Hello,

We had chosen to use deal.ii since most of it looked templated in terms of 
accepting the float type. However, I noticed that the FEValuesBase returns 
a 'double', instead of a generic 'value_type' for the Jacobian term.

Our goal would be to differentiate through the entire residual (and 
possibly the time step) with respect to the solution variables and *mesh 
points*. Therefore, the general element Jacobian (and its determinant) 
would need to be differentiated. I see that the Jacobian entries already 
have differentiation functions, however, that would entail manually 
assembling the derivative of the residual with respect to the mesh point. 
We would like to instead have a function that assembles the residual 
right-hand side and pass the AD types to that high-level function to obtain 
dRdX.

Looking at the current implementation in fe_values, mapping_q_generic, and 
local integrators, I see that it looks relatively easy to template the 
value type since it is already partially templated.

Is there a reason why those functions were not templated? Is there anything 
we should be careful about before we start templating those functions?

Doug

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