Third M.I.T. Conference on Computational Fluid and Solid Mechanics June 14–17, 2005  

A new stable discontinuous Galerkin approximation for non-linear conservation laws on adaptively refined grids

Andreas Dednera, Charalambos Makridakisb, Mario Ohlbergera,*
aAbteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Str.10, D-79104 Freiburg, Germany  bDepartment of Applied Mathematics, University of Crete, GR-71409 Heraklion, Greece
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ABSTRACT
We propose an a-posteriori error estimate for the semi-discrete discontinuous Galerkin (DG) method of arbitrary order in arbitrary space dimensions. For stabilization of the scheme a general framework of projections is introduced. Finally it is demonstrated numerically how the a posteriori error estimate is used for defining appropriate projection operators and in order to design an efficient grid adaption strategy. Numerical experiments show the gain in efficiency in comparison with computations on uniform grids.

Keywords:  Discontinuous Galerkin; Higher order; Adaptive methods; Error estimate; Finite element; Conservation laws
* Corresponding author. Tel.: +49 76 12035642; Fax: +49 76 12035632; E-mail: mario@mathematik.uni-freibrug.de