×

Augmented coupling interface method for solving eigenvalue problems with sign-changed coefficients. (English) Zbl 1203.65239

Summary: We propose an augmented coupling interface method on a Cartesian grid for solving eigenvalue problems with sign-changed coefficients. The underlying idea of the method is the correct local construction near the interface which incorporates the jump conditions. The method, which is very easy to implement, is based on a finite difference discretization. The main ingredients of the proposed method comprise (i) an adaptive-order strategy of using interpolating polynomials of different orders on different sides of the interfaces, which avoids the singularity of the local linear system and enables us to handle complex interfaces; (ii) when the interface condition involves the eigenvalue, the original problem is reduced to a quadratic eigenvalue problem by introducing an auxiliary variable and an interfacial operator on the interface; (iii) the auxiliary variable is discretized uniformly on the interface, the rest of variables are discretized on an underlying rectangular grid, and a proper interpolation between these two grids are designed to reduce the number of stencil points. Several examples are tested to show the robustness and accuracy of the schemes.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
92D25 Population dynamics (general)
65N06 Finite difference methods for boundary value problems involving PDEs

Software:

IIMPACK
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Li, Z.; Wang, W.; Chern, I.; Lai, M., New formulations for interface problems in polar coordinates, SIAM Journal on Scientific Computing, 25, 224-245 (2003) · Zbl 1040.65087
[2] Wang, W., A jump condition capturing finite difference scheme for elliptic interface problems, SIAM Journal on Scientific Computing, 25, 1479-1496 (2004) · Zbl 1061.65110
[3] Chen, Z.; Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik, 79, 175-202 (1998) · Zbl 0909.65085
[4] Glimm, J.; Mcbryan, O., A computational model for interfaces, Advances in Applied Mathematics, 6, 422-435 (1985)
[5] Huang, J.; Zou, J., A mortar element method for elliptic problems with discontinuous coefficients, IMA Journal of Numerical Analysis, 22, 549-576 (2002) · Zbl 1014.65117
[6] Li, Z.; Lin, T.; Wu, X., New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, 96, 61-98 (2003) · Zbl 1055.65130
[7] Tornberg, A.; Engquist, B., Regularization techniques for numerical approximation of PDEs with singularities, Journal of Scientific Computing, 19, 527-552 (2003) · Zbl 1035.65085
[8] Tornberg, A.; Engquist, B., Numerical approximations of singular source terms in differential equations, Journal of Computational Physics, 200, 462-488 (2004) · Zbl 1115.76392
[9] Peskin, C., The immersed boundary method, Acta Numerica, 11, 479-517 (2002) · Zbl 1123.74309
[10] Peskin, C., Numerical analysis of blood-flow in heart, Journal of Computational Physics, 25, 220-252 (1977) · Zbl 0403.76100
[11] Leveque, R.; LI, Z., The immersed interface method for elliptic-equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 31, 1019-1044 (1994) · Zbl 0811.65083
[12] Li, Z.; Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM Journal on Scientific Computing, 23, 339-361 (2001) · Zbl 1001.65115
[13] Li, Z.; Ito, K., The Immersed Interface Method, Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (2006), SIAM Frontiers in Applied Mathematics
[14] Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), Journal of Computational Physics, 152, 457-492 (1999) · Zbl 0957.76052
[15] Wiegmann, A.; Bube, K., The explicit-jump immersed interface method: finite difference methods for pdes with piecewise smooth solutions, SIAM Journal on Numerical Analysis, 37, 827-862 (2000) · Zbl 0948.65107
[16] Berthelsen, P., A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, Journal of Computational Physics, 197, 364-386 (2004) · Zbl 1052.65100
[17] Zhou, Y.; Zhao, S.; Feig, M.; Wei, G., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, Journal of Computational Physics, 213, 1-30 (2006) · Zbl 1089.65117
[18] Chern, I. L.; Shu, Y. C., A coupling interface method for elliptic interface problems, Journal of Computational Physics, 225, 2138-2274 (2007) · Zbl 1123.65108
[19] Chen, T.; Strain, J., Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems, Journal of Computational Physics, 227, 7503-7542 (2008) · Zbl 1157.65064
[20] Chang, C. C.; Chern, R. L.; Chang, C. C.; Hwang, R. R., Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures, Physical Review B, 72, 205112 (2005)
[21] Belgacem, F., Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications, Pitman Research Notes in Mathematics Series, vol. 368 (1997), Addison-Wesley-Longman · Zbl 0882.35002
[22] Cantrell, R. S.; Cosner, C., Diffusive logistic equations with indefinite weights: population models in a disrupted environments, Proceedings of the Royal Society Edinburgh, 112A, 293-318 (1989) · Zbl 0711.92020
[23] Cantrell, R. S.; Cosner, C., The effects of spatial heterogeneity in population dynamics, Journal of Mathematical Biology, 29, 315-338 (1991) · Zbl 0722.92018
[24] Lou, Y.; Yanagida, E., Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight and applications to population dynamics, Japan Journal of Industrial and Applied Mathematics, 23, 275-292 (2006) · Zbl 1185.35059
[25] Kao, C. Y.; Lou, Y.; Yanagida, E., Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Mathematical Biosciences and Engineering, 5, 315-335 (2008) · Zbl 1167.35426
[26] Ritchie, R. H., Plasma losses by fast electrons in thin films, Physical Review, 106, 874-881 (1957)
[27] Barnes, W. L.; Dereux, A.; Ebbesen, T. W., Surface plasmon subwavelength optics, Nature, 424, 824 (2003)
[28] Baida, F. I.; Belkhir, A.; Labeke, D. V.; Lamrous, O., Subwavelength metallic coaxial waveguides in the optical range: role of the plasmonic modes, Physical Review B, 74, 205419 (2006)
[29] Chang, C. C.; Shu, Y.-C.; Chern, I.-L., Solving guided wave modes in plasmonic crystals, Physical Review B, 78, 035133 (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.