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; similar to) d x H1(similar to similar to) d x L2(similar to) 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(similar to) d, we derive an a priori error bound in a weaker norm, that is, the L2(similar to) dxd x H1(similar to) d x L2(similar to similar to) norm. (c) 2016 Wiley Periodicals, Inc.