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Div-Curl Problems and H^1-regular Stream Functions in 3D Lipschitz Domains

by M. Kirchhart and E. Schulz

(Report number 2020-30)

Abstract
We consider the problem of recovering the divergence-free velocity field $\mathbf{U}\in\mathbf{L}^2(\Omega)$ of a given vorticity $\mathbf{F}=\text{curl}\mathbf{U}$ 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 $\mathbf{U} = \text{curl}\mathbf{A}$. 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.

Keywords: div-curl system, stream function, vector potential, vorticity, non-smooth domain

@Techreport{KS20_903,
  author = {M. Kirchhart and E. Schulz},
  title = {Div-Curl Problems and H^1-regular Stream Functions in 3D Lipschitz Domains},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-30.pdf },
  year = {2020}
}

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