The role of the error function in a singularly perturbed convection-diffusion problem in a rectangle with corner singularities. (English) Zbl 1174.35032

López, José L.; Pérez Sinusía, Ester

Proc. R. Soc. Edinb., Sect. A, Math. 137, No. 1, 93-109 (2007).

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 \[ -\varepsilon\Delta u+ v\cdot\nabla u= 0,\quad x\in\Omega\subset \mathbb R^2, \]
\[ u|_{\partial\Omega}= f(\widetilde x),\quad \widetilde x\in\partial\Omega, \] 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.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35J25	Boundary value problems for second-order elliptic equations
35B25	Singular perturbations in context of PDEs
35C20	Asymptotic expansions of solutions to PDEs

Keywords:

convection-diffusion problems; singularly perturbed equation; asymptotic expansion