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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an/2011017/
[1] Finite element solution of boundary value problems, theory and computation. SIAM, Philidelphia, PA (2001). | MR | Zbl
and ,[2] Mixed and hybrid finite element methods. Springer-Verlag, New York, NY (1991). | MR | Zbl
and ,[3] Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J. Math. Phys. 35 (1994) 127-145. | MR | Zbl
and ,[4] The condition number of equivalence transformations that block diagonalize matrix pencils. SIAM J. Num. Anal. 20 (1983) 599-610. | MR | Zbl
,[5] Partial differential equations. American Mathematical Society, Providence, RI (1998). | MR | Zbl
,[6] Analytical and numerical studies of a finite element PML for the Helmholtz equation. J. Comp. Acoust. 8 (2000) 121-137. | MR
, and ,[7] On modifications of newton's second law and linear continuum elastodynamics. Proc. R. Soc. A 463 (2007) 855-880. | MR
and ,[8] Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin-Shtrikman type. Proc. R. Soc. Lond. 466 (2010) 3013-3032. | MR | Zbl
and ,[9] Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency. Proc. R. Soc. A 465 (2009) 367-396. | MR | Zbl
, and ,[10] Sound absorbing media with two types of acoustic losses. Acoust. Phys. 56 (2010) 33-36.
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