[firedrake] hybridisation and tensor-product multigrid

Tue Mar 17 14:08:37 GMT 2015

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 <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 <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/ <http://people.bath.ac.uk/em459/>
>>> 
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>>> 
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>> 
>> 
>> 
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