[firedrake] hybridisation and tensor-product multigrid

[Miklos Homolya]

Hi,

I'm not quite sure, but I think BrokenElement is probably what you need.
BrokenElement keeps all the dofs and transformations and their meaning, 
but re-associates all dofs with the cell interior.

Regards,
Miklos


On 17/03/15 13:29, Eike Mueller wrote:
> Hi again,
>
>> 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.
>
> we would also need a discontinuous HDiv space, i.e. e.g. discontinuous 
> versions of RT0 and BDFM1. How can I get those?
>
> Thanks,
>
> Eike
>
>> 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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