Broken FEEC projections on polar domains

[alisa-kirkinskaia]

This PR introduces broken FEEC projections for C0 and C1 sequences (conga_projections.py) in 2D. The formulas for the projections can be found here on pages 19-23.

• Added projections C0PolarProjection_V0, C0PolarProjection_V1, C0PolarProjection_V2 onto C0 sequence, and analogously onto C1 sequence.

• dot methods of the projections are implemented for parallel execution for arbitrary domain decompositions in the angular and/or radial dimensions.

• Added unit tests for the projections checking:

  • the projection property P(P(x)) = P(x)
  • tosparse methods against reference matrix operators
  • consistency between dot and tosparse methods

• Poisson example
  Example of run: mpirun -n 2 python poisson_2d.py -S -n 16 24 -d 2 2 -t disk -D 0.2 -m 'C0conga'
  Exact solution, approximate solution and error plot:

• TE Maxwell example
  Example of run: mpirun -n 2 python maxwell_2d.py -S -n 16 20 -d 2 2 -T 1 -D 0.2 -s 1
  Plot of exact solution and approximate solution at final time T = 1:

[jowezarek]

Why is this necessary? Can you not use mass matrices instead?

[jowezarek]

As discussed, I'll look into this next week. If no "quick" solution is found, we can keep this code as it works (and is also somewhat instructive).

[jowezarek]

Is this class worth keeping in the library? It's only used in the maxwell_2d.py example, and I'd argue that one could remove the scipy option from that file. Either way, I think one could use a MatrixFreeLinearOperator in the example instead, for example like so:

lambda_strong = lambda x : array_to_psydac(strong_curl_sp @ x.toarray(), W2.coeff_space)

weak_curl_sp = sp_inv(M1.tosparse()) @ strong_curl_sp.T @ M2.tosparse()
lambda_weak = lambda x : array_to_psydac(weak_curl_sp @ x.toarray(), W1.coeff_space)

if scipy:
   step_faraday_2d = MatrixFreeLinearOperator(W1.coeff_space, W2.coeff_space, lambda_strong)
   step_ampere_2d = MatrixFreeLinearOperator(W2.coeff_space, W1.coeff_space, lambda_weak)