Optimal Error Estimate for the Div Least-squares Method with Data f∈L2 and Application to Nonlinear Problems

The div least-squares methods have been studied by many researchers for the secondorder elliptic equations, elasticity, and the Stokes equations, and optimal error estimates have been obtained in the H(div) × H1 norm. However, there is no known convergence rate when the given data f belongs only to L2 space. In this paper, we will establish an optimal error estimate in the L2 × H1 norm with the given data f ∈ L2 and, hence, fill a theoretical gap of least-squares methods. As a consequence of this estimate, we will provide a convergence analysis for the linearization process on solving Navier–Stokes equations, which uses the div least-squares method for solving the corresponding Stokes equations.