The ill-conditioned Poisson matrix developed for the five-point central finite difference approximation to the solution of Poisson’s equation is positive definite, so the preconditioned method...

The ill-conditioned Poisson matrix

developed for the five-point central finite difference approximation to the solution of Poisson’s equation is positive definite, so the preconditioned

method applies. To use

we will need to construct the block tridiagonal matrix. Fortunately, MATLAB builds the sparse

<sup>2

2</sup>
matrix with the command

Modify the Poisson equation solver sorpoisson referenced in Section 20.5 to use the preconditioned

method, where

The function declaration should be function [x y u] = cgpoisson(n,f,g,numiter).

Test cgpoisson using the following problem with

maxiter

and tol
⁻¹⁰

The following Poisson problem describes the electrostatic potential field induced by charges in space, where

is a potential field and

is a charge density function.

Assume that the electrostatic potential fields are induced by approximately

randomly placed point charges with strength

The edges are grounded, so
on

Note that the right-hand side function

affects only the points inside the boundary. Solve the problem using

and draw a surface plot and a contour plot of the result

shows sample plots
One way of coding the right-hand side is