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Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation


Authors: Bernardo Cockburn and Vincent Quenneville-Bélair
Journal: Math. Comp. 83 (2014), 65-85
MSC (2010): Primary 65M60, 65N30, 65M15, 65M22
DOI: https://doi.org/10.1090/S0025-5718-2013-02743-3
Published electronically: July 16, 2013
MathSciNet review: 3120582
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Abstract: We present the first a priori error analysis of the hybridizable discontinuous Galerkin methods for the acoustic equation in the time-continuous case. We show that the velocity and the gradient converge with the optimal order of $ k+1$ in the $ L^2$-norm uniformly in time whenever polynomials of degree $ k \geq 0$ are used. Finally, we show how to take advantage of this local postprocessing to obtain an approximation to the original scalar unknown also converging with order $ k+2$ for $ k \geq 1$. This puts on firm mathematical ground the numerical results obtained in J. Comput. Phys. 230 (2011), 3695-3718.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Vincent Quenneville-Bélair
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: vqb@math.umn.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02743-3
Keywords: Discontinuous Galerkin methods, hybridization, superconvergence, hyperbolic problems
Received by editor(s): October 14, 2011
Received by editor(s) in revised form: January 26, 2012
Published electronically: July 16, 2013
Additional Notes: The first author was partially supported by the National Science Foundation (Grant DMS-1115331) and by the Minnesota Supercomputing Institute.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.