CHAPTER 22 - DIRECT SOLUTION OF ELLIPTIC FINITE-DIFFERENCE EQUATIONS

Abstract

This chapter discusses the direct solution of elliptic finite-difference equations. The classification of partial differential equations into elliptic, parabolic, and hyperbolic types depends only on the coefficients of the equation and not on any auxiliary conditions that can be imposed on the solution. However, problems of each type are normally associated with particular forms of the auxiliary conditions, and well-behaved solutions do not necessarily exist for arbitrary conditions. An elliptic problem is represented in finite differences by a set of simultaneous equations for the function values at mesh points that are linear if the original equation is linear in the function and its derivatives. Nonlinear equations are usually solved by some form of linearization, together with an iterative process such as Newton's method, so that the finite-difference representation gives a set of simultaneous linear equations. To obtain reasonable accuracy, it is often necessary to take several hundred or several thousand mesh points in the region. Any efficient method of solution must take advantage of the simple form of the matrix of coefficients.