POINTWISE A POSTERIORI ERROR ESTIMATES FOR THE STOKES EQUATIONS IN
POLYHEDRAL DOMAINS

STIG LARSSON and ERIK D. SVENSSON

Preprint 2006. 

Abstract. We derive pointwise a posteriori residual-based error
estimates for finite element solutions to the Stokes equations in
polyhedral domains. The estimates relies on the regularity of the of
Stokes equations and provide an upper bound for the pointwise error in
the velocity field on polyhedral domains. Whereas the estimates
provide upper bounds for the pointwise error in the gradient of the
velocity field and the pressure only for a restricted class of
polyhedral domains, convex polyhedral domains in $R^2$, and polyhedral
domains with angles at edges < $3\pi/4$ in $R^3$. In the cause of this
study we also derive $L^q$ a posteriori error estimates, generalizing
well known $L^2$ estimates.

Department of Mathematical Sciences, Chalmers University of
Technology, SE-412 96 G\"oteborg, Sweden

E-mail address: erik.svensson@chalmers.se, stig@chalmers.se