Accurate simulations of elasticity properties can be constructed by solving second order elliptic boundary value problems which have been approximated using finite elements. This talk will examine the process of converting the given PDE into a weaker form and applying the Galerkin method. In addition, novel MATLAB programs will be introduced, which will display a visual depiction of an object after force is applied, given a subdivision of the shape into regular or irregular triangles, a Dirichlet boundary condition, and a two dimensional force function.