by Schieweck, F, Skrzypacz, P..

**Series:** 2012-02, Preprints

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

**Abstract:**

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.

Whereas

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.

**Keywords:**

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