A mimetic discretization of the Stokes problem with selected edge bubbles. (English) Zbl 1352.76021
Summary: 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 \(P_1-P_0\) (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.
MSC:
76D07 | Stokes and related (Oseen, etc.) flows |
76M10 | Finite element methods applied to problems in fluid mechanics |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |