Least-Squares Method for the Oseen Equation

This article studies the least-squares finite element method for the linearized, stationary Navier–Stokes equation based on the stress-velocity-pressure formulation in d dimensions (d = 2 or 3). The least-squares functional is simply defined as the sum of the squares of the L2 norm of the residuals. It is shown that the homogeneous least-squares functional is elliptic and continuous in the H(div; ) ×H 1( ) × L2( ) norm. This immediately implies that the a priori error estimate of the conforming least-squares finite element approximation is optimal in the energy norm. The L2 norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right-hand side f belongs only to L2( ) , we derive an a priori error bound in a weaker norm, that is, the L2( )d×d × H 1( ) × L2( ) norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1289–1303, 2016