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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 L²-orthogonal local basis functions can be constructed. L²-orthogonal basis functions lead to a diagonal mass matrix which is advantageous for time discretization. We use the discontinuous Galerkin method of polynomial order one for the discretization in time which is superconvergent of order three at the endpoints of the time intervals. 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.

A Local Projection Stabilization Method with Shock Capturing and Diagonal Mass Matrix for Solving Non-stationary Transport Dominated Problems