Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T07:57:29.666Z Has data issue: false hasContentIssue false
  • English
  • Français

A numerical minimization scheme for the complexHelmholtz equation

Published online by Cambridge University Press:  22 July 2011

Russell B. Richins and
David C. Dobson
Russell B. Richins
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, 48824 Michigan, USA. richins@math.msu.edu
David C. Dobson
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, 84112 Utah, USA. dobson@math.utah.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Keywords

Variational methodsHelmholtz equationfinite element methods

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012, pp. 39 - 57
Copyright
© EDP Sciences, SMAI, 2011

References

O. Axelsson and V.A. Barker, Finite element solution of boundary value problems, theory and computation. SIAM, Philidelphia, PA (2001).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York, NY (1991).
Cherkaev, A.V. and Gibiansky, L.V., Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J. Math. Phys. 35 (1994) 127–145. CrossRef
Demmel, J., The condition number of equivalence transformations that block diagonalize matrix pencils. SIAM J. Num. Anal. 20 (1983) 599–610. CrossRef
L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI (1998).
Harari, I., Slavutin, M. and Turkel, E., Analytical and numerical studies of a finite element PML for the Helmholtz equation. J. Comp. Acoust. 8 (2000) 121–137. CrossRef
Milton, G.W. and Willis, J.R., On modifications of newton's second law and linear continuum elastodynamics. Proc. R. Soc. A 463 (2007) 855–880. CrossRef
Milton, G.W. and Willis, J.R., 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. CrossRef
Milton, G.W., Seppecher, P. and Bouchitté, G., Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency. Proc. R. Soc. A 465 (2009) 367–396. CrossRef
Tyutekin, V.V. and Tyutekin, Y.V., Sound absorbing media with two types of acoustic losses. Acoust. Phys. 56 (2010) 33–36. CrossRef