Xia Ji
Abstract
Transmission eigenvalue problem has important applications in inverse scattering. Since the problem is non-self-adjoint, the computation of transmission eigenvalues needs special treatment. Based on a fourth-order reformulation of the transmission eigenvalue problem, a mixed finite element method is applied. The method has two major advantages: 1) the formulation leads to a generalized eigenvalue problem naturally without the need to invert a related linear system, and 2) the nonphysical zero transmission eigenvalue, which has an infinitely dimensional eigenspace, is eliminated. To solve the resulting non-Hermitian eigenvalue problem, an iterative algorithm using restarted Arnoldi method is proposed. To make the computation efficient, the search interval is decided using a Faber-Krahn type inequality for transmission eignevalues and the interval is updated at each iteration. The algorithm is implemented using Matlab. The code can be easily used in the qualitative methods in inverse scattering and modified to compute transmission eigenvalues for other models such as elasticity problem.
Supplemental Material
Available for Download
Software for A Mixed Finite Element Method for Helmholtz Transmission Eigenvalues
The proof is given in an electronic appendix, available online in the ACM Digital Library.
- Argyris, J. H., Fried, I., and Scharpf, D. W. 1968. The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72, 701--709.Google Scholar
Cross Ref
- Cakoni, F. and Gintides, D. 2010. New results on transmission eigenvalues. Inverse Prob. Imaging 4, 1, 39--48.Google ScholarCross Ref
- Cakoni, F. and Hadder, H. 2009. On the existence of transmission eigenvalues in an inhomogenous medium. Appl. Anal. 88, 4, 475--493.Google ScholarCross Ref
- Cakoni, F., Colton, D., Monk, P., and Sun, J. 2010. The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems 26, 7, 074004.Google ScholarCross Ref
- Ciarlet, P. G. 2002. The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics 40, SIAM, Philadelphia, PA. Google ScholarDigital Library
- Ciarlet, P. G. and Raviart, P. A. 1974. A mixed finite element method for the biharmonic equation. In Proceedings of the Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations. C. de Boor Ed., Academic Press, New York, 125--145.Google Scholar
- Colton, D., Päivärinta, L., and Sylvester, J. 2007. The interior transmission problem. Inverse Prob. Imaging 1, 1, 13--28.Google ScholarCross Ref
- Colton, D., Monk, P., and Sun, J. 2010. Analytical and computational methods for transmission eigenvalues. Inverse Prob. 26, 4, 045011.Google ScholarCross Ref
- Golub, G. H. and Loan, C. F. V. 1989. Matrix Computations 2nd Ed. Johns Hopkins University Press, Baltimore, MD.Google Scholar
- Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L. 2011a. The interior transmission problem and bounds on transmission eigenvalues. Math. Res. Lett. 18, 2, 279--293.Google ScholarCross Ref
- Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L. 2011b. Transmission eigenvalues for elliptic operators. arXiv:1007.0503 physics.math-ph.Google Scholar
- Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L. 2011c. Transmission eigenvalues for operators with constant coefficients. arXiv:1004.5105 physics.math-ph.Google Scholar
- Hsiao, G., Liu, F., Sun, J., and Xu, L. 2011. A coupled BEM and FEM for the interior transmission problem in acoustic. J. Comput. Appl. Math. 3235, 17, 5213--5221.Google ScholarCross Ref
- Kirsch, A. 2009. On the existence of transmission eigenvalues. Inverse Prob. Imaging 3, 2, 155--172.Google ScholarCross Ref
- Monk, P. 1987. A mixed finite element method for the biharmonic equation. SIAM J. Numer. Anal. 24, 4, 737--749. Google ScholarDigital Library
- Päivärinta, L. and Sylvester, J. 2008. Transmission eigenvalues. SIAM J. Math. Anal. 40, 2, 738--753.Google ScholarCross Ref
- Saad, Y. 1980. Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl. 34, 269--295.Google ScholarCross Ref
- Sun, J. 2010. A new family of high regularity elements. Num. Meth. Partial Diff. Eq. 28, 1, 1--16.Google Scholar
- Sun, J. 2011a. Estimation of transmission eigenvalues and the index of refraction from cauchy data. Inverse Prob. 27, 1, 015009.Google ScholarCross Ref
- Sun, J. 2011b. Iterative methods for transmission eigenvalues. SIAM J. Numer. Anal. 49, 5, 1860--1874. Google ScholarDigital Library
Index Terms
Algorithm 922: A Mixed Finite Element Method for Helmholtz Transmission Eigenvalues
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