# Coupled Mixed Poisson

## 1. Models

The equations of the model are:

\begin{align} \boldsymbol{u} + k\nabla p &= \boldsymbol{f}\\ \frac{1}{M}\frac{\partial p}{\partial t} + \nabla\cdot \boldsymbol{u} &= g \end{align}

coupled with a system of ODE:

\begin{align} \frac{\partial \boldsymbol{y}}{\partial t} + \underline{\underline{A}}(\boldsymbol{y},t)\boldsymbol{y} = s(\boldsymbol{y},t) + b(\boldsymbol{y},t) \end{align}

where $\Pi_1\in \boldsymbol{y}$ and the boundary condition:

$\int_\Gamma \boldsymbol{u}\cdot\boldsymbol{n} = \frac{P_I - \Pi_1}{R_b} \quad \text{ on } \Gamma\\ |\Gamma|p - \int_\Gamma p = 0 \quad \text{ on } \Gamma$

with $P_I = p_{|\Gamma}$.

Most of the configuration is the same as for the Mixed Poisson toolbox, except for the boundary condition where the coupling happens.

## 2. Boundary Conditions

### 2.1. Coupling

On the flux/current-density/heat-flux:

Example of a Coupling boundary condition
``````"Coupling":
{
"buffer":
{
"markers":"top", (1)
"capacitor": "Cbuffer.C", (2)
"resistor": "Rbuffer.R", (3)
"circuit": "\$cfgdir/test3d0d_linear/test3d0d_linear.fmu", (4)
"buffer": "Pi_1.phi" (5)
}
}``````
 1 marker for $\Gamma$ 2 name of the variable in the FMU for $C_b$ 3 name of the variable in the FMU for $R_b$ 4 path of the FMU 5 name of the variable in the FMU for $\Pi_1$