# Minimizing a nonlinear functional over finite elements

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.