Elliptic Equations

Abstract

In the case of elliptic equations the canonical form is Laplace’s equation which is therefore the basis of the work in this chapter. With elliptic equations in the plane, the boundary conditions are specified round a closed curve, and the finite difference schemes then lead to a large set of linear algebraic equations for the complete set of unknowns. Elliptic equations are like boundary value problems in ordinary differential equations in which there is no step-by-step procedure such as those employed with parabolic equations in Chapter 2 and hyperbolic equations in Chapter 3. Hence most of the difficulty in the solution of elliptic equations lies in the solution of large sets of algebraic equations, and in the representation of curved boundaries. Iterative methods for the solution of linear algebraic equations have been considered in Chapter 1 and further examples of their application will arise here.