182 - Elliptic Equations: Finite Differences

Abstract

This chapter focuses on elliptic equations applicable to elliptic partial differential equations. A numerical approximation of the solution is obtained. By use of finite differences, a simultaneous system of equations may be determined. The solution of this algebraic system yields a numerical approximation to the differential equation. The method is simply to use finite differences everywhere, and solve the resulting set of simultaneous equations. Because elliptic equations are boundary value problems, the solution at all points in the domain must be determined simultaneously. If the original elliptic equation and the boundary conditions are linear in the independent variable, then the resulting system of equations will be linear. The most common type of elliptic systems has linear equations and linear boundary conditions. For this type of elliptic system, a standard linear equation solver may be used. If the system of linear equations is too large to solve directly, an iterative method may be used.