This work is a numerical study of Burgers' equation with Neumann boundary conditions. The goal is to determine the long term behavior of solutions. We develop and test two separate finite element and Galerkin schemes and then use those schemes to compute the response to various initial conditions and Reynolds numbers. It is known that for sufficiently small initial data, all steady state solutions of Burgers' equation with Neumann boundary conditions are constant. The goal here is to investigate the case where initial data is large. Our numerical results indicate that for certain initial data the solution of Burgers' equation can approach non-constant functions as time goes to infinity. In addition, the numerical results raise some interesting questions about steady state solutions of nonlinear systems. / Master of Science