QuadratureElement: construct directly from a QuadratureRule

[pbrubeck]

Enables quadrature spaces on a QuadratureRule defined by a finat element

from finat.quadrature import QuadratureRule
V = FunctionSpace(mesh, "CG", degree, variant="integral")

fe = V.finat_element
_, ps = fe.dual_basis
scheme = QuadratureRule(ps, numpy.full(len(ps.points), numpy.nan), ref_el=fe.cell)

Q = FunctionSpace(mesh, "Quadrature", degree, quad_scheme=scheme)

[pbrubeck]

This approach applied to RT1 gives a scalar FunctionSpace with the same DOFs and entity associations/continuity as Crouzeix-Raviart. The reference values of RT1 live on this space, but maybe CrossMeshInterpolator would want a fully-discontinuous vector-valued quadrature space.

[pbrubeck]

This approach applied to RT1 gives a scalar FunctionSpace with the same DOFs and entity associations/continuity as Crouzeix-Raviart. The reference values of RT1 live on this space, but maybe CrossMeshInterpolator would want a fully-discontinuous vector-valued quadrature space.

Maybe we shouldn't care about the discontinuities of the tangential components, as they are not involved in the DOFs. It should be fine as long as the normal component gets interpolated into the vector-valued quadrature space.

[rckirby]

Looks fine to me, do we have a use case? Or is this just a convenience function to go straight from rules to elements?

Also, I'm not sure what it means to apply this to RT1?

[pbrubeck]

Looks fine to me, do we have a use case? Or is this just a convenience function to go straight from rules to elements?

The use case is cross-mesh interpolation, which currently only works on target spaces that have point evaluation DOFs only. This PR enables creating an auxiliary point-evaluation-only space using the PointSet in the dual basis of the target element, for which cross-mesh interpolation is implemented. From there, we can further interpolate into the actual target space by computing the DOFs using the point data at the quadrature points.

[pbrubeck]

Also, I'm not sure what it means to apply this to RT1?

For cross-mesh interpolation into RT1 we need to generate a vertex only mesh on the quadrature points used by RT1. On this vertex-only mesh we then create a vector-valued P0 space. The interpolation is performed in two steps first go from the source space into the P0 space, and then evaluate the RT1 dofs from the P0 point data.