Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T16:14:19.823Z Has data issue: false hasContentIssue false
  • English
  • Français

Tailored Finite Point Method for Numerical Solutions of Singular Perturbed Eigenvalue Problems

Published online by Cambridge University Press:  03 June 2015

Houde Han ,
Yin-Tzer Shih  and
Chih-Ching Tsai
Houde Han*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Yin-Tzer Shih*
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
Chih-Ching Tsai*
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
*
Email: hhan@math.tsinghua.edu.cn
Corresponding author. Email: yintzer shih@nchu.edu.tw
Email: jet.tsai98004@gmail.com
Get access

Abstract

We propose two variants of tailored finite point (TFP) methods for discretizing two dimensional singular perturbed eigenvalue (SPE) problems. A continuation method and an iterative method are exploited for solving discretized systems of equations to obtain the eigen-pairs of the SPE. We study the analytical solutions of two special cases of the SPE, and provide an asymptotic analysis for the solutions. The theoretical results are verified in the numerical experiments. The numerical results demonstrate that the proposed schemes effectively resolve the delta function like of the eigenfunctions on relatively coarse grid.

Keywords

65N2535B2574G1581Q05Singular perturbationtailored finite pointSchrödinger equationeigenvalue problem
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 3 , June 2014 , pp. 376 - 402
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allgower, E. L. AND Georg, K., An Introduction to Numerical Continuation Methods, SIAM Publications, Philadelphia, 2003.CrossRefGoogle Scholar
[2]Avila, A. and Jeanjean, L., A result on singularly perturbed elliptic problems, Commun. Pure App. Anal., 4.2 (2005), pp. 343–358.Google Scholar
[3]Babuska, I. AND Osborn, J. E., Eigenvalue Problems, Handbook of Numerical Analysis, Finite Element Methods (Part 1), Vol. 2, pp. 641–787, Ed. by Ciarlet, P. G. and Lious, J. L., Amsterdam, 1991.Google Scholar
[4]Golub, G. H. AND Van Loan, C. F., Matrix Computations, 3rd ed., Johns Hopkins University Press, 1996.Google Scholar
[5]Han, H. AND Huang, Z., Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions, J. Sci. Comput., 41 (2009), pp. 200–220.CrossRefGoogle Scholar
[6]Han, H. AND Huang, Z., Tailored finite point method for steady-state reaction-diffusion equations, Commun. Math. Sci., 8 (2010), pp. 887–899.Google Scholar
[7]Han, H. AND Huang, Z., Tailored finite point method based on exponential bases for convection- diffusion-reaction equation, Math. Comput., 82 (2013), pp. 213–226.Google Scholar
[8]Han, H. and Huang, Z., A tailored finite point method for the Helmholtz equation withhighwave numbers in heterogeneous medium, J. Comput. Math., 26 (2008), pp. 728–739.Google Scholar
[9]Han, H., Huang, Z. and Kellogg, R. B., A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput., 36 (2008), pp. 243–261.Google Scholar
[10]Han, H., Miller, J. J. H. and Tang, M., A parameter-uniform tailored finite point method for singularly perturbed linear ODE systems, J. Comput. Math., 31 (2013), pp. 422–438.Google Scholar
[11]Han, H., Tang, M. and Ying, W., Two uniform tailored finite point schemes for the two dimen-sional discrete ordinates transport equations with boundary and interface layers, Commun. Comput. Phys., 15 (2014), pp. 797–826.Google Scholar
[12]Han, H., Zhou, Z. and Zheng, C., Numerical solutions of an eigenvalue problem in unbounded domains, Numer. Math. J. Chinese Univ. (English Ser.), 14 (2006), pp. 1–13.Google Scholar
[13]Hsieh, P., Shih, Y. and Yang, S., A tailored finite point method for solving steady MHD duct flow problems with boundary layers, Commun. Comput. Phys., 10 (2011), pp. 161–182.Google Scholar
[14]Huang, Z. AND Yang, X., Tailored finite point method for first order wave equation, J. Sci. Comput., 49 (2011), pp. 351–366.Google Scholar
[15]Reed, M. AND Simon, B., Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, Inc., 1978.Google Scholar
[16]Schulten, K., Notes on Quantum Mechanics, Department of Physics and Beckman Institute, University of Illinois at UrbanaChampaign, 1999.Google Scholar
[17]Shankar, R., Principles of Quantum Mechanics, 2nd Ed., New York, Kluwer Academic/Plenum Publishers, 1994.Google Scholar
[18]Shih, Y., Kellogg, R. B. and Chang, Y., Characteristic tailored finite point method for convection-dominated convection-diffusion-reaction problems, J. Sci. Comput., 47 (2011), pp. 198–215.Google Scholar
[19]Shih, Y., Kellogg, R. B. and Tsai, P., A tailored finite point method for convection-diffusion- reaction problems, J. Sci. Comput., 43 (2010), pp. 239–260.Google Scholar
[20]Szyld, D. B., Criteria for combining inverse and Rayleigh quotient iteration, SIAM J. Numer. Anal., 53 (1988), pp. 1369–1375.Google Scholar
[21]Parlett, B. N., The Symmetric Eigenvalue Problem, PrenticeHall, Englewood Cliffs, NJ, 1980.Google Scholar