Vrftsjtryk

I am working on an error estimation strategy for a finite element code working on unstructured anisotropic meshes. In order to compute this error estimator I need the characteristic lengths of any hexahedron that is part of my mesh. In the case of a cartesian mesh I could get easily the values $Delta x$, $Delta y$ and $Delta z$, however in the generic case I need to both identify the principal axes of the element and the associated characteristic lengths. I guess a similar problem is found when trying to evaluate the CFL condition on highly anisotropic meshes.

In the case of a cartesian mesh my error indicator will look something like:

begin{equation}
e=Delta x vertfrac{partial{f}}{partial{x}}vert + Delta y vertfrac{partial{f}}{partial{y}}vert + Delta z vertfrac{partial{f}}{partial{z}}vert
end{equation}

whereas in the general case I am searching for something of the form

begin{equation}
e=h_1 vertfrac{partial{f}}{partial{n_1}}vert + h_2 vertfrac{partial{f}}{partial{n_2}}vert + h_3 vertfrac{partial{f}}{partial{n_3}}vert
end{equation}

where $n_1$, $n_2$ and $n_3$ are the principal axes and $h_1$, $h_2$ and $h_3$ are the corresponding characteristic lengths.

I was thinking about some form of enclosed ellipsoid but I haven't found any interesting solution yet.