Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T05:58:48.421Z Has data issue: false hasContentIssue false
  • English
  • Français

On the asymptotic solution of an elliptic interior layer problem

Published online by Cambridge University Press:  17 February 2009

N. G. Barton
N. G. Barton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W. 2033, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

An interior layer problem posed by an elliptic partial differential equation of the type ε∇2φ - x∂φ/∂y = f(x, y, ε), 0 < ε ≪ 1, is investigated. This equation arises, for example, in the theory of rotating fluids and the important feature of the problem is an interior layer of width O(ε1/3) in which the solution has a relatively large magnitude.

The paper considers the simplest case which involves an interior layer, that is, where the domain is rectangular and f(x, y, ε) = εA for A constant. A leading approximation is derived and it is shown to be asymptotic to the exact solution in nearly all of the domain as ε → 0. The error estimates are derived using an a priori estimate for the solution of elliptic equations and a technique which optimizes the estimates is introduced. The applicability and limitations of the estimation technique are discussed briefly.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 19 , Issue 4 , December 1976, pp. 493 - 512
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Abramowitz, M. & Stegun, I. A., Handbook of mathematical functions, Dover, New York (1965).Google Scholar
[2]Agmon, S., ‘The Lp approach to the Dirichiet problem’ Annali Scu. Norm. Sup.—Pisa, 13 (1959), 405–448.Google Scholar
[3]Barton, N. G., Interior layers in rotating fluids, Ph.D. thesis, University of Western Australia (1973).Google Scholar
[4]Barton, N. G., ‘An inequality for the pointwise estimation of solutions of second order elliptic partial differential equations: J. Inst. Maths Applics, 14 (1974), 325–333.CrossRefGoogle Scholar
[5]Barton, N. G., ‘A device for the numerical solution of Fredholm integral equations of the first kind’, Applied Mathematics Preprint No. 83, Department of Mathematics, University of Queensland (1975).Google Scholar
[6]Barton, N. G., ‘An example of the modification of ocean currents by bottom topography’, to appear in Tellus, 28 (1976), 261–265.CrossRefGoogle Scholar
[7]Berger, M. S. & Fraenkel, L. E., ‘On the asymptotic solution of a non-linear Dirichlet problem’, J. Math. Mech., 19 (1970), 553–585.Google Scholar
[8]Brink, K. H., Veronis, G. & Yang, C. C., ‘The effect on ocean circulation of a change in the sign of β, Tellus, 25 (1973), 518–521.CrossRefGoogle Scholar
[9]Carrier, G. F., Krook, M. & Pearson, C. E., Functions of a complex variable, McGraw-Hill, New York (1966).Google Scholar
[10]Cook, L. P. & Ludford, G. S. S., ‘The behaviour as ε → +0 of solutions to εnabla;2w = ∂w/∂y in y ≪ 1 for discontinuous boundary data, SIAM J. Math. Anal., 2 (1971), 567–594.CrossRefGoogle Scholar
[11]Cook, L. P., Ludford, G. S. S. & Walker, J. S., ‘Corner regions in the asymptotic solution of ε∇2u = ∂u/∂y with reference to MHD duct flow’, Proc. Cam. Phil. Soc., 72 (1972), 117–122.CrossRefGoogle Scholar
[12]Courant, R. & Hubert, D., Methods of mathematical physics, Interscience, Vol 2, New York (1962).Google Scholar
[13]de Villiers, J. M., ‘A uniform asymptotic expansion of the positive solution of a non-linear Dirichlet problem’, Proc. London Math. Soc. (3), 27 (1973), 701–722.CrossRefGoogle Scholar
[14]Eckhaus, W. & de Jager, E. M., ‘Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type’, Arch. Rational Mech. Anal. 23 (1966–1967), 26–86.CrossRefGoogle Scholar
[15]Eckhaus, W., ‘Boundary layers in linear elliptic singular perturbation problems’, SIAM Review, 14 (1972), 225–270.CrossRefGoogle Scholar
[16]Fandry, C. B. & Leslie, L. M., ‘A note on the effect of latitudinally varying bottom topography on the winddriven ocean circulation’, Tellus, 24 (1972), 164–167.CrossRefGoogle Scholar
[17]Johnson, J. A., Fandry, C. B. & Leslie, L. M., ‘On the variation of ocean circulation produced by bottom topography’, Tellus, 23 (1971), 113–122.CrossRefGoogle Scholar
[18]Kamenkovich, V. M. & Mitrofanov, V. A., ‘An example of ocean-bottom topography on currents’, translated from Doklady, Akad. Nauk, SSSR, 199 (1971), 78–81.Google Scholar