# Bercovier-Engelman test

The exact solution of Bercovier-ENgelman [BE79] is defined on $\Omega=$0, 1]^3$ by: \[\begin{eqnarray} u_1 &=& -256y(y-1)(2y-1)x^2(x-1)^2\\ u_2 &=& 256x(x-1)(2x-1)y^2(y-1)^2\\ u_3 &=& 0\\ p &=& (x-0.5)(y-0.5) \end{eqnarray}$

External forces are calculated using exact solution in the Stokes / Navier-Stokes equations.

## Dirichlet conditions

We compare the approximated solution and the exact solution of Bercovier-Engelman, respectively for pressure (Figure 1 and Figure 2) and velocity (Figure 3 and Figure 4).

Figure 1. Pressure on the unit cube. Computed solution
Figure 2. Pressure on the unit cube. Exact solution
Figure 3. Velocity field on the unit cube on the cut z=0.5. Computed solution
Figure 4. Velocity field on the unit cube on the cut z=0.5. Exact solution

FreeFem++ algorithm:

• Stokes_BE_dirichlet.edp

• Navier-Stokes_BE_dirichlet.edp

## Mixed conditions

We compare the approximated solution and the exact solution of Bercovier-Engelman, respectively for pressure (Figure 5 and Figure 6) and velocity (Figure 7 and Figure 8).

Figure 5. Pressure on the unit cube. Computed solution
Figure 6. Pressure on the unit cube. Exact solution
Figure 7. Velocity field on the unit cube on the cut z=0.5. Computed solution
Figure 8. Velocity field on the unit cube on the cut z=0.5. Exact solution