Higher order solution for a singularly perturbed diffusion-convection problem in two dimensions

Dialla Konaté
Virginia Tech
email: dkonate@vt.edu

Abstract: We consider the following problem

where and

We assume

Problem (), a version of the Stokes problem, i.e a linearized version of the Navier-Stokes problem, is elliptic with a dominant convection term. Its limiting problem obtained in setting is an hyperbolic one given by

So, for problem () exhibits a boundary layer phenomenon along the portion of the boundary which make difficult the computation of a good'' approximation solution to The current paper is intended to construct an approximation solution to up to any higher order where is an arbitrary and prescribed non zero natural number. In the literature computable solutions are usually of order
Our strategy comes like this:
1- In first place we compute a th order outer expansion to
2- We compute a th order corrector which operates only within the boundary layer
3- We adjust the outer expansion in subtracting from it the corrector.
4- We make use of the Hilbert spaces method to show that we get an approximation solution which converges strongly and uniformly to and that this uniform convergence is valid all over the reference geometric domain including both the sub-domains inside and outside the boundary layer.
This strategy provides also a precise localization of the boundary layer.