The purpose of this project is to derive stability estimates for a finite element method for linear, elliptic partial differential equation in non-divergence form. The thesis begins by introducing basic definitions of the Sobolev spaces used and the corresponding norms of these spaces. We then define the finite element method of the problem. We dedicate one chapter to working out what the finite element method reduces to when we are in one dimension. Chapter 4 involves preliminary lemmas which we will lead up to the proof of the main result. The proof of these lemmas, including the main result, involve a common theme of using inverse estimates, interpolation estimates, and various other inequalities. Once we prove the main result, we then prove existence, uniqueness, and error estimates of the solution of the finite element method. The last chapter is dedicated to numerical experiments. We choose three test problems in the one dimensional case and discuss error and convergence rates of each, as well as whether each problem supports the theoretical estimates.
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