Derivation and Solution of Elliptic Difference Equations

The basis aims in this chapters are to derive finite difference approximations for certain elliptic differential equations, to study the properties of the associated matrix equations, and to describe rigorous iterative methods for the solution of such matrix equations, In certain cases, the derived properties of the associated matrix equations are sufficiently powerful to allow us to prove rather easily (Theorem 6.2) that these finite difference approximations become more accurate with finer mesh spacings, but results in this area, especially in higher dimensions, require detailed developments which we omit. Rather, in this chapter we concentrate on useful methods for deriving finite difference approximations of boundary-value problems which apply to internal interfaces, which are of interest in reactor physics.