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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.
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Next: Flow around thin
Up: The Laplace equation.
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