Test continuity of polynomial basis and its gradients along an edge on line (2D) or on a face in - plane (3D) between two elements aligned with the coordinate system, stack one on top of the other. The evaluation occurs in several points shifted by a very small amount from the boundary between the elements into the top and the bottom element.
For H1 space, the basis should be continuous. The components of its gradient parallel to the edge/face should be continuous as well, while the perpendicular component should have the same absolute value, but different sign in the top and the bottom element.
All connectivity permutations of the two elements are tested.
The serendipity basis implementation is a pure python proof-of-concept. Its order in continuity tests is limited to 2 on 3_8 elements to decrease the tests run time.
Test the second partial derivatives of basis functions using finite differences.