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

Abstract

We consider the problem of recovering the divergence-free velocity field $\mathbf{U}\in {\mathbf{L}}^2(\Omega )$ of a given vorticity $\mathbf{F}=\mathrm{curl}\mathbf{U}$ on a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^3$. To that end, we solve the "div-curl problem" for a given $\mathbf{F}\in {\mathbf{H}}^{-1}(\Omega )$. The solution is expressed in terms of a vector potential (or stream function) $\mathbf{A}\in {\mathbf{H}}^1(\Omega )$ 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.

References in corpus (1)

• Analytic Integration of the Newton Potential over Cuboids and an Application to Fast Multipole Methods

Cited by in corpus (1)

• Locally Structure-Preserving div-curl operators for high order Discontinuous Galerkin schemes