On the eigenvectors of a finite-difference approximation to the Sturm-Liouville eigenvalue problem

Gekeler, Eckart

doi:10.1090/S0025-5718-1974-0356524-1

On the eigenvectors of a finite-difference approximation to the Sturm-Liouville eigenvalue problem
HTML articles powered by AMS MathViewer

by Eckart Gekeler PDF

Math. Comp. 28 (1974), 973-979 Request permission

Abstract:

This paper is concerned with a centered finite-difference approximation to to the nonselfadjoint Sturm-Liouville eigenvalue problem \[ \begin {array}{*{20}{c}} {L[u] = - {{[a(x){u_x}]}_x} - b(x){u_x} + c(x)u = \lambda u,\quad 0 < x < 1,} \hfill \\ {u(0) = u(1) = 0.} \hfill \\ \end {array} \] It is shown that the eigenvectors ${W_p}$ of the $M \times M$-matrix ($\Delta x = 1/(M + 1)$ mesh size), which approximates L, are bounded in the maximum norm independent of M if they are normalized so that $|{W_p}{|_2} = 1$.

References

H. Bückner, Über Konvergenzsätze, die sich bei der Anwendung eines Differenzenverfahrens auf ein Sturm-Liouvillesches Eigenwertproblem ergeben, Math. Z. 51 (1948), 423–465 (German). MR 30820, DOI 10.1007/BF01185778
Alfred Carasso, Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators, Math. Comp. 23 (1969), 717–729. MR 258291, DOI 10.1090/S0025-5718-1969-0258291-7
Alfred Carasso and Seymour V. Parter, An analysis of “boundary-value techniques” for parabolic problems, Math. Comp. 24 (1970), 315–340. MR 284019, DOI 10.1090/S0025-5718-1970-0284019-9
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338

Methoden der Mathematischen Physik.

16

John Gary, Computing eigenvalues of ordinary differential equations by finite differences, Math. Comp. 19 (1965), 365–379. MR 179926, DOI 10.1090/S0025-5718-1965-0179926-X
Eckart Gekeler, Long-range and periodic solutions of parabolic problems, Math. Z. 134 (1973), 53–66. MR 341886, DOI 10.1007/BF01219091
E. Gekeler, $A$-convergence of finite difference approximations of parabolic initial-boundary value problems, SIAM J. Numer. Anal. 12 (1975), 1–12. MR 423827, DOI 10.1137/0712001
Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502