Div–curl problems and H1‐regular stream functions in 3D Lipschitz domains

Mathematical Methods in the Applied Sciences, vol. 45, pp. 1097–1117

Abstract

We consider the problem of recovering the divergence‐free velocity field U ∈ L2(Ω) of a given vorticity on a bounded Lipschitz domain . To that end, we solve the ‘div–curl problem’ for a given F ∈ H−1(Ω). The solution is expressed in terms of a vector potential (or stream function) A ∈ H1(Ω) such that . 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.

Authors 2

1. Matthias Kirchhart corresponding Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics RWTH Aachen University Aachen Germany

   Applied and Computational Mathematics, RWTH Aachen University, Aachen, Germany

2. Erick Schulz

   ETH Zurich · Seminar for Applied Mathematics

   Affiliation as printed

   Seminar in Applied Mathematics ETH Zürich Zürich Switzerland