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).

First kind of coupled problem : solution appearing in the weak formulation

Mathematics

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, $H^1_0(\Omega)$ is the space of $H^1(\Omega)$ functions which vanish at the boundary $\Gamma$.

GetDP

In GetDP, this kind of problem is very easy to solve. The user must introduce two function space (one per solution), and solve each problem in the right order. There is no trap ! Here are the files. The .gmsh file quite the same as in the previous examples : it has no special interest.

Second kind of coupled problem : solution used as a Dirichlet boundary condition

Mathematics

We still consider problem (\ref{eq:problemU}) satisfied by $u$, but now, the second problem is to find $v$, solution of \begin{equation} \begin{cases}\label{eq:problemV2} -\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}. The solution $u$ is now used as a Dirichlet boundary condition.

GetDP

With GetDP, a dirichlet boundary condition is generally imposed through the "Constraint" term. Here, we propose to solve this problem using the function "AssignFromResolution". The procedure is the following :

• Construct only ONE function space that will contain $u$ and then $v$
• Solve problem 1 (and save $u$)
• Transfert solution of problem 1 to problem 2
• Solve problem 2 (and erase $u$ by $v$)

Remarks

• The solution $u$ is no more stored at the end of the process. To avoid this, we propose here to save $u$ in an auxiliary function.
• A Dirichlet boundary condition can also be imposed using a Lagrange Multiplier, that also avoid to use "TransfertSolution" ...