[firedrake] hybridisation and tensor-product multigrid

Tue Mar 17 14:16:50 GMT 2015

Miklos: this is a bug, rather than by design.

On 17 March 2015 at 15:14, Homolya, Miklós <m.homolya14 at imperial.ac.uk>
wrote:

>  I'm afraid BrokenElement is not currently functional on extruded (i.e.
> OuterProductElements). They should still work on quadrilaterals though.
>
>
> On 17/03/15 14:08, Eike Mueller wrote:
>
> Hi Miklos,
>
>  thanks, that’s probably what I’m looking for. It works if I just create
> a BDFM1 element on a 2d grid, and then say
>
>  U1 = FiniteElement('BDFM',triangle,2)
> U1_broken = BrokenElement(U1)
>
>  However, the 3d code below which builds elements on an extruded mesh
> does not work. I’ve also tried to first create BrokenElement versions of U1
> and V1, and then combine them, but that doesn’t work either.
>
>  Thanks,
>
>  Eike
>
>  PS: I started writing a class for locally assembling matrices (for this
> class it does not matter whether the function spaces are discontinuous or
> not, but the multiplying them locally of course only makes sense if the
> spaces are discontinuous), see here:
> https://github.com/firedrakeproject/firedrake-helmholtzsolver/blob/hybridization/source/bandedmatrix/locallyassembledmatrix.py
> It has methods for assembling a UFL form, appling the locally assembled
> matrices to a vector, adding and multiplying locally assembled matrices and
> calculating the inverse (using dgels from LAPACK).
>
>  from firedrake import *
>
> host_mesh = UnitIcosahedralSphereMesh(0)
> mesh = ExtrudedMesh(host_mesh,layers=4,extrusion_type='radial')
>
> U1 = FiniteElement('BDFM',triangle,2)
> U2 = FiniteElement('DG',triangle,1)
> V0 = FiniteElement('CG',interval,2)
> V1 = FiniteElement('DG',interval,1)
>
> W2_elt = HDiv(OuterProductElement(U1,V1))+HDiv(OuterProductElement(U2,V0))
> W2_broken_elt = BrokenElement(W2_elt)
>
> W = FunctionSpace(mesh,W2_broken_elt)
>
>  WARNING: Creating an EnrichedElement,
>          if you intended to create a MixedElement use '*' instead of '+'.
> Traceback (most recent call last):
>   File "brokenelement.py", line 14, in <module>
>     W = FunctionSpace(mesh,W2_broken_elt)
>   File
> "/Users/eikemueller/PostDocBath/EllipticSolvers/PyOP2/pyop2/caching.py",
> line 160, in __new__
>     obj = make_obj()
>   File
> "/Users/eikemueller/PostDocBath/EllipticSolvers/PyOP2/pyop2/caching.py",
> line 141, in make_obj
>     obj.__init__(*args, **kwargs)
>   File
> "/Users/eikemueller/PostDocBath/EllipticSolvers/firedrake/firedrake/functionspace.py",
> line 540, in __init__
>     super(FunctionSpace, self).__init__(mesh, element, name, dim=1)
>   File
> "/Users/eikemueller/PostDocBath/EllipticSolvers/firedrake/firedrake/functionspace.py",
> line 52, in __init__
>     self.flattened_element = self.fiat_element.flattened_element()
> AttributeError: Discontinuized instance has no attribute
> 'flattened_element'
>
>    --
>
>  Dr Eike Hermann Mueller
> Lecturer in Scientific Computing
>
> Department of Mathematical Sciences
> University of Bath
> Bath BA2 7AY, United Kingdom
>
> +44 1225 38 5557
> e.mueller at bath.ac.uk
> http://people.bath.ac.uk/em459/
>
>  On 17 Mar 2015, at 13:34, Miklos Homolya <m.homolya14 at imperial.ac.uk>
> wrote:
>
>  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> 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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