Div–curl problems and H 1 ‐regular stream functions in 3D Lipschitz domains

We consider the problem of recovering divergence-free velocity field ${\mathbf U}\in\mathbf{L}^2(\Omega)$ a given vorticity F}=\mathrm{curl}\,{\mathbf U}$ on bounded Lipschitz domain $\Omega\subset\mathbb{R}^3$. To that end, we solve "div-curl problem" for F}\in{\mathbf H}^{-1}(\Omega)$. The solution is expressed in terms vector potential (or stream function) A}\in{\mathbf H}^1(\Omega)$ such U}=\mathrm{curl}\,{\mathbf A}$. After discussing existence and uniqueness solutions associated potentials, propose well-posed construction function. A numerical method based this presented, experiments confirm resulting approximations display higher regularity than those another common approach.