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@bath.ac.uk
http://people.bath.ac.uk/em459/
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