WIP: Export of discontinuous fields to VTK

[KnutAM]

Until VTK supports discontinuous fields, this patch, building on top of #692 makes a vtk grid that is geometrically discontinuous as well, allowing discontinuous interpolations to be visualized.

TODO

• Add tests using apply_analytical! to validate continuity when it should be
• Document how to use this, perhaps a how-to postprocess with L2-projection of fluxes (which could be extended to Hdiv with WIP: New vector interpolations and mappings #798) ?

[KnutAM]

Test case

Generated to validate the discontinuous, tests for different reference shapes will be needed before merge...

DG tutorial (dg penalization variation)

The results looks quite strange, any idea if this is reasonable @AbdAlazezAhmed / @termi-official ? (I'm wondering if something is off in the export, but test case looks correct)

$\mu = 4.0 / h_e = (1 + O)^D / h_e$ (same result for $\mu = 2.0 / h_e $

$\mu = 1.0 / h_e $

$\mu = 0.5 / h_e $

[termi-official]

Test case

Generated to validate the discontinuous, tests for different reference shapes will be needed before merge...

Any reason not to use apply_analytical! for the first tests?

DG tutorial (dg penalization variation)

The results looks quite strange, any idea if this is reasonable @AbdAlazezAhmed / @termi-official ? (I'm wondering if something is off in the export, but test case looks correct)

Looks correct for me. The parameter which you tune is basically a penalty on the jump in the gradients between the elements. Please notice in the first figure in the section above that your gradients are essentially continuous with large jumps between elements. Just look at the formula for 1D and I think you will understand what happens. :) Also, a too small value for the penalty parameter makes the forumlation unstable, which is why you see this weird checkerboarding pattern.

[KnutAM]

Any reason not to use apply_analytical! for the first tests?

Since it uses the nodal coordinate, the result wouldn't be discontinuous. But it is a good point adding that to validate the the result is continuous when it should!

Please notice in the first figure in the section above that your gradients are essentially continuous with large jumps between elements.

Yes, that was the idea of the test to generate such a case :)