|
|
Re: Finite Difference vs. Finite Element
Posted:
Apr 28, 2000 2:37 PM
|
|
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...
>
> 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.
|
|
