ABSTRACT
In this chapter, we switch the discussion of finite difference method (FDM) to time-independent problems. In Sec. 4.1, we construct a difference scheme for solving the two-dimensional Poisson’s equation. Then in Sec. 4.2, we briefly introduce both direct methods and iterative methods for solving a general system of linear equations, such as that resulting from the difference scheme for the Poisson’s equation. Sec. 4.3 is devoted to the error analysis of the difference method for solving elliptic equations. In Sec. 4.4, we extend the construction of difference methods to some other elliptic problems. Finally, we present an examplary MATLAB code for solving the Poisson’s equation.
Suppose that Ω is a bounded domain of R2 with boundary ∂Ω. The equation
