Tutorial/Coupled problems

The considered problem

Here is explained how to use the solution of a first problem as a data for a second problem. Two kind of problem are studied here, depending if the solution of the first problem directly appears in the weak formulation (for example, as a source or a Neumann boundary), or if it is used a constraint (Dirichlet boundary condition). Let us begin by the first example with the very simple following example. The computation domain is the unit square $\Omega = [0,1]\times[0,1]$ with boundary $\Gamma$ and unit outwardly directed normal $\mathbf{n}$.

The first coupled problem read as follows : \begin{equation} \begin{cases}\label{eq:problemU} u = C & \text{in } \Omega,\\ \end{cases} \end{equation} where $C$ is a constant. Then, find the solution $v$ of the second problem \begin{equation} \begin{cases}\label{eq:problemV} -\Delta v + v = 0 & \text{in } \Omega,\\ \displaystyle{\frac{\partial v}{\partial \mathbf{n}} = u} & \text{on }\Gamma, \end{cases} \end{equation} where $u$ is the solution of problem (\ref{eq:problemU}).