On stability of the P n mod /P n element for incompressible flow problems

Abstract

It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuska and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of nth order accuracy in the energy norm called P n mod elements. For n ≤ 3 we show that the stability condition holds if the velocity space is constructed using the P n mod elements and the pressure space consists of continuous piecewise polynomial functions of degree n.