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.