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Rational functions for pyramid #895

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@jmv2009 jmv2009 commented Jan 9, 2025

This commit generates the proper orthonormal rational functions for the pyramid. See #894

Still doing more checks on the derivatives, especially around p=q.

The Lagrangian elements have a problem with these functions at the top vertex (nan), even though the limit is well defined.

These are not yet the orthonormal rational functions that we want though. The first generated proper rational function here are (2x-1+z)(2y-1+z)/(1-z). This should be x y/(1-z) as in Cockburn and Fu and https://defelement.org/elements/examples/pyramid-lagrange-equispaced-1.html However, for orthogonality purposes, the base should then be from -1,-1,zmin to 1,1,zmin and the peak at 0,0,1. This will keep the admissiblity/compatibility. Alternatively it should be (x-1/2)(y-1/2)/(1-z) with the regular base peak at 1/2,1/2,1. So in the end the tilted, asymmetric pyramid is a bad choice. To get the 1 and 2-forms going without immediately needing to apply piola transformations doesn't seems to work.

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