Singular Perturbation Problems and Multiple Scales

Wednesday, January 9, 2002 - 3:30pm - 3:55pm
Keller 3-180
Singular perturbation problems (SPPs) arise in many application areas, such as in chemical kinetics, fluid dynamics and system control, plate and shell problems, etc. Such problems usually contain one or more small parameters in the equations. Solutions of these problems undergo rapid changes within very thin layers near the boundary or inside the problem domain. Such sharp transitions require very fine meshes inside those thin layers to resolve the fine scales.

In this talk, we will first review several numerical techniques developed in the past, especially finite element methods (FEM). Then we introduce some highly non-uniform anisotropic meshe which can be used to solve SPPs efficiently. However, such highly non-uniform mesh complicates the error analysis, which frequently assumes quasi-uniformity in the classical finite element analysis. Here we will present the special techniques, which can be used to prove the global uniform convergence and superconvergence. Finally, numerical experiments supporting the theoretical analysis will be presented.