On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws

Giesselmann, J. and Pryer, T. (2014) On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws. In: Fuhrmann, J., Ohlberger, M. and Rohde, C. (eds.) Finite volumes for complex applications VII-methods and theoretical aspects: FVCA 7, Berlin, June 2014. Springer proceedings in mathematics and statistics, 77. Springer International, Cham, Switzerland, pp. 313-321. ISBN 9783319056838 doi: 10.1007/978-3-319-05684-5_30

Abstract/Summary

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the L1 contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in L2, based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.