This thesis studies the convergence of a singularly perturbed two-dimensional problem of the convection-diflfusion type. The problem is solved using the bilinear finite element method on a Shishkin Mesh. This thesis will consider the results of two separate types of Shishkin Meshes, as well as a quick consideration of the uniform mesh and its shortcomings. Results will show a superconvergence rate close to 0 using a discrete energy norm. Results will also consider stability of the method by examining the condition number of the element stiffness matrix.