We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.
Mots-clés : variational methods, Helmholtz equation, finite element methods
@article{M2AN_2012__46_1_39_0,
author = {Richins, Russell B. and Dobson, David C.},
title = {A numerical minimization scheme for the complex {Helmholtz} equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {39--57},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/m2an/2011017},
mrnumber = {2846366},
zbl = {1272.65095},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011017/}
}
TY - JOUR AU - Richins, Russell B. AU - Dobson, David C. TI - A numerical minimization scheme for the complex Helmholtz equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 39 EP - 57 VL - 46 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011017/ DO - 10.1051/m2an/2011017 LA - en ID - M2AN_2012__46_1_39_0 ER -
%0 Journal Article %A Richins, Russell B. %A Dobson, David C. %T A numerical minimization scheme for the complex Helmholtz equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 39-57 %V 46 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011017/ %R 10.1051/m2an/2011017 %G en %F M2AN_2012__46_1_39_0
Richins, Russell B.; Dobson, David C. A numerical minimization scheme for the complex Helmholtz equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 39-57. doi : 10.1051/m2an/2011017. http://www.numdam.org/articles/10.1051/m2an/2011017/
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