Tutorial/Coupled problems

The considered problem

Here, the two academics problem of the Laplace equation with Neumann boundary condition and with Dirichlet boundary condition are coupled, in te sense that the solution of the first problem is data for the second problem. The computation domain is still the unit square $\Omega = [0,1]\times[0,1]$ with boundary $\Gamma$ and unit outwardly directed normal $\mathbf{n}$. The problems read as follows : firstly, find $u$ such that \begin{equation} \begin{cases}\label{eq:problemU} -\Delta u + u = f & \text{in } \Omega,\\ \displaystyle{\frac{\partial u}{\partial \mathbf{n}} = 0} & \text{on }\Gamma, \end{cases} \end{equation} where $\displaystyle{\Delta = \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} }$ is the Laplace operator and $f$ is given by $$ \forall (x,y)\in [0,1]^2,\qquad f(x,y) = (1+2\pi^2)\cos(\pi x)\cos(\pi y). $$ Then, find the solution $v$ of the second problem \begin{equation} \begin{cases}\label{eq:problemV} -\Delta v + v = 0 & \text{in } \Omega,\\ \displaystyle{v = u} & \text{on }\Gamma, \end{cases} \end{equation} where $u$ is the solution of problem (\ref{eq:problemU}).