A mesh refinement method for
Author:
Stephen F. McCormick
Journal:
Math. Comp. 36 (1981), 485498
MSC:
Primary 65N25; Secondary 65F15, 65L15, 65R99
MathSciNet review:
606508
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 twopoint boundary value problems.
 [1]
E.
L. Allgower and S.
F. McCormick, Newton’s method with mesh refinements for
numerical solution of nonlinear twopoint boundary value problems,
Numer. Math. 29 (1977/78), no. 3, 237–260. MR 0468200
(57 #8038)
 [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
(55 #11653)
 [4]
L. C. Bernard & F. J. Helton, Computation of the Ideal Linearized Magneto Hydrodynamic (MHD) Spectrum in Toroidal Axisymmetry, General Atomic report no. GAA15257, La Jolla, Calif., January 1979.
 [5]
Achi
Brandt, Multilevel adaptive solutions to
boundaryvalue problems, Math. Comp.
31 (1977), no. 138, 333–390. MR 0431719
(55 #4714), http://dx.doi.org/10.1090/S0025571819770431719X
 [6]
W.
Hackbusch, On the computation of approximate eigenvalues and
eigenfunctions of elliptic operators by means of a multigrid method,
SIAM J. Numer. Anal. 16 (1979), no. 2, 201–215.
MR 526484
(80d:65065), http://dx.doi.org/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 twopoint 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
(80c:65168)
 [9]
B.
N. Parlett and D.
S. Scott, The Lanczos algorithm with selective
orthogonalization, Math. Comp.
33 (1979), no. 145, 217–238. MR 514820
(80c:65090), http://dx.doi.org/10.1090/S00255718197905148203
 [10]
H.
R. Schwarz, Two algorithms for treating
𝐴𝜒=𝜆𝐵𝜒, Comput. Methods Appl.
Mech. Engrg. 12 (1977), no. 2, 181–199. MR 0494878
(58 #13661)
 [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
(56 #13563)
 [12]
R. Underwood, An Iterative Block Lanczos Method for the Solution of Large Sparse Symmetric Eigenproblems, Ph.D. Thesis, Stanford University, STANCS75496, 1975.
 [13]
Eugene
L. Wachspress, Iterative solution of elliptic systems, and
applications to the neutron diffusion equations of reactor physics,
PrenticeHall, Inc., Englewood Cliffs, N.J., 1966. MR 0234649
(38 #2965)
 [1]
 E. L. Allgower & S. F. McCormick, "Newton's method with mesh refinements for numerical solution of nonlinear twopoint boundary value problems," Numer. Math., v. 29, 1978, pp. 237260. MR 0468200 (57:8038)
 [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]
 K. Atkinson, "Convergence rates for approximate eigenvalues of compact integral operators," SIAM J. Numer. Anal., v. 12, 1975, pp. 213222. MR 0438746 (55:11653)
 [4]
 L. C. Bernard & F. J. Helton, Computation of the Ideal Linearized Magneto Hydrodynamic (MHD) Spectrum in Toroidal Axisymmetry, General Atomic report no. GAA15257, La Jolla, Calif., January 1979.
 [5]
 A. Brandt, "Multilevel adaptive solutions to boundary value problems," Math. Comp., v. 31, 1977, pp. 333390. MR 0431719 (55:4714)
 [6]
 W. Hackbusch, "On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multigrid method," SIAM J. Numer. Anal., v. 16, 1979, pp. 201215. MR 526484 (80d:65065)
 [7]
 D. E. Longsine & S. F. McCormick, "Simultaneous Rayleigh quotient optimization methods for ." (Submitted.)
 [8]
 S. F. McCormick, A Revised Mesh Refinement Strategy for Numerical Solution of TwoPoint Boundary Value Problems, Lecture Notes in Math., Vol. 679, SpringerVerlag, Berlin and New York, 1978, pp. 1523. MR 515566 (80c:65168)
 [9]
 B. N. Parlett & D. S. Scott, "The Lanczos algorithm with selective orthogonalization," Math. Comp., v. 33, 1979, pp. 217238. MR 514820 (80c:65090)
 [10]
 H. R. Schwarz, "Two algorithms for treating ," Comput. Methods Appl. Mech. Engrg., v. 12, 1977, pp. 181199. MR 0494878 (58:13661)
 [11]
 G. W. Stewart, "A bibliographical tour of the large, sparse generalized eigenvalue problem," Sparse Matrix Computations (J. R. Bunch and D. J. Rose, Eds.), Academic Press, New York, 1976, pp. 113130. MR 0455324 (56:13563)
 [12]
 R. Underwood, An Iterative Block Lanczos Method for the Solution of Large Sparse Symmetric Eigenproblems, Ph.D. Thesis, Stanford University, STANCS75496, 1975.
 [13]
 E. L. Wachspress, Iterative Solution of Elliptic Systems, and Applications to the Newton Diffusion Equations of Reactor Physics, PrenticeHall, Englewood Cliffs, N. J., 1966. MR 0234649 (38:2965)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65N25,
65F15,
65L15,
65R99
Retrieve articles in all journals
with MSC:
65N25,
65F15,
65L15,
65R99
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106065084
PII:
S 00255718(1981)06065084
Article copyright:
© Copyright 1981
American Mathematical Society
