Div-Curl Problems and $\mathbf{H}^1$-regular Stream Functions in 3D Lipschitz Domains

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

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