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

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.