Have you considered looking into our FEPointEvaluation [1] and related
"nonmatching" facilities?

I have the feeling that we already provide the right data structures for
your interpolation/evaluation needs - in particular with an efficient
and fast implementation. An example usage can be found in step-19.

Best,
Matthias

[1]  https://www.dealii.org/current/doxygen/deal.II/classFEPointEvaluation.html

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