io: fix saving/loading of HDiv/HCurl functions on a high-order mesh #3838
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Fix #3781.
As @colinjcotter suggested on slack, we map HDiv/HCurl function defined on a high-order mesh to a DG function defined on an affine mesh using the contravariant/covariant Piola map.
Notes:
If we attach the Piola map to the vector DG element (using
WithMapping
), the DG DoF values will then depend on the reference-physical element map, which is inconvenient for checkpointing.We need to have users pass a representation of the affine mesh explicitly as the mere CG1 function on a high-order mesh does not necessarily represent a sane affine mesh; e.g., a single CG2 mesh cell can look like the following, and CG1 mesh connecting vertices would collapse onto a line:
We need to have users pass the quadrature degree used when projecting to the DG space.