A mesh refinement method for

Author:
Stephen F. McCormick

Journal:
Math. Comp. **36** (1981), 485-498

MSC:
Primary 65N25; Secondary 65F15, 65L15, 65R99

MathSciNet review:
606508

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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 . 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.

**[1]**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**0468200****[2]**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).**[3]**Kendall Atkinson,*Convergence rates for approximate eigenvalues of compact integral operators*, SIAM J. Numer. Anal.**12**(1975), 213–222. MR**0438746****[4]**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.**[5]**Achi Brandt,*Multi-level adaptive solutions to boundary-value problems*, Math. Comp.**31**(1977), no. 138, 333–390. MR**0431719**, 10.1090/S0025-5718-1977-0431719-X**[6]**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**, 10.1137/0716015**[7]**D. E. Longsine & S. F. McCormick, "Simultaneous Rayleigh quotient optimization methods for ." (Submitted.)**[8]**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****[9]**B. N. Parlett and D. S. Scott,*The Lanczos algorithm with selective orthogonalization*, Math. Comp.**33**(1979), no. 145, 217–238. MR**514820**, 10.1090/S0025-5718-1979-0514820-3**[10]**H. R. Schwarz,*Two algorithms for treating 𝐴𝜒=𝜆𝐵𝜒*, Comput. Methods Appl. Mech. Engrg.**12**(1977), no. 2, 181–199. MR**0494878****[11]**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****[12]**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.**[13]**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**

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0606508-4

Article copyright:
© Copyright 1981
American Mathematical Society