On finite element approximations of the streamfunction‐vorticity and velocity‐vorticity equations

Abstract We consider finite element methods for vorticity formulations of viscous incompressible flows. In two‐dimensional settings the familiar streamfunction‐vorticity formulation is examined. We focus on its accuracy, especially when using low‐order elements, and on its use with a variety of boundary conditions and in multiply connected domains. In three dimensions the velocity‐vorticity formulation is shown to be preferable, and a promising algorithm using this formulation is presented. We close by considering the recovery of the pressure field once the streamfunction or velocity fields are known. In particular we describe and analyse an algorithm for recovering the pressure which is based on well known methods for the primitive variable formulation and which requires no boundary conditions on the pressure at solid walls.