[firedrake] hybridisation and tensor-product multigrid

[Miklos Homolya]

Andrew said a long time ago, that trace elements are basically facet 
elements dotted with the facet normal. This latter approach probably 
works on quads.

On 16/03/15 13:25, Colin Cotter 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 
> <mailto: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 <tel:%2B44%201225%2038%206241>
>     e.mueller at bath.ac.uk <mailto:e.mueller at bath.ac.uk>
>     http://people.bath.ac.uk/em459/
>
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>
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