From: Arian Novruzi <ariannovruzi@idirect.com>
Subject: Re: Finite Difference vs. Finite Element
Date: Thu, 27 Apr 2000 21:42:36 GMT
Newsgroups: sci.math.num-analysis
Summary: [missing]

[MIME wrappers deleted --djr]

telford@xenon.triode.net.au wrote:

> Arian Novruzi <ariannovruzi@idirect.com> wrote:
> > This is a multi-part message in MIME format.
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>
> >> ..
> >>
> >> You can accomplish this with finite differences also.  In our PDE class we
> >> took the fourier transform of a modified difference scheme which had a
> >> diffusion term and adjusted the parameters to obtain the correct decay and
> >> dispersion for the waves.  After that, everything ran just fine...  As a
> >> matter of fact, controlling these things in finite differences was far easier
> >> than for finite elements.
>
> > Leaving out the technical problems related to the geometry, the main difference
> > between FEM and FDM (for one problem of order 2 for example) is:
> > -in FEM we need only H^1 regularity. In this case L^2 error is of order h.
> > If the solution is H^m the L^2 error is of order h^m ...
> > -in FDM we need C^2 regularity. Then, the C^0 error is h^2. If the solution is
> > C^m the error is h^m ...
>
> Can you explain to me what that means for a physical problem?
>
>         - Tel

If you can not make a mathematical analysis of regularity but by physical
considerations
you know that the solution is regular enough (at least C^2 := two times continuosly
derivable) doesn't matter which method you use. Practically the error is the same
(but the error norm  is different - however involving the some derivatives).
If your solution is not C^2 (for example if your domain has "fissures" or your data
are not regular enough) then the method which works well is FEM. This method gives
the error with the Sobolev norms. For example, if you have to solve -\Delta u = f
with homogenous boundary data, if u is H**{1+\alpha} with 0<alpha<1 (basically
this means u is 1+\alpha times derivable) than the H^1 error (= \Int (u-u_h)**2 +
(Du-Du_h)**2) is bounded by C h**\alpha.

Arian.

[deletia --djr]
==============================================================================

From: Peter Hansbo <hansbo@solid.chalmers.se>
Subject: Re: Finite Difference vs. Finite Element
Date: Fri, 28 Apr 2000 19:37:33 +0100
Newsgroups: sci.math.num-analysis
Summary: [missing]

Maybe it should be pointed out that Sobolev theory
just happens to be the right framework for finite elements.
This should not be construed as an intrinsic difference between
finite differences and finite elements.
After all, on structured grids, finite elements + quadrature \subset finite
differences.
On unstructured grids the problem is how to define finite differences...

[quote of Novruzi paragraph deleted --djr]