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Mathematics of Computation

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Adaptive multiresolution discontinuous Galerkin schemes for conservation laws


Authors: Nune Hovhannisyan, Siegfried Müller and Roland Schäfer
Journal: Math. Comp. 83 (2014), 113-151
MSC (2010): Primary 35L65, 65M12, 65M60, 65T60, 74S05
Published electronically: July 10, 2013
MathSciNet review: 3120584
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Abstract: A multiresolution-based adaptation concept is proposed that aims at accelerating discontinuous Galerkin schemes applied to non-linear hyperbolic conservation laws. Opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement. Instead of this, a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. A central mathematical problem addressed in this work is then to show at least for scalar one-dimensional problems that choosing an appropriate threshold value, the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh. Numerical comparisons demonstrate the efficiency of the concept.


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Additional Information

Nune Hovhannisyan
Affiliation: Faculty of Informatics and Applied Mathematics, Alex Manoogian 1, Yerevan 0025, Armenia
Email: alnune03@yahoo.com

Siegfried Müller
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany
Email: mueller@igpm.rwth-aachen.de

Roland Schäfer
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany
Email: schaefer@igpm.rwth-aachen.de

DOI: https://doi.org/10.1090/S0025-5718-2013-02732-9
Received by editor(s): September 13, 2010
Received by editor(s) in revised form: October 20, 2011, March 6, 2012, and June 19, 2012
Published electronically: July 10, 2013
Additional Notes: This work has been performed with funding by the Deutsche Forschungsgemeinschaft in the Collaborative Research Center SFB 401 “Flow Modulation and Fluid-Structure Interaction at Airplane Wings” of RWTH Aachen University, Germany.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.