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

Abstract

We consider the problem of recovering the divergence-free velocity field of a given vorticity on a bounded Lipschitz domain. To that end, we solve the div-curl problem and express the solution in terms of a vector potential (stream function). 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.