[firedrake] hybridisation and tensor-product multigrid

[Colin Cotter]

I'm happy to use FacetElement in this way, but there is one tiny wrinkle
which hopefully we can figure out, namely that in TraceElement we dotted
with the local normal. If we do this with FacetElement, we have to dot with
the global normal, which leads to possible sign changes (you have to either
choose "+" or "-"). This is fine for implementing hybridisation, but it has
implications for the preconditioner, since the trace variable is used as an
approximation to the DG variable solution. If we don't choose the sign
correctly, then having a positive FacetElement basis coefficient could lead
to a negative value of the trace variable, and then we can't map directly
from FacetElement basis coefficients to CG basis coefficients in the
preconditioner.

To make this work, you need to consistently choose "+" or "-" for the
normal. Is it the convention that positive H(div) basis coefficient leads
to positive value of u.n("+")? I guess this is the real question I wanted
to ask Marie yesterday, but hadn't formulated it right in my head.

cheers
---cjc

On 16 March 2015 at 13:46, Andrew McRae <a.mcrae12 at imperial.ac.uk> wrote:

> 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/
>>>
>>> _______________________________________________
>>> firedrake mailing list
>>> firedrake at imperial.ac.uk
>>> https://mailman.ic.ac.uk/mailman/listinfo/firedrake
>>>
>>
>>
>
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