A two-grid discretization scheme for semilinear elliptic eigenvalue problems

SIAM Journal on Scientific Computing

Volumn 27, Issue 4, 2006, Pages 1287-1304

  Chien, C S ^a   Jeng, B W ^a  

^a NATIONAL CHUNG HSING UNIVERSITY   (Taiwan)

Author keywords

Continuation method; Finite element method; Linear eigenvalue problems; Nonlinear eigenvalue problems; Singular points; Two grid scheme

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BIFURCATION (MATHEMATICS); COMPUTATION THEORY; DISCRETE TIME CONTROL SYSTEMS; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; PARAMETER ESTIMATION; PROBLEM SOLVING;

CONTINUATION METHOD; LINEAR EIGENVALUE PROBLEMS; NONLINEAR EIGENVALUE PROBLEMS; SINGULAR POINTS; TWO-GRID SCHEME;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 33746104462     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/030602447     Document Type: Article

References (32)

1. E. L. ALLGOWER AND K. BÖHMER, Application of the mesh independence principle to mesh refinement strategies, SIAM J. Numer. Anal., 24 (1987), pp. 1335-1351.
2. E. L. ALLGOWER AND K. GEORG, Introduction to Numerical Continuation Methods, SIAM, Philadelphia, PA, 2003.
3. R. E. BANK AND T. F. CHAN, PLTMGC: A multi-grid continuation program for parameterized nonlinear elliptic systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 540-559.
4. J. H. BOLSTAD AND H. B. KELLER, A multigrid continuation method for elliptic problems with folds, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 1081-1104.
5. T. F. CHAN AND H. B. KELLER, Arc-length continuation and multi-grid techniques for nonlinear elliptic eigenvalue problems, SIAM J. Sci. Stat. Comput., 3 (1982), pp. 173-194.
6. S.-L. CHANG AND C.-S. CHIEN, A multigrid-Lanczos algorithm for the numerical solutions of nonlinear eigenvalue problems, Int. J. Bifurcation and Chaos, 13 (2003), pp. 1217-1228.
7. S.-L. CHANG, C.-S. CHIEN, AND B.-W. JENG, Tracing the solution surface with folds of a two parameter system, Int. J. Bifurcation and Chaos, 15 (2005), pp. 2689-2700.
8. C.-S. CHIEN AND S.-L. CHANG, Application of the Lanczos algorithm for solving the linear systems that occur in continuation problems, Numer. Linear Algebra Appl., 10 (2003), pp. 335-355.
9. C.-S. CHIEN AND B.-W. JENG, Symmetry reductions and a posteriori finite element error estimators for bifurcation problems, Int. J. Bifurcation and Chaos., 15 (2005), pp. 2091-2107.
10. S. COSTINER AND S. TA'ASAN, Adaptive multigrid techniques for large-scale eigenvalue problems: Solutions of the Schrödinger problem in two and three dimensions, Phys. Rev. E, 51 (1995), pp. 3704-3717.
11. S. COSTINER AND S. TA'ASAN, 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), pp. 1181-1192.
12. E. J. DOEDEL AND H. SHARIFI, Collocation methods for continuation problems in nonlinear elliptic PDEs, Notes Numer. Fluid Mech., 74 (2000), pp. 105-118.
13. F. A. DUL, MINRES and MINERR are better than SYMMLQ in eigenpair computations, SIAM J. Sci. Comput., 19 (1998), pp. 1767-1782.
14. A. I. FEDOSEYEV, M. J. FRIEDMAN, AND E. J. KANSA, Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method, Int. J. Bifurcation and Chaos, 10 (2000), pp. 481-492.
15. R. GLOWINSKI, H. B. KELLER, AND L. REINHART, Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 793-832.
16. R. ILLNER, O. KAVIAN, AND H. LANGE, Stationary solutions of quasi-linear SchrödingerPoisson systems, J. Differential Equations, 145 (1998), pp. 1-16.
17. H. B. KELLER, Lectures on Numerical Methods in Bifurcation Problems, Springer-Verlag, Berlin, 1987.
18. P. L. LIONS, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), pp. 441-467.
19. J.-L. LIU AND W. C. RHEINBOLDT, A posteriori finite element error estimators for parametrized nonlinear boundary value problems, Numer. Funct. Anal. Optim., 17 (1996), pp. 605-637.
20. Z. MEI, Path following around corank-2 bifurcation points of a semilinear elliptic problem with symmetry, Computing, 47 (1991), pp. 69-85.
21. Z. MEI, Solution branches of a semilinear elliptic problem at corank-2 bifurcation points, Manuscripta Math., 75 (1992), pp. 395-409.
22. H. D. MITTELMANN AND H. WEBER, Multi-grid solution of bifurcation problems, SIAM J. Sci. Stat, Comput., 6 (1985), pp. 49-60.
23. C. C. PAIGE AND M. A. SAUNDERS, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
24. O. SÁNCHEZ AND J. SOLER, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), pp. 179-204.
25. C. F. SMITH, A. F. PETERSON, AND R. MITTRA, A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields, IEEE Trana. Antennas and Propagation, 37 (1989), pp. 1490-1493.
26. S. TA'ASAN, Multigrid methods for locating singularities in bifurcation problems, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 51-62.
27. S.-H. WANG, On the Evolution and Qualitative Behaviors of Bifurcation Curve for a Boundary Value Problem, preprint, National Tsing Hua University, Hsinchu, Taiwan, 2004.
28. H. WEBER, An efficient technique for the computation of stable bifurcation branches, SIAM J. Sci. Stat. Comput., 5 (1984), pp. 332-348.
29. H. WEBER, Multigrid bifurcation iteration, SIAM J. Numer. Anal., 22 (1985), pp. 262-279.
30. J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), pp. 231-237.
31. J. XU, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777.
32. J. Xu AND A. ZHOU, A two-grid discretization scheme for eigenvalue problems, Math. Comp., 70 (1999), pp. 17-25.