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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.
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Abstract

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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.

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

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