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

摘要

We consider a singularly perturbed convection–diffusion equation, -εΔu+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<2π. 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.