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Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number


Authors: Xiaobing Feng and Yulong Xing
Journal: Math. Comp. 82 (2013), 1269-1296
MSC (2010): Primary 65N12, 65N15, 65N30, 78A40
DOI: https://doi.org/10.1090/S0025-5718-2012-02652-4
Published electronically: October 30, 2012
MathSciNet review: 3042564
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Abstract: This paper develops and analyzes two local discontinuous Galerkin (LDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are stable for all positive wave number $ k$ and all positive mesh size $ h$. Energy norm and $ L^2$-norm error estimates are derived for both LDG methods in all mesh parameter regimes including pre-asymptotic regime (i.e., $ k^2 h\gtrsim 1$). To analyze the proposed LDG methods, they are recast and treated as (nonconforming) mixed finite element methods. The crux of the analysis is to show that the sesquilinear form associated with each LDG method satisfies a coercivity property in all mesh parameter regimes. These coercivity properties then easily infer the desired discrete stability estimates for the solutions of the proposed LDG methods. In return, the discrete stabilities not only guarantee the well-posedness of the LDG methods but also play a crucial role in the error analysis. Numerical experiments are also presented in the paper to validate the theoretical results and to compare the performance of the proposed two LDG methods.


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Additional Information

Xiaobing Feng
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: xfeng@math.utk.edu

Yulong Xing
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996 – and – Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830
Email: xingy@math.utk.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02652-4
Keywords: Helmholtz equation, time harmonic waves, local discontinuous Galerkin methods, stability, error estimates
Received by editor(s): October 16, 2010
Received by editor(s) in revised form: August 25, 2011, and November 10, 2011
Published electronically: October 30, 2012
Additional Notes: The work of the first author was partially supported by the NSF grants DMS-0710831 and DMS-1016173. The research of the second author was partially sponsored by the Office of Advanced Scientific Computing Research; U.S. Department of Energy. The work of the second author was performed at the ORNL, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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