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A Local Projection Stabilization method with shock capturing and diagonal mass matrix for solving non-stationary transport dominated problems

by Schieweck, F, Skrzypacz, P..

Series: 2012-02, Preprints

65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

We consider a time-dependent convection diffusion equation in the
transport dominated case.
As a stabilization method in space we propose a new variant of Local
Projection Stabilization (LPS) which uses special enriched bubble functions
such that $L^2$-orthogonal local basis functions can be constructed.
$L^2$-orthogonal basis functions lead to a diagonal mass matrix which is
advantageous for time discretization.
We use the discontinuous Galerkin method of order one for the
discretization in time.
In order to avoid the remaining oscillations in the LPS-solution we
add for each time step
in the space discretization an extra shock capturing term which acts only
locally on those mesh cells where an error-indicator is relatively large.
The novelty in the shock capturing term is that the scaling factor in
front of the additive diffusion term is computed from a
low order post-processing error.
As a result we obtain both, an oscillation-free discrete solution and
the information about the local regions where this solution is still
inaccurate due to some smearing. The latter information can be used to
create in each time step an adaptively refined space mesh.
the numerical experiments are restricted to one space dimension
the proposed ideas work also in the multi-dimensional spatial case.
The numerical tests show that the discrete solution with shock capturing
is oscillation-free and of optimal accuracy in the regions outside of
the shock.

Local Projection Stabilization, discontinuous Galerkin time discretization, shock capturing, post-processing, error indicator