[...lots of code...]
But I am not sure if that is even remotely correct.

This is really too broad a question. If you try it, what concretely goes wrong?


How does fe_values_N differ from fe_values_TE? But how should I initialize them else? Should I use FESystem instead? If yes, how should I initialize them? Furthermore, can I use one dof_handler for both variables?

The canonical way to go for coupled problems is to use an FESystem, one DoFhandler, using fe_values[variable_name] in assembly, and block vectors/block matrices for the linear algebra. Almost all tutorial programs use this scheme, and for good reason.

Best
 W.

--
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Wolfgang Bangerth          email:                 [email protected]
                           www: http://www.math.colostate.edu/~bangerth/

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