An efficient algorithm for the Schrödinger-Poisson eigenvalue problem

Journal of Computational and Applied Mathematics

Volumn 205, Issue 1, 2007, Pages 509-532

  Chang, S L ^a   Chien, C S ^b   Jeng, B W ^b  

^a SOUTHERN TAIWAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Taiwan)

^b NATIONAL CHUNG HSING UNIVERSITY   (Taiwan)

Author keywords

RQI; Schr dinger eigenvalue problem; Schr dinger Poisson system; Two grid method; Wave functions

Indexed keywords

ALGORITHMS; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; WAVE FUNCTIONS;

TWO-GRID METHODS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 34247324404     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.05.013     Document Type: Article

References (28)

1. Allgower E.L., and Chien C.-S. Continuation and local perturbation for multiple bifurcations. SIAM J. Sci. Statist. Comput. 7 (1986) 1265-1281
2. Allgower E.L., and Georg K. Introduction to Numerical Continuation Methods (2003), SIAM, Philadelphia, PA
3. Bank R.E., and Chan T.F. PLTMGC: a multigrid continuation program for parameterized nonlinear elliptic systems. SIAM J. Sci. Statist. Comput. 7 (1986) 540-559
4. Chan T.F., and Elman H.C. Fourier analysis of iterative methods for elliptic problems. SIAM Rev. 31 (1989) 20-49
5. Chien C.-S., and Jeng B.-W. A two-grid discretization scheme for semilinear elliptic eigenvalue problems. SIAM J. Sci. Comput. 27 (2006) 1287-1304
6. Chu M.T. A simple application of the homotopy method to symmetric eigenvalue problems. Linear Algebra Appl. 59 (1984) 85-90
7. Collatz L. Functional Analysis and Numerical Mathematics (1966), Academic Press, New York
8. Costiner S., and Ta'asan S. Adaptive multigrid techniques for large-scale eigenvalue problems: Solutions of the Schrödinger problem in two and three dimensions. Phys. Rev. E 51 (1995) 3704-3717
9. Costiner S., and Ta'asan S. Simultaneous multigrid techniques for nonlinear eigenvalue problems: solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions. Phys. Rev. E 52 (1995) 1181-1192
10. Dul F.A. MINRES and MINERR are better than SYMMLQ in eigenpair computations. SIAM J. Sci. Comput. 19 (1998) 1767-1782
11. Golub G.H., and Van Loan C.F. Matrix Computations. third ed. (1996), Johns Hopkins University Press, Baltimore, MD
12. Govaerts W.J.F. Numerical Methods for Bifurcations of Dynamical Equilibria (2000), SIAM, Philadelphia, PA
13. Hwang L.J., and Li T.Y. Parallel homotopy algorithm for symmetric large sparse eigenproblems. J. Comput. Appl. Math. 60 (1995) 77-100
14. Hwang T.-M., Lin W.-W., Wang W.-C., and Wang W. Numerical simulation of three dimensional pyramid quantum dot. J. Comput. Phys. 196 (2004) 208-232
15. Illner R., Kavian O., and Lange H. Stationary solutions of quasi-linear Schrödinger-Poisson systems. J. Differential Equations 145 (1998) 1-16
16. Kerkhoven T., Galick A.T., Ravaioli U., Arends J.H., and Saad Y. Efficient numerical simulation of electron states in quantum wires. J. Appl. Phys. 68 (1990) 3461-3469
17. Li T.Y., and Rhee N.H. Homotopy algorithm for symmetric eigenvalue problems. Numer. Math. 55 (1989) 265-280
18. Livne O.E., and Brandt A. Multiscale eigenbasis calculations: N eigenfunctions in O (N log N). Multiscale and Multiresolution Methods, Lecture Notes in Computational Science and Engineering vol. 20 (2002), Springer, Berlin 347-357
19. Lui S.H., and Golub G.H. Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators. Numer. Algorithms 10 (1995) 363-378
20. Mei Z. Path following around corank-2 bifurcation points of a semi-linear elliptic problems with symmetry. Computing 47 (1991) 69-85
21. Paige C.C., and Saunders M.A. Solutions of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12 (1975) 617-629
22. Parlett B.N. The Rayleigh quotient iteration and some generalizations for nonnormal matrices. Math. Comput. 28 (1974) 679-693
23. Parlett B.N. The Symmetric Eigenvalue Problem (1980), Prentice-Hall, Englewood Cliffs, NJ
24. Rheinboldt W.C. Methods for Solving Systems of Nonlinear Equations (1974), SIAM, Philadelphia, PA
25. Sánchez O., and Soler J. Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Statist. Phys. 114 (2004) 179-204
26. Stakgold I. Green's Functions and Boundary Value Problems (1979), Wiley, Chichester
27. Wang W., Hwang T.-M., Lin W.-W., and Liu J.-L. Numerical methods for semiconductor heterostructures with band nonparabolicity. J. Comput. Phys. 190 (2003) 141-158
28. Xu J., and Zhou A. A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70 (1999) 17-25