Finite Difference and Finite Element Methods for Solving Elliptic Partial Differential Equations

Elliptic partial differential equations appear frequently in various fields of science and engineering. These involve equilibrium problems and steady state phenomena. The most common examples of such equations are the Poisson's and Laplace equations. These equations are classified as second order linear partial differential equations. Most of these physical problems are very hard to solve analytically, instead, they can be solved numerically using computational methods. In this thesis, boundary value problems involving Poisson's and Laplace equations with different types of boundary conditions will be solved numerically using the finite difference method (FDM) and the finite element method (FEM). The discretizing procedure transforms the boundary value problem into a linear system of n algebraic equations. Some iterative techniques, namely: the Jacobi, the Gauss-Seidel, Successive over Relaxation (SOR), and the Conjugate Gradient method will be used to solve such linear system. Numerical results show that the finite difference method is more efficient than the finite element method for regular domains, whereas the finite element method is more accurate for complex and irregular domains. Moreover, we observe that the SOR iterative technique gives the most efficient results among the other iterative schemes.