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

HTML articles powered by AMS MathViewer

- by T. M. Hagstrom and H. B. Keller PDF
- Math. Comp.
**48**(1987), 449-470 Request permission

## 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.

## References

- Shmuel Agmon,
*Lectures on elliptic boundary value problems*, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR**0178246** - S. Agmon and L. Nirenberg,
*Properties of solutions of ordinary differential equations in Banach space*, Comm. Pure Appl. Math.**16**(1963), 121–239. MR**155203**, DOI 10.1002/cpa.3160160204
R. Aris, - Catherine Bandle,
*Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems*, Arch. Rational Mech. Anal.**58**(1975), no. 3, 219–238. MR**454336**, DOI 10.1007/BF00280742
A. Bayliss, C. Goldstein & E. Turkel, "An iterative solution method for the Helmholtz equation." (To appear.)
- Ju. M. Berezans′kiĭ,
*Expansions in eigenfunctions of selfadjoint operators*, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR**0222718** - George J. Fix and Samuel P. Marin,
*Variational methods for underwater acoustic problems*, J. Comput. Phys.**28**(1978), no. 2, 253–270. MR**502898**, DOI 10.1016/0021-9991(78)90037-2 - I. C. Gohberg and M. G. Kreĭn,
*Introduction to the theory of linear nonselfadjoint operators*, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR**0246142** - Charles I. Goldstein,
*A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains*, Math. Comp.**39**(1982), no. 160, 309–324. MR**669632**, DOI 10.1090/S0025-5718-1982-0669632-7 - Bertil Gustafsson and Heinz-Otto Kreiss,
*Boundary conditions for time-dependent problems with an artificial boundary*, J. Comput. Phys.**30**(1979), no. 3, 333–351. MR**529999**, DOI 10.1016/0021-9991(79)90119-0 - Thomas Hagstrom and H. B. Keller,
*Exact boundary conditions at an artificial boundary for partial differential equations in cylinders*, SIAM J. Math. Anal.**17**(1986), no. 2, 322–341. MR**826697**, DOI 10.1137/0517026
T. Hagstrom, - Herbert B. Keller,
*Numerical solution of two point boundary value problems*, Regional Conference Series in Applied Mathematics, No. 24, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR**0433897** - Herbert B. Keller and Donald S. Cohen,
*Some positone problems suggested by nonlinear heat generation*, J. Math. Mech.**16**(1967), 1361–1376. MR**0213694** - Marianela Lentini and Herbert B. Keller,
*Boundary value problems on semi-infinite intervals and their numerical solution*, SIAM J. Numer. Anal.**17**(1980), no. 4, 577–604. MR**584732**, DOI 10.1137/0717049 - Heinz-Otto Kreiss,
*Difference approximations for boundary and eigenvalue problems for ordinary differential equations*, Math. Comp.**26**(1972), 605–624. MR**373296**, DOI 10.1090/S0025-5718-1972-0373296-3
M. Lentini,

*The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts*, vol. I, Clarendon Press, Oxford, 1975.

*Reduction of Unbounded Domains to Bounded Domains for Partial Differential Equation Problems*, Ph.D. Thesis, Applied Mathematics, California Institute of Technology, Pasadena, Calif., 1983. A. Jepson,

*Asymptotic Boundary Conditions for Ordinary Differential Equations*, Part I, Ph.D. Thesis, Applied Mathematics, California Institute of Technology, Pasadena, Calif., 1980. A. Jepson & H. B. Keller, "Boundary value problems on semi-infinite intervals. I. Linear problems,"

*Numer. Math.*(To appear.)

*Boundary Value Problems Over Semi-Infinite Intervals*, Ph.D. Thesis, Applied Mathematics, California Institute of Technology, Pasadena, Calif., 1978.

## Additional Information

- © Copyright 1987 American Mathematical Society
- 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