[firedrake] hybridisation and tensor-product multigrid

[Andrew McRae]

Officially I'd prefer you to use FacetElement and write dot(u, n).

Trace elements were a quick hack.  Their FIAT implementation essentially
does this, though on the reference cell, inside the tabulate() function.

If anyone wants to spend half-a-day extending FIAT's trace.py to work with
arbitrary H(div) elements then be my guest.

On 16 March 2015 at 14:25, Cotter, Colin J <colin.cotter at imperial.ac.uk>
wrote:

>    Hi Eike,
>    If you take a look at the test_hybridisation_inverse branch, in
> tests/regression/test_hybridisation_schur, you'll see a hacked up attempt
> at doing this for simplices. It's a bit fiddly because you need to assemble
> the form multiple times, once as a mixed system and once as a single block,
> so I'm thinking of making a tool to automate some of this by doing
> automated substitutions in UFL. Lawrence and I said we might try to sketch
> out how to do this.
>
>  Another slight problem is that we don't have trace elements for
> quadrilaterals or tensor product elements at the moment. Our approach to
> trace spaces is also rather hacked up, we extract the facet basis functions
> from an H(div) basis and the tabulator returns DOFs by dotting the local
> basis functions by the local normal.
>
>  Andrew: presumably you didn't implement them because you anticipated some
> fiddliness for tensor-products?
>
>  cheers
>  --cjc
>
> On 16 March 2015 at 08:49, Eike Mueller <E.Mueller at bath.ac.uk> wrote:
>
>> Dear firedrakers,
>>
>> I have two questions regarding the extension of a hybridised solver to a
>> tensor-product approach:
>>
>> (1) In firedrake, is there already a generic way of multiplying locally
>> assembled matrices? I need this for the hybridised solver, so for example I
>> want to (locally) assemble the velocity mass matrix M_u and divergence
>> operator D and then multiply them to get, for example:
>>
>> D^T M_u^{-1} D
>>
>> I can create a hack by assembling them into vector-valued DG0 fields and
>> then writing the necessary operations to multiply them and abstract that
>> into a class (as I did for the column-assembled matrices), but I wanted to
>> check if this is supported generically in firdrake (i.e. if there is
>> support for working with a locally assembled matrix representation). If I
>> can do that, then I can see how I can build all operator that are needed in
>> the hybridised equation and for mapping between the Lagrange multipliers
>> and pressure/velocity. For the columnwise smoother, I then need to extract
>> bits of those locally assembled matrices and assemble them columnwise as
>> for the DG0 case.
>>
>> (2) The other ingredient we need for the Gopalakrishnan and Tan approach
>> is a tensor-product solver in the P1 space. So can I already
>> prolongate/restrict in the horizontal-direction only in this space? I
>> recall that Lawrence wrote a P1 multigrid, but I presume this is for a
>> isotropic grid which is refined in all coordinate directions. Again I can
>> probably do it 'by hand' by just L2 projecting between the spaces, but this
>> will not be the most efficient way. Getting the columnwise smoother should
>> work as for the DG0 case: I need to assemble the matrix locally and then
>> pick out the vertical couplings and build them into a columnwise matrix,
>> which I store as a vector-valued P1 space on the horizontal host-grid.
>>
>> Thanks a lot,
>>
>> Eike
>>
>> --
>> Dr Eike Hermann Mueller
>> Lecturer in Scientific Computing
>>
>> Department of Mathematical Sciences
>> University of Bath
>> Bath BA2 7AY, United Kingdom
>>
>> +44 1225 38 6241
>> e.mueller at bath.ac.uk
>> http://people.bath.ac.uk/em459/
>>
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>>
>
>
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