Asymptotic analysis of a singular perturbation problem

In a rectangle 0 ≤ x ≤ a and 0 ≤ y ≤ b, the Dirichlet problem is studied for an elliptic differential equation of the form -epsilon Δu/sub epsilon/ + p (delta/delta x)u/sub epsilon/ + pu/sub epsilon/ = f(x,y) were epsilon is a small parameter 0 < epsilon << 1, Δ is the Laplace operator, p is a positive number, p is a nonnegative number, and all of the input data are smooth. A constructive procedure is established for obtaining an asymptotic approximation of arbitrary order with respect to epsilon of this singular perturbation problem, and also, a proof is given of its uniform validity in the closed rectangle by 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 layers along the characteristic boundaries y = 0 and y = b. Both ordinary corner layers and elliptic corner layers 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