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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
MathSciNet review: 0315918
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Abstract: Finite-difference methods using the five-point discrete Laplacian and suitable boundary modifications for approximating $ (1) - \Delta u = \lambda f(x,u)$ in a plane domain $ D,u = 0$ on its boundary are considered. It is shown that if (1) has an isolated solution, $ u$, then the discrete problem has a solution, $ {U_h}$, for which $ {U_h} - u = O({h^2})$. If the discrete problem has solutions, $ {U_h}$, such that $ \vert{U_h}\vert \leqq M$ as $ h$ tends to zero, then (1) has a solution, $ u$, satisfying $ \vert u\vert \leqq M$. Let $ {\lambda ^\ast}$ be a critical value of $ \lambda $ so that (1) has positive solutions for $ \lambda \leqq \lambda ^\ast$ but not for $ \lambda > {\lambda ^\ast}$, then the discrete problem has an analogous critical value $ \lambda _h^\ast$ and, under suitable conditions, $ \lambda _h^\ast - {\lambda ^\ast} = O({h^{4/3 - \epsilon}}),\epsilon > 0$. Computed results for the case $ f(x,u) = {e^u}$ and $ D$ the unit square are given.

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Keywords: Finite-difference methods, nonlinear eigenvalue problems, nonlinear elliptic problems, Laplacian, bifurcation, error estimates, Newton method
Article copyright: © Copyright 1972 American Mathematical Society

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