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The role of the error function in a singularly perturbed convection-diffusion problem in a rectangle with corner singularities. (English) Zbl 1174.35032
The authors deal with an two-dimensional linear elliptic convection-diffusion problems: find a function $u\in C(\overline\Omega)\cap D^2(\Omega)$ such that $$-\varepsilon\Delta u+ v\cdot\nabla u= 0,\quad x\in\Omega\subset \Bbb R^2,$$ $$u|_{\partial\Omega}= f(\widetilde x),\quad \widetilde x\in\partial\Omega,$$ where $\varepsilon$ is a small positive parameter, $V$ is the convection vector, $\widetilde x$ is a variable which lives in $\partial\Omega$, and $D^2(\Omega)$ is the set of functions with partial derivatives up to order two defined in all points of $\Omega$. The authors derive asymptotic expansion of the solution for $\varepsilon\to 0$. Moreover, they derive also asymptotic approximations near the points of discontinuity of the boundary condition.

35J25Second order elliptic equations, boundary value problems
35B25Singular perturbations (PDE)
35C20Asymptotic expansions of solutions of PDE
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