Q2-P1 stokes element on FEniCSx

variational formulation
1

Hello all,

I want to reproduce the results of this paper, where the authors use the space Q2-P1 to solve the stationary Stokes equation.

In this space, P1 is discontinuous across elements and could be implemented in two ways:

  1. Mapping from a reference element (“mapped”)
  2. Defining it directly in the geometrical elements (“unmapped”)

In my code, I implemented the spaces according to

Q2 = VectorElement('CG',msh.ufl_cell(),2)
P1 = FiniteElement('DPC',msh.ufl_cell(),1)

W   = FunctionSpace(msh,MixedElement([Q2,P1]))

which seems to lead to the “mapped” version of P1. How could I implement the “unmapped” version?

2

In DOLFINx we use code generation to generate the assembly kernels, i.e. Computing the maps from the reference element to the physical element etc.

I dont know how you would define the unmapped version, maybe @mscroggs has an idea.

3

FEniCSx doesn’t support elements defined directly on the physical cells. (I’m 99% sure FEniCS doesn’t support this either.)

It might be possible to implement this with FEniCSx by using a custom kernel instead of using the kernels generated by FFCx.

4

To add to this, I’m not sure what the motivation to implement elements is, as it removes alot of the benefits of finite elements.

5

Thanks for your replies. I’m fairly inexperienced with FEniCSx, so using custom kernels seems a little out of reach for me. Do you know some forum threads or tutorials that can guide me through this?

About the motivation for using the unmapped P1, the article I cited argues that, for some types of quadrilateral meshes, the unmapped version achieves better convergence rates than the mapped. It is a somewhat edge case, but I found it interesting to test.

7

I am not aware of an easy way to do it with DOLFIN or DOLFINx. I’ll let you know if I hear of a way to do it.

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