Convection-diffusion equations in a circle: the compatible case
The authors consider the following 2D singular perturbation problem \[ \begin{cases}-\varepsilon \Delta u^{\varepsilon}-\partial_y u^{\varepsilon}=f(x,y) &\text{in } D,\\ u^{\varepsilon}=0 &\text{on } \partial D,\end{cases}\tag{E} \] where \(D\) is the unit disc \(D=\{(x,y): x^2+y^2\leq 1\}\). They study the convergence of the family \(u^{\varepsilon}\) when \(\varepsilon\rightarrow 0\). As the points \((\pm 1,0)\) are characteristic for the limit problem, the boundary-layer estimates require special study near these points. The authors prove various convergence results and corrector formulas depending on the degree of compatibility conditions on the data at these characteristic points.
scientific article; zbMATH DE number 5925831
scientific article