This is a year old post but I'm dealing with the same issue so I decided to 
continue here instead creating a new one.

In my case I solve a simple laplace problem, that represent groundwater 
flow with non zero dirichlet boundary conditions
My goal is to estimate how much water comes in and out from the dirichlet 
BC nodes.

The way I could do this, is after I solve the reduced version of the full 
linear system A*U = F 
to calculate
DBC = A*U-F

When I call 
system_rhs.reinit (locally_owned_dofs, mpi_communicator);
system_matrix.reinit (locally_owned_dofs,
                          locally_owned_dofs,
                          dynamic_sparsity_pattern,
                          mpi_communicator);


to initialize the A and F, are these the reduced (without the dofs of DBC) 
or the full ones?
                          
is
system_matrix.vmult(output, system_rhs);

equivalent to ?
A*U (where A includes the rows and columns of dirichlet BC)

Thank you

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