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 = or 3). The least-squares functional is simply defined as the sum of the squares of the L 2 norm of the residuals. It is shown that the homogeneous least-squares functional is elliptic and continuous in the H(div; !) d × H 1 (!) d × L 2 (!) 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 L 2 norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right-hand side f belongs only to L 2 (!) d , we derive an a priori error bound in a weaker norm, that is, the L 2 (!) d×d × H 1 (!) d × L 2 (!) norm. © 2016 Wiley