Bruno.

Let's say I solve a first equation for Velocity, and I would like to use 
> this velocity in another equation for advection-diffusion of say 
> Temperature.
> I would first set-up my DOF and FE system for my velocity equation and 
> solve it and set-up by DOF and FE system for my Temperature.
> However, how would I be able to set up an additional space to store my 
> velocity at the nodes that are related to my Temperature FESystem without 
> introducing additional unknown in my  temperature equations (Since velocity 
> is known after all) and without assuming that both my temperature and 
> velocity equation are numbered in the same exact fashion? 
>

For assemling a bilinear form you typically don't nees the support points 
of the ansatz functions but their evaluations in the quadrature points.
In your case, I would simply use two DoFHandler objects (one for each 
system of equations) and loop over both of them for the temperature 
equations.
https://www.dealii.org/current/doxygen/deal.II/step_31.html#BoussinesqFlowProblemassemble_temperature_system
 
might be helpful.

Best,
Daniel

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