### 2011-01

by Juhnke, D., Tobiska, L..

**Series:** 2011-01, Preprints

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

**Abstract:**

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

enrichments.

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.

**Keywords:**

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