I am looking for a reference (an url, a book, or a paper) that could help me in discretization and minimization of the following cost function $J\left(\omega\right)$, over 3D tetrahedrons (finite elements). I am familiar with finite element analysis for partial differential equations such as Poisson equation in 3D using linear elements and have developed FEM codes before. However, $J$ in the following functional is new for me for which I don't have any experience.

$$\inf_\omega J\left(\omega\right), \quad J\left(\omega\right)=\int_V\big\{|\omega-\omega_g\|^2+\alpha\left(\|\nabla\times \omega\|^2+ 2\langle\Delta\omega_g,\omega\rangle\right)\big\}{\rm d}V$$

$\omega$ is an unknown 3D vector field and $\omega_g$ is a known 3D vector field.

I would really appreciate any tips and hints.


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