Computational Methods for Singularly Perturbed Systems

The di culty encountered when solving singularly perturbed differential equations is that errors introduced in layers pollute the solution in smooth regions. Since a priori control of the errors in layers is di cult, special methods must be designed to reduce or eliminate polluting errors. Successful methods add dissipation to a computational scheme to enlarge layers to the mesh spacing. We focus on a method of using special quadrature rules to con ne spurious pollution e ects, such as excess di usion and non-physical oscillations, to layers. In particular, we indicate that Radau and Lobatto quadrature are useful for, respectively, convection-di usion and reaction-di usion systems. With large errors con ned to small regions, an adaptive technique can successfully improve accuracy. The quadrature approach is suitable for use with adaptive methods that both adjust meshes and vary method orders. We describe the key aspects of such an adaptive strategy and present several applications. We also demonstrate an equivalence between the quadrature-based methods and a generalized Galerkin least-squares stabilization.