| This thesis investigates numerical methods for solving stationary and evolutionary initial and boundary value problems arising from convection-dominated convection diffusion equations. In particular, we develop and give comprehensive analysis for several stabilization schemes involving multiscale discretization of the spatial domain. Initially, we present a finite element scheme which adds a parameterized artificial viscosity term acting only on the fine scales of a finite element mesh. Analysis is completed for the semi-discrete case and the fully discrete case with a Crank-Nicholson time discretization. An analysis of parameter selection is completed for several common finite element spaces. The ideas behind this scheme are then extended to finite differences. We present a comprehensive analysis of this formulation using Fourier methods. This method is then coded in Matlab and results are presented for a number of benchmark problems taken from the literature. |
