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Abstract
The usual variational (or weak) formulations of the Helmholtz equation are signindefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be signdefinite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new signdefinite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a starshaped domain with impedance boundary conditions or the exterior of a starshaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise from simply multiplying the Helmholtz equation by particular test functions and integrating by parts.
Original language  English 

Pages (fromto)  274312 
Number of pages  39 
Journal  Siam Review 
Volume  56 
Issue number  2 
Early online date  8 May 2014 
DOIs  
Publication status  Published  2014 
Keywords
 Helmholtz equation
 high frequency
 coercivity
 signdefiniteness
 Morawetz identity
 frequencyexplicit analysis
 Finite element method
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 1 Finished

Post Doc Fellowship  New Methods and Analysis for Wave Propagation Problems
Engineering and Physical Sciences Research Council
1/04/11 → 31/03/14
Project: Research council