About the numerical solution.

  To solve the steady heat conduction in the plate numerically, we can restrict the number of points again to a finite set of mesh points as in figure 15. But in this case, there is no time coordinate that only allows influences for later times. The mesh point values depend on the boundary conditions everywhere. Typically, the mesh point values are all determined at the same time.

A considerable amount of work has been done to reduce the effort needed for such a solution. For example, for the present problem, a Fourier series expansion would be very helpful, especially since there is a very good algorithm, the Fast Fourier transform, to determine and evaluate Fourier series. Iterative methods, in which an initial guess for the solution is systematically improved, can also be very effective. The ``multigrid'' method is an iterative method which is extremely efficient here.

  

Figure 15: Typical numerical solution of the Laplace equation.