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On the role of boundary conditions for CIP stabilization of higher order finite elements

by Schieweck, F..

Series: 2007-41, Preprints

65N15 Error bounds
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65N50 Mesh generation and refinement

We investigate the
Continuous Interior Penalty (CIP) stabilization method
for higher order finite elements applied to a convection diffusion
equation with a small diffusion parameter $\varepsilon$.
Performing numerical experiments,
it turns out that strongly imposed Dirichlet boundary conditions lead to
relatively bad numerical solutions.
However, if the Dirichlet boundary conditions are imposed
on the inflow part of the boundary in a weak sense and additionally on
the whole boundary in an $\varepsilon$-weighted weak sense due to Nitsche
then one obtains reasonable numerical results.
In many cases, this holds even in the limit case where the parameter of
the CIP stabilization is zero, i.e., where the standard Galerkin
discretization is applied.
We present an analysis which explains this effect.

diffusion-convection-reaction equation, finite elements, Nitsche type boundary conditions, Continuous Interior Penalty method, error estimates