Better support for high-order meshes

This is just random ideas and possible future directions regarding high-order meshes. We have ex31.py for quadratic meshes but it’s a bit convoluted because you go back and forth between InteriorBasis and Mesh.

I was reading https://mfem.org/mesh-format-v1.x/ and experimented with similar ideas in scikit-fem. I created a new module with classes Topology, Dofnum (extracted from InteriorBasis._build_dofnum) and Geometry.

Then you describe the traditional triangular mesh as follows:

>>> from skfem import *
>>> from topology import *
>>> m = MeshTri()
>>> t = m.t
>>> p = m.p
>>> topo = Topology(t)
>>> dofnum = Dofnum(topo, ElementTriP1())
>>> geom = Geometry(p, dofnum)
>>> geom.p
array([[0., 1., 0., 1.],
       [0., 0., 1., 1.]])
>>> geom.t
array([[0, 1],
       [1, 2],
       [2, 3]])

This Geometry is approximately API compatible with traditional skfem.Mesh so you can build a MappingIsoparametric and, using that, an InteriorBasis:

>>> mapping = MappingIsoparametric(geom, ElementTriP1())
>>> basis = InteriorBasis(m, ElementTriP2(), mapping)

Eventually I’d abstract out the mapping step so that geom is directly compatible with InteriorBasis constructor and returns the correct mapping via geom.mapping().

Anyhow, now a quadratic mesh would be described simply as follows:

>>> dofnum = Dofnum(topo, ElementTriP2())
>>> geom = Geometry(array([[0. , 1. , 0. , 1. , 0.5, 0. , 0.5, 1. , 0.5],
...        [0. , 0. , 1. , 1. , 0. , 0.5, 0.5, 0.5, 1. ]]), dofnum)
>>> mapping = MappingIsoparametric(geom, ElementTriP2())
>>> basis = InteriorBasis(m, ElementTriP2(), mapping)

Pretty cool? Obviously all this should be invisible to the user but I wonder if this warrants a rewrite of skfem.mesh so that we can easily define MeshTri2 or something to refer to the quadratic mesh.

[kinnala]

Just did some serious progress on this in #493

Loading a quadratic mesh from Gmsh using meshio and solving the Laplace equation.

[kinnala]

Closing this shortly and opening new ones that are more detailed.