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), 449470
MSC:
Primary 65N99; Secondary 35A35, 35J25
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 semiinfinite "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 finitedifference scheme and present the results of numerical experiments on the solution of the Bratu problem in a twodimensional 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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 S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, N. J., 1965. MR 0178246 (31:2504)
 [2]
 S. Agmon & L. Nirenberg, "Properties of solutions of ordinary differential equations in Banach space," Comm. Pure Appl. Math., v. 16, 1963, pp. 121239. MR 0155203 (27:5142)
 [3]
 R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, vol. I, Clarendon Press, Oxford, 1975.
 [4]
 C. Bandle, "Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems," Arch. Rational Mech. Anal., v. 58, 1975, pp. 219238. MR 0454336 (56:12587)
 [5]
 A. Bayliss, C. Goldstein & E. Turkel, "An iterative solution method for the Helmholtz equation." (To appear.)
 [6]
 J. Berezanskiĭ, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monos., vol. 17, Amer. Math. Soc., Providence, R. I., 1968. MR 0222718 (36:5768)
 [7]
 G. Fix & S. Marin, "Variational methods for underwater acoustic problems," J. Comput. Phys., v. 28, 1978, pp. 253270. MR 0502898 (58:19799)
 [8]
 I. Gohberg & M. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monos., vol. 18, Amer. Math. Soc., Providence, R. I., 1969. MR 0246142 (39:7447)
 [9]
 C. Goldstein, "A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains," Math. Comp., v. 39, 1982, pp. 309324. MR 669632 (84e:65112)
 [10]
 B. Gustafsson & H.O. Kreiss, "Boundary conditions for time dependent problems with an artificial boundary," J. Comput Phys., v. 30, 1979, pp. 333351. MR 529999 (80i:65096)
 [11]
 T. Hagstrom & H. B. Keller, "Exact boundary conditions at an artificial boundary for partial differential equations in cylinders," SIAM J. Math. Anal., v. 17, 1986, pp. 322341. MR 826697 (87g:35022)
 [12]
 T. Hagstrom, Reduction of Unbounded Domains to Bounded Domains for Partial Differential Equation Problems, Ph.D. Thesis, Applied Mathematics, California Institute of Technology, Pasadena, Calif., 1983.
 [13]
 A. Jepson, Asymptotic Boundary Conditions for Ordinary Differential Equations, Part I, Ph.D. Thesis, Applied Mathematics, California Institute of Technology, Pasadena, Calif., 1980.
 [14]
 A. Jepson & H. B. Keller, "Boundary value problems on semiinfinite intervals. I. Linear problems," Numer. Math. (To appear.)
 [15]
 H. B. Keller, Numerical Solution of Two Point Boundary Value Problems, No. 24, CBMS/NSF Regional Conference Series on Applied Mathematics, SIAM, Philadelphia, Pa., 1976. MR 0433897 (55:6868)
 [16]
 H. B. Keller & D. S. Cohen, "Some positone problems suggested by nonlinear heat generation," J. Math. Mech., v. 16, 1967, pp. 13611376. MR 0213694 (35:4552)
 [17]
 H. B. Keller & M. Lentini, "Boundary value problems on semiinfinite intervals and their numerical solution," SIAM J. Numer. Anal., v. 17, 1980, pp. 577604. MR 584732 (81j:65092)
 [18]
 H.O. Kreiss, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605624. MR 0373296 (51:9496)
 [19]
 M. Lentini, Boundary Value Problems Over SemiInfinite Intervals, Ph.D. Thesis, Applied Mathematics, California Institute of Technology, Pasadena, Calif., 1978.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708786845
PII:
S 00255718(1987)08786845
Keywords:
Asymptotic boundary conditions,
asymptotic expansions,
artificial boundaries,
unbounded domains
Article copyright:
© Copyright 1987
American Mathematical Society
