2nd order Nedelec interpolation

[gijswl]

As discussed in issue #1195 I wanted to implement higher-order Hcurl interpolations. These require multiple degrees of freedom interior to the faces, which raises the issue of ensuring continuity between neighbouring elements when their local dofs do not match up.

The first commit (building on the work in PR #1196) includes the shape functions of Nedelec{RefTetrahedron,2} and minor additions to the interpolation tests to properly test this new interpolation. As a proof of concept I've implemented the facedof transformation inside apply_mapping!, but this need not be its final place. For convenience I've just commented out the restriction to a single facedof, although we would probably want to check if the correct face permutations/transformations are implemented.

As a result, the interpolation and continuity tests are passing, but I have my doubts about the actual continuity. Insofar as I can see, test_continuity.jl tests for continuity in the shape functions (i.e. the transformation Nξ to Nx), but does not consider dNdx, which is important for forming the shape_curl values during element assembly. @KnutAM do you have any comments on if we need to further transform dNdx and if this should be part of these tests?

[codecov]

Codecov Report

Attention: Patch coverage is 97.90210% with 3 lines in your changes missing coverage. Please review.

Project coverage is 88.15%. Comparing base (c718c73) to head (2fd726b).
Report is 2 commits behind head on master.

Files with missing lines	Patch %	Lines
src/interpolations.jl	93.61%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1199      +/-   ##
==========================================
- Coverage   94.03%   88.15%   -5.88%     
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  Files          39       40       +1     
  Lines        6555     7210     +655     
==========================================
+ Hits         6164     6356     +192     
- Misses        391      854     +463     

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[KnutAM]

Insofar as I can see, test_continuity.jl tests for continuity in the shape functions (i.e. the transformation Nξ to Nx), but does not consider dNdx, which is important for forming the shape_curl values during element assembly.

Just as we don't expect dNdx to be continuous for H1Conformity, we don't expect curl(N(x)) to be continuous for HcurlConformity, that would require a different type of interpolation. HcurlConformity implies that $\mathrm{curl}(u)$ is bounded (i.e. in L2), but it may be discontinuous.

[gijswl]

we don't expect curl(N(x)) to be continuous for HcurlConformity

Agreed, but since I am now applying a linear transformation to Nξ based on the orientation of the face, we should also make sure dNdξ is updated to account for the transformed orientation, right? Intuitively I feel like we have to do something more than apply the same transformation matrix to dNdξ, but I can't substantiate why.

[KnutAM]

Yes, that is true.

This should be covered by

Ferrite.jl/test/test_cellvalues.jl

Line 191 in c718c73

@testset "Non-identity mapping gradients" begin

, but I see now that we didn't include the new interpolations that you added to this test, having some in 3d makes sense, and then especially also the higher order ones. Since the logic is the same for all interpolations, its not required to do it for every single one, but for each new with different properties (e.g. dimension, order, linear transformation, etc.).