This example shows the application of the Poisson equation in a thermodynamic simulation. The equation system consists of four points from which two are boundary points with homogeneous Dirichlet boundary conditions. Furthermore a constant right hand source term is given which equals unity. The four points are given on a straight line in equal distances according to Figure 4.1. The governing equation yields for the interior points and for the boundary points.

In the equation system two unknown variables are defined, namely temperature values for the two inner vertices. The discretization formula resulting from finite differences for the vertices yields
(4.15) 
(4.17) 
(4.18) 
The solution vector contains the single solution elements . These elements are applied to the quantities according to (4.12). This means, that the solution is added to the respective solution quantity . If the initial value of the quantity is zero, the solution quantity has the same value of the solution of the initially given problem. Otherwise, the solution contains an update vector which has to be added in order to obtain the correct solution. This is equivalent to the Newton method, however, since the equation is linear, the solution is obtained in a single step.
Michael 20080116