dolfin team mailing list archive

["Johan Jansson" <jjan@xxxxxxxxxxx>]

> On Fri, Feb 09, 2007 at 03:36:41PM +0100, Garth N. Wells wrote:
>> >
>> >Added test + demos for projection between non-matching meshes, seems
>> >to work ok. Point evaluation on the intersection of several cells now
>> >computes the average of the cells. Some optimizations still remain to
>> >do: i.e. exploit that we know which cell we want to evaluate a
>> >function on.
>> >
>>
>> Great! I look forward to testing it out.
>>
>> Garth
>>
>
> Does this work for general elements also?
>
> /Anders

Yes, this works for general elements. It requires access to basis function
evaluation though, and right now you have to implement the ones you need
manually.

The key part of the projection is:

integral_K f * v_i

where f is defined on finite element space A and v_i is test function i on
finite element space B (the target for projection).

There are two options for evaluating this:

1. Generate a new triangulation of the whole domain Omega which includes
both the triangulation of A and B. Then the above integral can be computed
exactly as a sum of integrals on the triangulation of K.

2. Allow f to be defined by cells which don't match K, and use an
approximation of f. The approximation we use is a local projection on K to
some finite element space C (typically B or a higher order of B). In
practice this will involve computing the local projection with the
quadrature of C (which is exact for C but not for f).

1 is attractive because it's exact, but it's likely far too expensive (it
involves generating a new temporary non-trivial mesh). 2 is what's
currently implemented. Some work remains to investigate the effects of the
quadrature error on the global projection, and how the quadrature performs
for piecewise discontinuous f. 2 is always exact for refinement, but not
for coarsening.

  Johan