A numerical minimization scheme for the complex Helmholtz equation

Russell B. Richins; David C. Dobson

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

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

Abstract

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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.

How to cite

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Richins, 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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  1. [1] O. Axelsson and V.A. Barker, Finite element solution of boundary value problems, theory and computation. SIAM, Philidelphia, PA (2001). Zbl0981.65130MR1856818
  2. [2] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York, NY (1991). Zbl0788.73002MR1115205
  3. [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
  4. [4] J. Demmel, The condition number of equivalence transformations that block diagonalize matrix pencils. SIAM J. Num. Anal.20 (1983) 599–610. Zbl0509.65020MR701100
  5. [5] L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI (1998). Zbl1194.35001MR1625845
  6. [6] I. Harari, M. Slavutin and E. Turkel, Analytical and numerical studies of a finite element PML for the Helmholtz equation. J. Comp. Acoust.8 (2000) 121–137. MR1766332
  7. [7] G.W. Milton and J.R. Willis, On modifications of newton's second law and linear continuum elastodynamics. Proc. R. Soc. A463 (2007) 855–880. Zbl05233344MR2293080
  8. [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. [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. [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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