Joel,
> [...]
> for (unsigned int q_index=0; q_index<n_q_points; ++q_index)
> for (unsigned int i=0; i<dofs_per_cell; ++i)
> {
> velocity[local_dof_indices[i]] = local_velocity_values[q_index];
> }
>
> Here you assign the same Tensor<1,dim> at all DoFs multiple times and you
end with velocity[*]=local_velocity_values[n_q_points]-1.
I don't think that this is what you want to do, is it? If I understand you
correctly, you try to assign the magnitude of the velocity in each degree
of freedom
to a different scalar Vector that represents a scalar FiniteElement Vector.
In this case, you need to use a Quadrature object that uses the
unit_support_points of the scalar FE [1] and your assignment would read

##
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for (unsigned int i=0; i<dofs_per_cell; ++i)
velocity[local_dof[indices[i]] = local_velocity_values(i).norm_sqr();
where velocity is a Vector that is related to fe_scalar and
local_velocity_values has the type std::vector<Vector<double> >.
Best,
Daniel
[1]
https://github.com/dealii/dealii/wiki/Frequently-Asked-Questions#how-to-get-the-mapped-position-of-support-points-of-my-element
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