Pseudo Residual-Gree Bubble Functions for the Stabilization of Convection-Diffusion Prolems

Abstract

Convection - diffusion - reaction problems may contain thin regions in which the solution varies abruptly. The plain Galerkin method may not work for such problems on reasonable discretizations, producing non-physical oscillations. The Residual - Free Bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless in special limit cases. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. In my thesis we will examine a stabilization technique, based on the RFB method and particularly designed to treat the most interesting case of small diffusion in one and two space dimensions for both steady and unsteady convection - diffusion - reaction problems. We replace the RFB functions by their cheap, but efficient approximations which retain the same qualitative behavior. We compare the method with other stabilized methods.