Analysis of Least-Squares Finite Element Methods for the Navier--Stokes Equations

Abstract

In this paper we study finite element methods of least-squares type for the stationary, incompressible Navier--Stokes equations in two and three dimensions. We consider methods based on velocity-vorticity-pressure form of the Navier--Stokes equations augmented with several nonstandard boundary conditions. Least-squares minimization principles for these boundary value problems are developed with the aid of the Agmon--Douglis--Nirenberg (ADN) elliptic theory. Among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions. Results of several computational experiments with least-squares methods which illustrate, among other things, the optimal convergence rates are also reported.