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

• Authors: José L. López;Ester Pérez Sinusía
• Venue: Journal of Computational and Applied Mathematics
• Year: 2005

We consider a singularly perturbed convection-diffusion equation, -εΔu + v→ ċ ∇rarr;u = 0 on an arbitrary sector shaped domain. Ω ≡ {(r,φ)|r 0,0 r and φ polar coordinates and 0 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.