Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems
Author:
R. B. Simpson
Journal:
Math. Comp. 26 (1972), 359375
MSC:
Primary 65N25
MathSciNet review:
0315918
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Abstract: Finitedifference methods using the fivepoint discrete Laplacian and suitable boundary modifications for approximating in a plane domain on its boundary are considered. It is shown that if (1) has an isolated solution, , then the discrete problem has a solution, , for which . If the discrete problem has solutions, , such that as tends to zero, then (1) has a solution, , satisfying . Let be a critical value of so that (1) has positive solutions for but not for , then the discrete problem has an analogous critical value and, under suitable conditions, . Computed results for the case and the unit square are given.
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 J. H. Bramble, & B. E. Hubbard, ``On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation,'' Numer. Math., v. 4, 1962, pp. 313327. MR 26 #7157. MR 0149672 (26:7157)
 [4]
 J. H. Bramble, ``Error estimates for difference methods in forced vibration problems,'' SIAM J. Numer. Anal., v. 3, 1966, pp. 112. MR 34 #969. MR 0201084 (34:969)
 [5]
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 H. B. Keller, ``Newton's method under mild differentiability conditions,'' J. Comput. System Sci., v. 4, 1970, pp. 1528. MR 40 #3710. MR 0250476 (40:3710)
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 J. B. Rosen, ``Approximate solution and error bounds for quasilinear elliptic boundary value problems,'' SIAM J. Numer. Anal., v. 7, 1970, pp. 80103. MR 41 #9452. MR 0264861 (41:9452)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197203159189
PII:
S 00255718(1972)03159189
Keywords:
Finitedifference methods,
nonlinear eigenvalue problems,
nonlinear elliptic problems,
Laplacian,
bifurcation,
error estimates,
Newton method
Article copyright:
© Copyright 1972
American Mathematical Society
