Asymptotic approximations for a singularly perturbed convection–diffusion problem with discontinuous data in a sector

We consider a singularly perturbed convection–diffusion equation, - ε Δ u + v → · ∇ → u = 0 on an arbitrary sector shaped domain, Ω ≡ { ( r , φ ) | r > 0 , 0 < φ < α } being r and φ polar coordinates and 0 < α < 2 π . We consider for this problem discontinuous Dirichlet boundary conditions at the corner of the sector: u ( r , 0 ) = 0 , u ( r , α ) = 1 . An asymptotic expansion of the solution is obtained from an integral representation in two limits: (a) when the singular parameter ε → 0 + (with fixed distance r to the discontinuity point of the boundary condition) and (b) when that distance r → 0 + (with fixed ε ). It is shown that the first term of the expansion at ε = 0 contains an error function. This term characterizes the effect of the discontinuity on the ε -behaviour of the solution and its derivatives in the boundary or internal layers. On the other hand, near discontinuity of the boundary condition r = 0 , the solution u ( r , φ ) of the problem is approximated by a linear function of the polar angle φ .