Hi,
On 01.02.05, John Peterson wrote:
> > On 01.02.05, KIRK, BENJAMIN (JSCEG) (NASA) wrote:
> > > Sure... Examples 3 and 4 use 9noded biquadratic quadrilaterals in 2D.
> > > Also, the middle node need not lie in the geometric center of the two
> > > vertices. By default, the secondorder elements in libMesh are treated
> > > isoparametrically, which allows them to approximate curved boundaries with a
> > > quadratic line segment.
> > thanks for the quick answer.
> >
> > I would like to solve a Stokes equation which is a vector equation.
> > With higherorder elements the ChristoffelSymbols in the equation are
> > not zero anymore. Is there an interface to get the
> > ChristoffelSymbols, similar to the integrationweights?
>
> Hm...it sounds like you are some kind of winkydink mathematician
> or something ;) There's no interface like that in libmesh,
> in fact I've never heard of Christoffel symbols...are you solving
> some kind of problem on a nonEuclidean manifold?
No, the space is perfectly Euclidean. But I try to calculate a free
surface boundary of a fluid governed by a Stokes equation. For this I
need to take the geometry of the elements, i.e. the Euclidean
coordinates of the nodes into account.
Now, if we use arbitrary curved reference coordinates xi_1 and xi_2
for an element, then we may simply calculate the covariant derivatives
of a scalar field phi as
phi,i = dphi / dxi_i
and we can get back to the Cartesian derivative in, let's say
xdirection:
dphi / dx = (dphi / dxi_i) g^{ij} (dx / dxi_j)
with a sum over i and j.
For a vecor field, however, the derivative is somewhat more
complicated. We have to take into account that the base
vector may change in the direction we want to calculate the
derivative.
So, if we have a velocity field in 2D (u,v) and want to calculate some
derivatives of the first component, then we will get the changes of
the coordinate system in that direction  they head in the direction
of the second base vector  and get terms with v also.
To summarize this, in curved coordinates there is a difference between
a field that is really a vector (with two coordinates) and two scalar
fields. In the derivatives they behave differently.
Michael.

Michael Schindler
mail: Theoretische Physik I
Universität Augsburg
86135 Augsburg
email: michael.schindler@...
http: http://www.physik.uniaugsburg.de/~schindmi
Tel: +49 (0)821 5983230
Fax: +49 (0)821 5983222
"A mathematician is a device for turning coffee into theorems"
Paul Erdös.
