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

\begin{displaymath}-\varepsilon \Delta u+\vec{b}(x,y)\centerdot \nabla u+c(x,y)u=f \ \textrm{in}\ \Omega ;\quad u=0 \ \textrm{on}\ \Gamma \eqno(1)\end{displaymath}

where $\,\Gamma = \Gamma _1\cup\Gamma _2\cup\Gamma _3\cup\Gamma _4,\quad $ $\Gamma _{\textrm{in}}=\Gamma _1\cup\Gamma _2,\quad $ $\Gamma _{\textrm{out}}=\Gamma _3\cup\Gamma _4,\, $ and $\,\vec{b}^T=(b_1,b_2).$

\begin{displaymath} \setlength{\unitlength}{1cm} \begin{picture}(8,8) \put(0,0)... ...6.5,0.2){$x$} \put(-0.4,4){$1$} \put(6,-0.4){$1$} \end{picture}\end{displaymath}

We assume

\begin{displaymath}\left\{\begin{array}{llll} (b_1,b_2)>(\beta _1,\beta _2)>(0,0... ...rac{1}{2}\textrm{div}b\ge\gamma >0\end{array} \right. \eqno(2) \end{displaymath}

Problem ($1$), 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 $\,\varepsilon =0\,$ is an hyperbolic one given by

\begin{displaymath}\vec{b}(x,y)\centerdot \nabla v+c(x,y)v=f \ \textrm{in}\ \Omega ;\quad v=0 \ \textrm{on}\ \Gamma _{\textrm{in}}\eqno(3).\end{displaymath}

So, for $\,0<\varepsilon <<1\,$ problem ($1$) exhibits a boundary layer phenomenon along the portion $\,\Gamma _{\textrm{out}}\,$ of the boundary $\,\Gamma \,$ which make difficult the computation of a ``good'' approximation solution to $\,u.\,$ The current paper is intended to construct an approximation solution to $\,u\,$ up to any higher order $\,{\cal{O}}({\varepsilon ^q})\,$ where $\,q\,$ is an arbitrary and prescribed non zero natural number. In the literature computable solutions are usually of order $\,{\cal{O}}({\varepsilon ^{1/2}}).\,$
Our strategy comes like this:
1- In first place we compute a $\,q$th order outer expansion to $\,u\,$
2- We compute a $\,q$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 $\,u\,$ and that this uniform convergence is valid all over the reference geometric domain $\,\Omega \,$ including both the sub-domains inside and outside the boundary layer.
This strategy provides also a precise localization of the boundary layer.