ABSTRACT
This chapter deals with discrete methods for elliptic problems. The simplest example of an elliptic partial differential equation is the Poisson equation in two space dimensions. Elliptic equations and systems of elliptic equations typically appear when modeling various steady-state (i.e., time-independent) processes and phenomena. The chapter addresses two questions. First, it shows that a simple central-difference scheme for the Dirichlet problem on a rectangular domain is consistent and stable, and as such, converges when the grid is refined. Then, the chapter provides a very brief and introductory account of the method of finite elements, including variational formulations of boundary value problems, the Ritz and Galerkin approximations, and basic concepts related to convergence.
