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

摘要

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.