewise polynomial functions. As recently shown, the two-level approach can
be considered also as a one-level method, however, with piecewise polynomial
Here, we study the question under which
condition a linearly independent $H^1$~function can serve as an enrichment
for the standard space of continuous, piecewise polynomials of degree $r$
leading to the same type of error estimates for the solution as the original
one-~and two-level approaches. Moreover, in the constant coefficient case,
we derive formulas for the user-chosen stabilization parameter which
guarantee that the piecewise linear part of the solution becomes nodal exact.
Finally, we choose exponential enrichments based on the asymptotic expansion
of the solution and show by numerical tests that compared to the classical
one-level variant of the LPS -- a considerable improvement of the accuracy
of the solution on non-layer adapted meshes can be achieved.

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