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

Published electronically:
October 30, 2012

MathSciNet review:
3042564

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 and all positive mesh size . Energy norm and -norm error estimates are derived for both LDG methods in all mesh parameter regimes including pre-asymptotic regime (i.e., ). 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.

**1.**Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini,*Unified analysis of discontinuous Galerkin methods for elliptic problems*, SIAM J. Numer. Anal.**39**(2001/02), no. 5, 1749–1779. MR**1885715**, 10.1137/S0036142901384162**2.**Susanne C. Brenner and L. Ridgway Scott,*The mathematical theory of finite element methods*, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR**2373954****3.**P. Castillo, B. Cockburn, I. Perugia and D. Schötzau.

Local discontinuous Galerkin method for elliptic problems.*Commun. Numer. Meth. Engrg.*, 18:69-75, 2002.**4.**Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov,*Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems*, SIAM J. Numer. Anal.**47**(2009), no. 2, 1319–1365. MR**2485455**, 10.1137/070706616**5.**Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu (eds.),*Discontinuous Galerkin methods*, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, Berlin, 2000. Theory, computation and applications; Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999. MR**1842160****6.**Bernardo Cockburn and Chi-Wang Shu,*The local discontinuous Galerkin method for time-dependent convection-diffusion systems*, SIAM J. Numer. Anal.**35**(1998), no. 6, 2440–2463 (electronic). MR**1655854**, 10.1137/S0036142997316712**7.**Peter Cummings and Xiaobing Feng,*Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations*, Math. Models Methods Appl. Sci.**16**(2006), no. 1, 139–160. MR**2194984**, 10.1142/S021820250600108X**8.**Jim Douglas Jr., Juan E. Santos, Dongwoo Sheen, and Lynn Schreyer Bennethum,*Frequency domain treatment of one-dimensional scalar waves*, Math. Models Methods Appl. Sci.**3**(1993), no. 2, 171–194. MR**1212938**, 10.1142/S0218202593000102**9.**Björn Engquist and Andrew Majda,*Radiation boundary conditions for acoustic and elastic wave calculations*, Comm. Pure Appl. Math.**32**(1979), no. 3, 314–358. MR**517938**, 10.1002/cpa.3160320303**10.**Xiaobing Feng and Haijun Wu,*Discontinuous Galerkin methods for the Helmholtz equation with large wave number*, SIAM J. Numer. Anal.**47**(2009), no. 4, 2872–2896. MR**2551150**, 10.1137/080737538**11.**Xiaobing Feng and Haijun Wu,*ℎ𝑝-discontinuous Galerkin methods for the Helmholtz equation with large wave number*, Math. Comp.**80**(2011), no. 276, 1997–2024. MR**2813347**, 10.1090/S0025-5718-2011-02475-0**12.**X. Feng and Y. Xing.

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

arXiv:1010.4563v1 [math.NA].**13.**R. Griesmaier and P. Monk.

Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation.*J. Scient. Computing*, 49:291-310, 2011.**14.**U. Hetmaniuk,*Stability estimates for a class of Helmholtz problems*, Commun. Math. Sci.**5**(2007), no. 3, 665–678. MR**2352336****15.**Ralf Hiptmair and Ilaria Perugia,*Mixed plane wave discontinuous Galerkin methods*, Domain decomposition methods in science and engineering XVIII, Lect. Notes Comput. Sci. Eng., vol. 70, Springer, Berlin, 2009, pp. 51–62. MR**2743958**, 10.1007/978-3-642-02677-5_5**16.**Frank Ihlenburg,*Finite element analysis of acoustic scattering*, Applied Mathematical Sciences, vol. 132, Springer-Verlag, New York, 1998. MR**1639879****17.**F. Ihlenburg and I. Babuška,*Finite element solution of the Helmholtz equation with high wave number. I. The ℎ-version of the FEM*, Comput. Math. Appl.**30**(1995), no. 9, 9–37. MR**1353516**, 10.1016/0898-1221(95)00144-N**18.**Teemu Luostari, Tomi Huttunen, and Peter Monk,*Plane wave methods for approximating the time harmonic wave equation*, Highly oscillatory problems, London Math. Soc. Lecture Note Ser., vol. 366, Cambridge Univ. Press, Cambridge, 2009, pp. 127–153. MR**2562508****19.**Béatrice Rivière,*Discontinuous Galerkin methods for solving elliptic and parabolic equations*, Frontiers in Applied Mathematics, vol. 35, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and implementation. MR**2431403****20.**O. C. Zienkiewicz,*Achievements and some unsolved problems of the finite element method*, Internat. J. Numer. Methods Engrg.**47**(2000), no. 1-3, 9–28. Richard H. Gallagher Memorial Issue. MR**1744287**, 10.1002/(SICI)1097-0207(20000110/30)47:1/3<9::AID-NME793>3.0.CO;2-P

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
65N12,
65N15,
65N30,
78A40

Retrieve articles in all journals with MSC (2010): 65N12, 65N15, 65N30, 78A40

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:
http://dx.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.