Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems

Author:
R. B. Simpson

Journal:
Math. Comp. **26** (1972), 359-375

MSC:
Primary 65N25

DOI:
https://doi.org/10.1090/S0025-5718-1972-0315918-9

MathSciNet review:
0315918

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Abstract: Finite-difference methods using the five-point 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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0315918-9

Keywords:
Finite-difference methods,
nonlinear eigenvalue problems,
nonlinear elliptic problems,
Laplacian,
bifurcation,
error estimates,
Newton method

Article copyright:
© Copyright 1972
American Mathematical Society