We are interested in solving \( -\nu \Delta \bvec{u} + \nabla p = \bvec{f},\quad \nabla \cdot \bvec{u} = 0 \) in curl-curl formulation in 2D.
First we introduce the following notations:
\[ u \times n = u_1 n_2 - u_2 n_!, \quad u = (u_1, u_2),\ n = (n_1, n_2) \]
The curl of a vectorial field \(\phi\)
\[ \mathrm{curl} \phi = (-\frac{\partial \phi_2}{\partial x_1}, -\frac{\partial \phi_1}{\partial x_2} ) \]
The curl of a scalar field $ \(\psi\)
\[ \nabla \times \psi = \frac{\partial \psi}{\partial x_1} -\frac{\partial \psi}{\partial x_2} \]
Using the above notations we easily verify that:
\[ \mathrm{curl}( \nabla \times \psi) = \Delta \psi + \nabla (div \psi) \]
We also need a partial integration formulae:
\[ \int_\Omega \bvec{v}\cdot \mathrm{curl}w = \int_\Omega w ( \nabla \times \bvec{v}) -\int_{\partial\Omega} w (\bvec{v}\times n) \]
Using the formulae above, the system can now be written as:
Then we recall the same method used to obtain the strong formulation for the laplacian problem (see section Laplacian Examples ): we multiply the first equation by a test function \(v\in H^1(\Omega)\) and we integrate on the domain \(\Omega\)
\[ \mu \int_\Omega \mathrm{curl}( \nabla \times \bvec{u}) \cdot \bf \bvec{v} - \mu \int_\Omega \nabla (div \bvec{u}) \cdot \bvec{v} + \nabla p \cdot \bvec{v} & = \int_\Omega \bvec{f}\cdot \bvec{v} \]
Then we apply the above partial integration formulae on the first term, and the green formulae on the rest of the terms, we obtain:
\[ \mu \int_\Omega ( \nabla \times \bvec{u})( \nabla \times \bvec{v}) - \mu \int_{\partial\Omega} (\nabla \times \bvec{u}) \cdot ( \bvec{v} \times \bvec{n})+ \mu \int_\Omega ( \nabla \cdot \bvec{u})( \nabla \times \bvec{v}) - \mu \int_{\partial\Omega} (\nabla \cdot \bvec{u})n \cdot \bvec{v} + \int_\Omega p \nabla \cdot \bvec{v} -\int_{\partial\Omega} p \mathbf n \cdot \bvec{v} = \int_\Omega \bvec{f} \cdot \bvec{v} \]