Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow

Abstract

We derive estimates for the error in a variational approximation of the lift and drag coefficients of a body immersed into a viscous flow governed by the Navier-Stokes equations. The variational approximation is based on computing a certain weighted average of a finite element approximation to the solution of the Navier-Stokes equations. Our main result is an a posteriori estimate that puts a bound on the error in the lift and drag coefficients in terms of the local mesh size, a local residual quantity, and a local weight describing the local stability properties of an associated dual problem. The weight may be approximated by solving the dual problem numerically. The error bound is thus computable and can be used for quantitative error estimation; we apply it to design an adaptive finite element algorithm specifically for the approximation of the lift and drag coefficients.