Laplace equation with an equation as boundary condition. MultiFieldFESpace

[DaniStauber]

Hi there!
The problem that i want to solve is
$$-\Delta u =-( \partial_x ^2 u + \partial_y ^2 u)= -f_1 $$ on $\Omega$ that is a circle of radius $R=40$.
With the boundary condition of
$$\partial_{\theta} ^2 u= f_2$$ on $\partial \Omega$, with the condition of being zero mean on the boundary, ie $\int_{\partial \Omega} u d\Gamma =0$

I used as a guide an exact solution of this problem to test the code.
$$u(\bar{x})= (y^2 -x^2)/R^2$$, so $$f_2(\bar{x})=0$$
$$f_1(\bar{x})= 4(x^2 -y^2)/R^2$$

I provide here a zip with the mesh and two notebooks that i want to compare. In "MWE_LaplaceEq_separatedFESpaces" i solve the problem in two parts. First solving the boundary problem and after that solving the problem on the entire domain. That way was successfull. The notebook "MWE_LaplaceEq_mixedFESpaces" trying to solve the problem with the same formulation of the previous one, but using MultiFieldsFESpaces, to solve one problem. But this note wasnt successful. I don't really catch the problem using MultiFieldFESpace, for me the formulations are the same.

My big goal is to solve the laplacian equation for a tensor in an arbitrary background metric. So will be a success for me to make work the notebook "MWE_LaplaceEq_mixedFESpaces" that try to solve the problem using MultiFieldFESpaces to solve the complete problem in one solve.
If anyone could help me to understand this, i will appreciate a lot. Thanks!

MWE.zip