A mesh refinement method for $Ax=\lambda Bx$
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- by Stephen F. McCormick PDF
- Math. Comp. 36 (1981), 485-498 Request permission
Abstract:
The aim of this paper is to introduce a simple but efficient mesh refinement strategy for use with inverse iteration for finding one or a few solutions of an ordinary or partial differential eigenproblem of the form $Ax = \lambda Bx$. The focus is upon the case where A and B are symmetric and B is positive definite, although the approaches have a very broad application. A discussion of the combined use of mesh refinement and a correction scheme multigrid technique is also provided. The methods are illustrated by numerical results from experiments with two-point boundary value problems.References
- E. L. Allgower and S. F. McCormick, Newton’s method with mesh refinements for numerical solution of nonlinear two-point boundary value problems, Numer. Math. 29 (1977/78), no. 3, 237–260. MR 468200, DOI 10.1007/BF01389210 E. L. Allgower, S. F. McCormick & D. V. Pryor, "A general mesh independence principle for Newton’s method applied to second order boundary value problems," J. Numer. Func. Anal. Opt. (To appear).
- Kendall Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Numer. Anal. 12 (1975), 213–222. MR 438746, DOI 10.1137/0712020 L. C. Bernard & F. J. Helton, Computation of the Ideal Linearized Magneto Hydrodynamic (MHD) Spectrum in Toroidal Axisymmetry, General Atomic report no. GA-A15257, La Jolla, Calif., January 1979.
- Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333–390. MR 431719, DOI 10.1090/S0025-5718-1977-0431719-X
- W. Hackbusch, On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method, SIAM J. Numer. Anal. 16 (1979), no. 2, 201–215. MR 526484, DOI 10.1137/0716015 D. E. Longsine & S. F. McCormick, "Simultaneous Rayleigh quotient optimization methods for $Ax = \lambda Bx$." (Submitted.)
- S. F. McCormick, A revised mesh refinement strategy for Newton’s method applied to nonlinear two-point boundary value problems, Numerical treatment of differential equations in applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer, Berlin, 1978, pp. 15–23. MR 515566
- B. N. Parlett and D. S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comp. 33 (1979), no. 145, 217–238. MR 514820, DOI 10.1090/S0025-5718-1979-0514820-3
- H. R. Schwarz, Two algorithms for treating $A\chi =\lambda B\chi$, Comput. Methods Appl. Mech. Engrg. 12 (1977), no. 2, 181–199. MR 494878, DOI 10.1016/0045-7825(77)90011-1
- G. W. Stewart, A bibliographical tour of the large, sparse generalized eigenvalue problem, Sparse matrix computations (Proc. Sympos., Argonne Nat. Lab., Lemont, Ill., 1975) Academic Press, New York, 1976, pp. 113–130. MR 0455324 R. Underwood, An Iterative Block Lanczos Method for the Solution of Large Sparse Symmetric Eigenproblems, Ph.D. Thesis, Stanford University, STAN-CS-75-496, 1975.
- Eugene L. Wachspress, Iterative solution of elliptic systems, and applications to the neutron diffusion equations of reactor physics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0234649
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Math. Comp. 36 (1981), 485-498
- MSC: Primary 65N25; Secondary 65F15, 65L15, 65R99
- DOI: https://doi.org/10.1090/S0025-5718-1981-0606508-4
- MathSciNet review: 606508