# Feel++

This test case is taken from [patera].

## Problem description

We consider an advection-diffusion problem in a rectangular parametrized domain \Omega_0(\mu) = ]0,L[\times]0,1[ representing a channel. The governing equation for the temperature T (passive-scalar field) is the advection-diffusion equation \eqref{advec-diff2D} with imposed Couette velocity \mathbf{u} = (y , 0). \mathbf{u} \cdot \nabla T - \frac{1}{Pe} \Delta T = 0 Note that Pe is the Péclet number.

``````\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- node[label=below:$\Gamma_{bottom}$]{} (6,0);
\draw (6,0) -- node[label=right:$\Gamma_{out}$]{} (6,1);
\draw (6,1) -- node[label=above:$\Gamma_{top}$]{} (0,1);
\draw (0,1)  -- node[label=left:$\Gamma_{in}$]{} (0,0);
%\draw [<->] (0,-0.1) -- node [label=below:L]{} (6,-0.1);
\end{tikzpicture}
\caption{ \label{omega} Study domain $\Omega$ with a fixed length $L$.}
\end{center}
\end{figure}``````

### Boundary conditions

We set

• on \Gamma_{in} and \Gamma_{top} , an homogeneous Dirichlet condition T = 0;

• at \Gamma_{bottom}, a non-homogeneous Neumann condition \nabla T \cdot \mathbf{n} = 1

• at \Gamma_{out}, an homogeneous Neumann condition \nabla T \cdot \mathbf{n} = 0

### Inputs

As inputs, We define the parameter set \mu by \mu=(L,Pe)~\in~[1~,~10] \times [0.1~,~100]

### Outputs

The output s(\mu) is the integral of the temperature over the heated surface \Gamma_{\mathrm{bottom}} defined by s(\mu) = \int_{\Gamma_{bottom}} T This output depends on the solution of the advection diffusion equation above and is dependent on the parameter set \mu.

Bibliography
• [patera] G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced Basis Approximation and A Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations — Application to Transport and Continuum Mechanics, Arch Comput Methods Eng, 2008