A Mimetic Discretization of the Stokes Problem with Selected Edge Bubbles

A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The mimetic discretization methodology can be understood as a generalization of the finite element method to meshes with general polygons/polyhedrons. In this paper, the mimetic generalization of the unstable P1 - P0 (and the "conditionally stable" Q1 - P0) finite element is shown to be fully stable when applied to a large range of polygonal meshes. Moreover, we show how to stabilize the remaining cases by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments. [PUBLICATION ABSTRACT]