| > <<< Note 1771.0 by POLAR::MOKHTAR >>>
> is finite element method to solve PDE the same as finite difference ?
No, but finite differences can be shown to be a subset of the
finite element method, as both really come from the use of
weighted residuals or variational principles.
To make this clear - suppose you have a finite difference scheme
in a planar region.
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-[1]-----[2]-----[3]-
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-[4]-----[5]-----[6]-
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-[7]-----[8]-----[9]-
Here, node 5 is connected by linear equations to nodes 2, 4, 6
by the finite difference scheme. For example, Laplaces equation
states that each node is the average of its neighbors.
If you hade a triangulated planar region using finite elements,
like this
[1]-----[2]-----[3]-----[4]
|\ | /|\ |
| \ | / | \ |
| \ | / | \ |
| \ | / | \ |
| \ | / | \ |
| \ | / | \ |
[5]-----[6]-----[7]-----[8]
|\ | /|\ |
| \ | / | \ |
| \ | / | \ |
| \ | / | \ |
| \ | / | \ |
| \ | / | \ |
[9]-----[A]-----[B]-----[C]
then node 6 would be related to all nodes which are vertices of all
triangles which have 6 as a vertex - that is 1, 2, 3, 5, 7, A.
So it is more general. The "connections" arise because on each
triangle we require that a miniature boundary value problem be
satisfied with the limited class of functions available which
interpolate on that triangle. Globally fitting all these together
leads to a system of linear equations and gives you a solution.
So you can see that the ideas are very similar, but finite elements
are much more flexible.
- Jim
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