A least-squares finite-element method for the Stokes equations with improved mass balances

We prove the convergence of a least-squares mixed finite-element method for the Stokes equations with zero residual of mass conservation. For the standard least squares mixed finite-element method, the equations for continuity of mass, and momentum are minimized in a global sense. Therefore, the mass may not be conserved at every point of the discretization. Recently, in [1], a modified least squares finite-element method is developed to enforce near zero residual of mass conservation. This is achieved by attaching a discrete divergence free constrain to the standard least squares finite-element method, and as a consequence, the number of equations is increased. In this paper, we take a different approach to improve the conservation of mass and reduce the number of the equations. This method does not require LBB condition on the finite-dimensional subspaces and the resulting bilinear form is symmetric and positive definite.