[petsc-dev] 2D finite elements in 3D ambient space

[Jed Brown]

Matthew Knepley <knepley at gmail.com> writes:
>> Unfortunately, something more is required for higher order accuracy,
>> since naively the coordinate section itself would have to be higher
>> order, and this would require lots of changes (the equivalent of
>> DMPlexComputeCellGeometry would be called once per quadrature point
>> instead of once per element).
>>
>
> I have never been convinced that isoperimetric stuff produces enough
> benefit for its complication. Polynomials are not good approximators
> for the Jacobian of these transforms. NURBS are so much better. 

These issues are orthogonal.  If the mapping is not affine, you need
separate Jacobians at each quadrature point.  Non-affine elements are
required for high-order accuracy with curved boundaries, and in many
cases when using quad and hex elements.

The value of NURBS is that (a) some coordinate transformations can be
represented exactly and (b) for certain problems, the rest solution can
be represented exactly in the ansatz space.  Quadrature error does not
magically vanish.
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