Convergence
To make sure of the similarity, we calculate the space errors on th epressure in the Lebesgue norm (\(L^2\)) and the space error on the velocity in the Lebesgue and Soboloev norm (\((H^1)\)):
\[\begin{eqnarray}
\Vert p-p_h\Vert_{L^2(\Omega)} &=& \left(\int_{\Omega}{(p-p_h)^2}\right)^{\frac{1}{2}}\\
\Vert\mathbf{u}-\mathbf{u}_h\Vert_{L^2(\Omega)} &=& \left(\int_{\Omega}{\sum_{i=1}^{3}{(u_i-u_{i,h})^2}}\right)^{\frac{1}{2}}\\
\Vert\mathbf{u}-\mathbf{u}_h\Vert_{H^1(\Omega)} &=& \left(\int_{\Omega}{\sum_{i=1}^{3}{\Vert u_i-u_{i,h}\Vert_{H^1(\Omega)}^2}}\right)^{\frac{1}{2}}\\
\end{eqnarray}\]
Where:
\[\Vert u_i-u_{i,h}\Vert_{H^1(\Omega)} = \Vert u_i-u_{i,h}\Vert_{L^2(\Omega)} +
\sum_{j=1}^{3}{\left\Vert\frac{\partial u_i}{\partial x_j}-\frac{\partial
u_{i,h}}{\partial x_j}\right\Vert_{L^2(\Omega)}}\]
Stokes
Dirichlet conditions
The obtained convergence orders for the velocity and the pressure errors according to the mesh step (see. Figure 1) lead to conclude that the proposed approach is good.
Figure 1. Convergence curves (logarithmic scale) for velocity in \(L^2\) (left: expected slope 3), velocity in \((H^1)\) (middle: expected slope 2) and pressure in \(L^2\) (right: expected slope 2).
Mixed conditions
The obtained convergence orders for the velocity and the pressure errors according to the mesh step (see. Figure 2) lead to conclude that the proposed approach is good.
Figure 2. Convergence curves (logarithmic scale) for velocity in \(L^2\) (left: expected slope 3), velocity in \((H^1)\) (middle: expected slope 2) and pressure in \(L^2\) (right: expected slope 2).
Navier-Stokes
Dirichlet conditions
Ethier-Steinmann benchmark only, Figure 3.
Figure 3. Convergence curves (logarithmic scale) for velocity in \(L^2\), velocity in \((H^1)\) (left: expected slope 3) and pressure in \(L^2\) (right: expected slope 2).
Mixed conditions
Ethier-Steinmann benchmark only, Figure 4.
Figure 4. Convergence curves (logarithmic scale) for velocity in \(L^2\), velocity in \((H^1)\) (left: expected slope 3) and pressure in \(L^2\) (right: expected slope 1).