This chapter studies finite difference methods for elliptic problems. It begins with a rather lengthy and general discussion of grid domains, grid functions, finite difference operators, and their consistency. We then introduce the notion of stability of a finite difference scheme and Lax’s principle: a consistent and stable scheme is convergent. Then we apply all these notions to elliptic operators in one and two dimensions, with the main focus being the Laplacian. We show the discrete maximum principle, energy arguments and how these can be used to attain stability and convergence in various norms. For more general operators we introduce the notions of homogeneous schemes and upwinding. For operators in divergence form we provide an analysis via energy arguments. For non divergence form operators we analyze the monotonicity and comparison principles of the arising schemes.

In this expository paper weanswertwo fundamental questions concerning discrete magnetic Schrödinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators, i.e., when they have self-adjoint realizations. If such self-adjoint realizations exist, we explore when they are unique.