[firedrake] hybridisation and tensor-product multigrid

[Eike Mueller]

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/