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} \Delta u = 0 & \text{in } \Omega,\\ u = C & \text{on }\Gamma, \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}. Obviously, solution $u$ is constant inside $\Omega$ and is equal to $C$ and there is no "real" need in solving problem \ref{eq:problemU}. The weak formulation of problems \ref{eq:problemU} and \ref{eq:problemV} read as :

\begin{equation}\label{eq:WeakFormulationU} \left\{\begin{array}{l} \text{Find } u\in H^1(\Omega) \text{ such that, '"`UNIQ-MathJax11-QINU`"' and}\\ \displaystyle{\forall u'\in H^1_0(\Omega), \qquad \int_{\Omega} \nabla u\cdot\nabla u' \;{\rm d}\Omega = 0}, \end{array}\right. \end{equation}

\begin{equation}\label{eq:WeakFormulationV} \left\{\begin{array}{l} \text{Find } v\in H^1(\Omega) \text{ such that, }\\ \displaystyle{\forall v'\in H^1(\Omega), \qquad \int_{\Omega} \nabla v\cdot\nabla v' \;{\rm d}\Omega + \int_{\Omega}vv' \;{\rm d}\Omega - \int_{\Gamma}uv'\;{\rm d}\Gamma = 0}, \end{array}\right. \end{equation} where $H^1(\Omega)$ is the classical Sobolev space and the functions $v$ are the test functions.