Asymptotically exact discontinuous Galerkin error estimates for linear symmetric hyperbolic systems

Authors:
Slimane Adjerid;Thomas Weinhart
Affiliations:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, United States;Multiscale Mechanics, CTW and MESA+, University of Twente, PO Box 217, 7500AE Enschede, The Netherlands
Venue:
Applied Numerical Mathematics
Year:
2014

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Abstract

We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p and p+1. We apply these asymptotic results to observe that projections of the error are pointwise O(h^p^+^2)-superconvergent in some cases. Then we solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.