Analysis of the L² least-squares finite element method for a velocity-vorticity problem arising in incompressible inviscid rotational flows

In this paper we analyze the L² least-squares finite element method for a stationary velocity-vorticity problem arising in incompressible inviscid rotational flows. Introducing the additional vorticity variable, we rewrite the governing equations of incompressible inviscid rotational flow in the velocity-vorticity-pressure formulation and then further split the formulation into the pressure and velocity-vorticity subsystems. After time-discretizing the time derivative and linearizing the non-linear terms, we reach the stationary velocity-vorticity system. The L² least-squares finite element approach is applied to generate accurate numerical solutions of the velocity-vorticity system with suitable boundary conditions. We show that this approach produces an optimal rate of convergence in the H¹ norm for velocity and suboptimal rate in the L² norm for vorticity. A numerical example is given which confirms the theoretical results.