 
Summary: LOCAL ERROR ESTIMATES FOR FINITE ELEMENT
DISCRETIZATIONS OF THE STOKES EQUATIONS
DOUGLAS N. ARNOLD* and XIAOBO LIU
Abstract. Local error estimates are derived which apply to most stable mixed finite element discretiza
tions of the stationary Stokes equations.
RŽesumŽe. Les estimations locales d'erreur obtenues s'appliquent `a la plupart des discrŽetisations stables
par ŽelŽements finis mixtes du probl`eme de Stokes stationnaire.
Key words. Stokes equations, mixed finite element method, local error estimates, interior error
estimates
AMS(MOS) subject classifications (1991 revision). 65N30, 65N15, 76M10, 76D07
1. Introduction. In this article we establish local error estimates for finite element
approximations to solutions of the Stokes equations. To fix ideas, consider a finite element
approximation to the Stokes equations on a polygonal domain. Suppose that the velocity
space contains (at least) all continuous piecewise polynomials of degree r 1 subordinate
to some triangulation of the domain which satisfy any essential boundary conditions, and
that the pressure space contains all continuous piecewise polynomials of degree r or of
degree r1 (in which case the continuity is dropped for r = 1). Suppose also that the usual
stability condition for Stokes elements is fulfilled. Specific examples for r = 1 include the
MINI finite element (continuous piecewise linears and bubble functions for the velocity and
continuous piecewise linears for the pressure) [1] and the P2 P0 finite element (continuous
