Unstable modes of the Q1-P0 element
摘要
. In this paper the unstable eigenmodes of Q1 --P0 velocity/pressure finite element approximation for incompressible flow problems are characterised. It is shown that the inf-sup stability constant is O(h) in two dimensions and O(h 2 ) in three dimensions. The basic tool in the analysis is the method of modified equations which is applied to finite difference representations of the underlying finite element equations. The asymptotic estimates are confirmed and supplemented by numerical experiments. 1. Introduction. It is universally recognised that discretisation schemes for Stokes and Navier-Stokes equations are subject to an inf-sup or div-stability condition, see [2]. The stability requirement is manifested in practical computations by the predominance of staggered grid finite volume discretisations, and the existence of unnatural velocity--pressure finite element combinations. These typically involve velocity bubble functions, or else have a macro-element definition of the veloci...