We consider several model problems from a class of elliptic perturbation equations in two dimensions. The domains, the differential operators, the boundary conditions, and so on, are rather simple, and are chosen in a way that the solutions can be obtained in the form of integrals or Fourier series. By using several techniques from asymptotic analysis (saddle point methods, for instance) we try to construct asymptotic approximations with respect to the small parameter that multiplies the differential operator of highest order. In particular we consider approximations that hold uniformly in the so-called boundary layers. We also pay attention to how to obtain a few terms in the asymptotic expansion by using direct methods based on singular perturbation methods.

Singular perturbations (msc 35B25), Solutions in closed form (msc 35C05), Asymptotic expansions (msc 35C20), Boundary value problems for second-order elliptic equations (msc 35J25), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60)
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Temme, N.M. (1997). Analytical methods for a selection of elliptic singular perturbation problems. Modelling, Analysis and Simulation [MAS]. CWI.