Analysis of finite element approximations of stokes equations with nonsmooth data

Abstract

In this paper we analyze the finite element approximation of the Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid-driven cavity problem, improving the error estimates obtained in Cai and Wang [Math. Comp., 78 (2009), pp. 771-787]. Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and efficiency and show some numerical examples which suggest that optimal order of convergence is obtained by an adaptive procedure based on our estimator.