Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains

Authors:
T. M. Hagstrom and H. B. Keller

Journal:
Math. Comp. **48** (1987), 449-470

MSC:
Primary 65N99; Secondary 35A35, 35J25

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878684-5

MathSciNet review:
878684

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Abstract: We present a derivation and implementation of asymptotic boundary conditions to be imposed on "artificial" boundaries for nonlinear elliptic boundary value problems on semi-infinite "cylindrical" domains. A general theory developed by the authors in [11] is applied to establish the existence of *exact* boundary conditions and then to obtain useful approximations to them. The derivation is based on the Laplace transform solution of the linearized problem at infinity. We discuss the incorporation of the asymptotic boundary conditions into a finite-difference scheme and present the results of numerical experiments on the solution of the Bratu problem in a two-dimensional stepped channel. We also touch on certain problems concerning the existence of solutions of this problem on infinite domains and conjecture on the behavior of the critical parameter value with respect to changes in the domain. Some numerical evidence supporting the conjecture is given.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878684-5

Keywords:
Asymptotic boundary conditions,
asymptotic expansions,
artificial boundaries,
unbounded domains

Article copyright:
© Copyright 1987
American Mathematical Society