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Approximation methods for nonlinear problems with application to two-point boundary value problems


Author: H. B. Keller
Journal: Math. Comp. 29 (1975), 464-474
MSC: Primary 65J05; Secondary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1975-0371058-7
MathSciNet review: 0371058
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Abstract: General nonlinear problems in the abstract form $ F(\chi ) = 0$ and corresponding families of approximating problems in the form $ {F_h}({\chi _h}) = 0$ are considered (in an appropriate Banach space setting). The relation between "isolation" and "stability" of solutions is briefly studied. The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution. Error estimates are obtained and Newton's method is shown to converge quadratically. These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear two-point boundary value problems.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0371058-7
Keywords: Nonlinear stability, Newton's method, two point boundary problems, finite difference schemes
Article copyright: © Copyright 1975 American Mathematical Society

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