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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 $ (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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  • [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31 #2504. MR 0178246 (31:2504)
  • [2] J. H. Bramble, ``On the convergence of difference approximations to weak solutions of Dirichlet's problem,'' Numer. Math., v. 13, 1969, pp. 101-111. MR 40 #3729. MR 0250495 (40:3729)
  • [3] 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. 313-327. 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. 1-12. MR 34 #969. MR 0201084 (34:969)
  • [5] K. O. Friedrichs & H. B. Keller, A Finite Difference Scheme for Generalized Neumann Problems, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965), Academic Press, New York, 1966, pp. 1-19. MR 34 #3803. MR 0203956 (34:3803)
  • [6] H. Fujita, ``On the nonlinear equations $ \Delta u + {e^u} = 0$ and $ \partial \upsilon /\partial t = \Delta \upsilon + {e^\upsilon }$,'' Bull. Amer. Math. Soc., v. 75, 1969, pp. 132-135. MR 39 #615. MR 0239258 (39:615)
  • [7] I. M. Gel'fand, ``Some problems in the theory of quasi-linear equations,'' Uspehi Mat. Nauk, v. 14, 1959, no. 2 (86), pp. 87-158; English transl., Amer. Math. Soc. Transl., (2), v. 29, 1963, pp. 295-381. MR 22 #1736; MR 27 #3921. MR 0110868 (22:1736)
  • [8] E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 #924. MR 0201039 (34:924)
  • [9] D. D. Joseph, ``Nonlinear heat generation and stability of the temperature distribution in conduction solids,'' Int. J. Heat Mass Transfer, v. 8, 1965, pp. 281-288.
  • [10] D. D. Joseph & E. M. Sparrow, ``Nonlinear diffusion induced by nonlinear sources,'' Quart. Appl. Math., v. 28, 1970, pp. 327-342. MR 0272272 (42:7153)
  • [11] H. B. Keller, ``Elliptic boundary value problems suggested by nonlinear diffusion processes,'' Arch. Rational Mech. Anal., v. 35, 1969, pp. 363-381. MR 41 #639. MR 0255979 (41:639)
  • [12] H. B. Keller, ``Newton's method under mild differentiability conditions,'' J. Comput. System Sci., v. 4, 1970, pp. 15-28. MR 40 #3710. MR 0250476 (40:3710)
  • [13] H. B. Keller & D. S. Cohen, ``Some positone problems suggested by nonlinear heat generation,'' J. Math. Mech., v. 16, 1967, pp. 1361-1376. MR 35 #4552. MR 0213694 (35:4552)
  • [14] J. R. Kuttler, ``Finite difference approximations of eigenvalues of uniformly elliptic operators,'' SIAM J. Numer. Anal., v. 7, 1970, pp. 206-232. MR 42 #8717. MR 0273841 (42:8717)
  • [15] T. Laetsch, ``A note on a paper of Keller and Cohen,'' J. Math. Mech., v. 18, 1968/ 69, pp. 1095-1100. MR 40 #1716. MR 0248464 (40:1716)
  • [16] Ja. G. Panovko & I. I. Gubanov, Stability and Oscillations of Elastic Systems. Paradoxes, Fallacies, and Concepts, ``Nauka", Moscow, 1964; English transl., Consultants Bureau, New York, 1965. MR 32 #4914; MR 33 #2025. MR 566209 (81b:73001)
  • [17] R. B. Simpson, ``Finite difference methods for mildly nonlinear eigenvalue problems,'' SIAM J. Numer. Anal., v. 8, 1971, pp. 190-211. MR 0286269 (44:3482)
  • [18] R. B. Simpson & D. S. Cohen, ``Positive solutions of nonlinear elliptic eigenvalue problems,'' J. Math. Mech., v. 19, 1970, pp. 895-910. MR 0435566 (55:8525)
  • [19] J. B. Rosen, ``Approximate solution and error bounds for quasi-linear elliptic boundary value problems,'' SIAM J. Numer. Anal., v. 7, 1970, pp. 80-103. MR 41 #9452. MR 0264861 (41:9452)
  • [20] L. V. Kantorovič & G. Akilov, Functional Analysis in Normed Spaces, Fizmatgiz, Moscow, 1959; English transl., Internat. Series of Monographs in Pure and Appl. Math., vol. 46, Macmillan, New York, 1964. MR 22 #9837; MR 35 #4699. MR 0119071 (22:9837)

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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

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