NDLI: Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Feng, Xiaobing Wu, Haijun
Copyright Year	2009
Abstract	This paper develops and analyzes some interior penalty discontinuous Galerkin (IPDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number k, optimal order (with respect to h) error estimate in the broken $H^1$-norm and suboptimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to k) in the error bounds. The novelties of the proposed IPDG methods include the following: First, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To this end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [P. Cummings, Analysis of Finite Element Based Numerical Methods for Acoustic Waves, Elastic Waves and Fluid-Solid Interactions in the Frequency Domain, Ph.D. thesis, The University of Tennessee, Knoxville, TN, 2001], [P. Cummings and X. Feng, Math. Models Methods Appl. Sci., 16 (2006), pp. 139160], [U. Hetmaniuk, Commun. Math. Sci., 5 (2007), pp. 665678].
Starting Page	2872
Ending Page	2896
Page Count	25
File Format	PDF
ISSN	00361429
DOI	10.1137/080737538
e-ISSN	10957170
Issue Number	4
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-08-14
Access Restriction	Subscribed
Subject Keyword	Stability and convergence of numerical methods absorbing boundary conditions Waves and radiation time harmonic waves Helmholtz equation Error bounds Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods discontinuous Galerkin methods error estimates
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in