# A numerical minimization scheme for the complex Helmholtz equation

Russell B. Richins; David C. Dobson

- Volume: 46, Issue: 1, page 39-57
- ISSN: 0764-583X

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topRichins, Russell B., and Dobson, David C.. "A numerical minimization scheme for the complex Helmholtz equation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.1 (2012): 39-57. <http://eudml.org/doc/273270>.

@article{Richins2012,

abstract = {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.},

author = {Richins, Russell B., Dobson, David C.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {variational methods; Helmholtz equation; finite element methods; error bound; numerical experiments},

language = {eng},

number = {1},

pages = {39-57},

publisher = {EDP-Sciences},

title = {A numerical minimization scheme for the complex Helmholtz equation},

url = {http://eudml.org/doc/273270},

volume = {46},

year = {2012},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2012

PB - EDP-Sciences

VL - 46

IS - 1

SP - 39

EP - 57

AB - 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.

LA - eng

KW - variational methods; Helmholtz equation; finite element methods; error bound; numerical experiments

UR - http://eudml.org/doc/273270

ER -

## References

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- [2] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York, NY (1991). Zbl0788.73002MR1115205
- [3] A.V. Cherkaev and L.V. Gibiansky, Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J. Math. Phys.35 (1994) 127–145. Zbl0805.49028MR1252102
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- [8] G.W. Milton and J.R. Willis, 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. Zbl1211.74180MR2684717
- [9] G.W. Milton, P. Seppecher and G. Bouchitté, Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency. Proc. R. Soc. A465 (2009) 367–396. Zbl1186.74044MR2471764
- [10] V.V. Tyutekin and Y.V. Tyutekin, Sound absorbing media with two types of acoustic losses. Acoust. Phys.56 (2010) 33–36.

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