The efficient computation of the solution to self-adjoint elliptic operators is the subject of this dissertation. Discretization of this equation by finite differences or finite elements yields a large, sparse, symmetric system of equations, Ax = b. We use the preconditioned conjugate gradient method with domain decomposition to develop an effective, vectorizable preconditioner which is suitable for solving large two-dimensional problems on vector and parallel machines. The convergence of the preconditioned conjugate gradient method is determined by the condition number of the matrix M('-1)A where A and M correspond to the matrix for the discretized differential equation and to the preconditioning matrix, respectively. By appropriately preconditioning the system Ax = b we can significantly reduce the computational effort that is required in solving for x. The basic approach in domain decomposition techniques is to break up the domain of integration into many pieces, solve the appropriate equation on each piece, then somehow construct the global solution from these local solutions. In this dissertation we formulate an effective preconditioner for two-dimensional elliptic partial differential equations using this notion of domain decomposition. We demonstrate that this method is efficient in its vectorized form and present numerical results to support this conclusion.