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Mathematics of Computation

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Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems


Authors: Slimane Adjerid and Thomas Weinhart
Journal: Math. Comp. 80 (2011), 1335-1367
MSC (2010): Primary 65M60, 65N35; Secondary 35L50
DOI: https://doi.org/10.1090/S0025-5718-2011-02460-9
Published electronically: January 25, 2011
MathSciNet review: 2785461
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Abstract: We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetrizable 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 show that projections of the error are pointwise $ \mathcal{O}(h^{p+2})$-superconvergent. 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.


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Additional Information

Slimane Adjerid
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Thomas Weinhart
Affiliation: Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands

DOI: https://doi.org/10.1090/S0025-5718-2011-02460-9
Keywords: Discontinuous Galerkin method, linear hyperbolic systems, Friedrichs symmetrizable systems, \textit{a posteriori} error estimation, superconvergence.
Received by editor(s): September 10, 2009
Received by editor(s) in revised form: June 17, 2010
Published electronically: January 25, 2011
Additional Notes: This research was partially supported by the National Science Foundation (Grant Numbers DMS 0511806, DMS 0809262).
Article copyright: © Copyright 2011 American Mathematical Society