On 7/2/21 10:44 AM, Simon Wiesheier wrote:
I scaled the x and y coordinates so that they are both between -1 and 1.
Actually this brought the condition numbers down to 1e-04.
I am just a little bit surprised that my results are nearly the same as those
obtained with reciprocal conditon numbers smaller than my machine accuracy,
i.e. about 1e-18.
>
> Would you say that one can continue work with reciprocal condition numbers of
> 1e-04 or is this still too big?
A condition number of 1e4 is quite moderate and should not cause any undue
problems.
The monomial basis you chose is known to be quite poorly conditioned. As Bruno
already remarked, a better conditioned basis is to use
1, (x-x0)/dx, (y-y0)/dy, (x-x0)^2/dx^2, ...
where x0,y0 are the center of the cell and dx, dy are the extents of the cell
in x and y direction (or you can choose dx=dy=h if your cells are all
reasonably well behaved). This basis is appropriate if you only need linear
and quadratic terms, but not if you go to higher polynomial degrees. In that
case, it is useful to go with something like Chebyshev or Legendre
polynomials. That's what all higher order finite element implementations are
based on.
Best
W.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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