1. Introduction
It is well known that the low-order mixed finite element P1/P0 (linear velocity, constant pressure) and P1/P1 (linear velocity and pressure) pairs do not satisfy the inf–sup condition (see, e.g., [1]). Since the low-order pairs remain a popular practical choice in mixed finite element approximation of incompressible models, several stabilized finite element methods have been developed in last two decades (see, e.g., [2–8]). The stabilized methods aim to relax the continuity equation so as to allow application of unstable pairs by adding extra stabilization terms. Bochev and his co-workers [4] pointed out that the unstable pairs satisfy the weaker form of the discrete inf–sup condition, and terms − e∈Eh he∥[ph]∥2 e 1 2 and − T∈τh hT ∥∇ ph∥T (1) reflect the inf–sup ‘deficiency’ of unstable P1/P0 and P1/P1 pairs, respectively. Stabilized methods introduce stabilization terms to counterbalance these key terms. In this paper, we propose an interesting property of the stabilized methods, that is, these key terms in (1) can be bounded by true errors. The observation provides useful arguments in a posteriori error estimates for the stabilization of low-order mixed finite element elements for the Stokes problem.
