A locally conservative,discontinuous least‐squares finite element method for the Stokes equations

Conventional least‐squares finite element methods (LSFEMs) for incompressible flows conserve mass only approximately. For some problems, mass loss levels are large and result in unphysical solutions. In this paper we formulate a new, locally conservative LSFEM for the Stokes equations wherein a discrete velocity field is computed that is point‐wise divergence free on each element. The central idea is to allow discontinuous velocity approximations and then to define the velocity field on each element using a local stream‐function. The effect of the new LSFEM approach on improved local and global mass conservation is compared with a conventional LSFEM for the Stokes equations employing standard C⁰ Lagrangian elements. Copyright © 2011 John Wiley & Sons, Ltd.