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A numerical minimization scheme for the complex Helmholtz equation

  • Russell B. Richins (a1) and David C. Dobson (a2)

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.

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[2] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York, NY (1991).
[3] 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) 127145.
[4] Demmel, J., The condition number of equivalence transformations that block diagonalize matrix pencils. SIAM J. Num. Anal. 20 (1983) 599610.
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[6] 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) 121137.
[7] Milton, G.W. and Willis, J.R., On modifications of newton's second law and linear continuum elastodynamics. Proc. R. Soc. A 463 (2007) 855880.
[8] 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) 30133032.
[9] 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) 367396.
[10] Tyutekin, V.V. and Tyutekin, Y.V., Sound absorbing media with two types of acoustic losses. Acoust. Phys. 56 (2010) 3336.

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A numerical minimization scheme for the complex Helmholtz equation

  • Russell B. Richins (a1) and David C. Dobson (a2)

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