On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

Abstract

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L 2 least-squares scheme or a weighted L 2 least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L 2 least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L 2 least-squares scheme seems to be better than the iterative L 2 least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.