Asymptotic approximation of singularly perturbed convection-diffusion problems with discontinuous derivatives of the Dirichlet data

Abstract

We consider a singularly perturbed convection-diffusion equation, − ϵ △ u + v → ⋅ ∇ → u = 0 -\epsilon \bigtriangleup u +\overrightarrow v\cdot \overrightarrow \nabla u=0 , defined on two domains: a quarter plane, ( x , y ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) (x,y)\in (0,\infty )\times (0,\infty ) , and a half plane, ( x , y ) ∈ ( − ∞ , ∞ ) × ( 0 , ∞ ) (x,y)\in (-\infty ,\infty )\times (0,\infty ) . We consider for these problems Dirichlet boundary conditions with discontinuous derivatives at some points of the boundary. We obtain for each problem an exact representation of the solution in the form of an integral. From this integral we derive an asymptotic expansion of the solution when the singular parameter ϵ → 0 + \epsilon \to 0^+ (with fixed distance r r to the points of discontinuity of the boundary condition). It is shown that, in both problems, the first term of the expansion contains the primitive of an error function. This term characterizes the effect of the discontinuities on the ϵ − \epsilon - behaviour of the solution and its derivatives in the boundary or internal layers.