Giorgos,

You can find a discussion for how constraints are handled in deal.II in
https://www.dealii.org/developer/doxygen/deal.II/group__constraints.html.

Thank you for your feedback.
> I'll try to explain what I try to calculate with the help of a standard
> FEM textbook:
>
> [...]
>
> In the book the global system is partitioned so that the first row
> contains the known U1 dofs and the second row the unknown the dofs U2.
> They solve the system for the U2 and then they calculate F1.
>

In short: we normally don't condense the linear system but use a diagonal
matrix for the constrained rows instead (and modify the right-hand side
accordingly).


> I think that the F1 vector  contains, among others, the fluxes from the
> imposed BCs.
>
In case you only have Dirichlet boundary conditions, K11 is diagonal, K12
is zero and F1 contains the (possibly scaled) Dirichlet boundary values.

Could you clarify in mathematical terms what you mean by "fluxes"? Maybe,
we are just using different terms fo the same thing.

In case you are interested in n\cdot In u at the boundary, you can use
FEValues::get_function_gradients() and FEValues::get_normal_vectors() with
a quadrature formel that contains
the points you are interested in on a cell in reference coordinates. This
is the approach VectorTools::point_gradient() is using for returning the
gradient for a single point in real coordinates.

Best,
Daniel

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