On mass-conserving least-squares methods

Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.