Asymptotic Analysis of a Singular Perturbation Problem

Abstract

We study, in a rectangle $0 < x < a$ and $0 < y < b$, the Dirichlet problem for an elliptic differential equation of the form \[ - \epsilon \Delta u\epsilon + p\frac{{\partial u\epsilon }}{{\partial x}} + qu\epsilon = f(x,y)\] where $\epsilon $ is a small parameter $0 < \epsilon \ll 1$, $\Delta $ is the Laplacian operator, p is a positive number, q is a nonnegative number and all of the input data are smooth. We establish a constructive procedure for obtaining an asymptotic approximation of arbitrary order with respect to $\epsilon $ of this singular perturbation problem, and also give a proof of its uniform validity in the closed rectangle by the use of the maximum principle and exponential estimates of all boundary or corner layer functions. The corner singularities of parabolic boundary layer functions are removed by introducing elliptic boundary layer functions along the characteristic boundaries $y = 0$ and $y = b$. Both ordinary corner layer functions and elliptic corner layer functions are employed at the outflow corners $(a,0)$ and $(a,b)$. An application is made to settle a long-standing problem in the magnetohydrodynamic flow in a rectangular duct.