Analysis of finite element approximations of Stokes equations with non-smooth data

TL;DR

This paper studies finite element methods for Stokes equations with non-smooth boundary data, providing error estimates, an a posteriori error estimator, and demonstrating improved convergence in numerical experiments.

Contribution

It introduces and analyzes regularization techniques and an a posteriori error estimator for finite element approximation of Stokes equations with non-smooth data.

Findings

Almost optimal error estimates for regularized problems

Reliability and efficiency of the a posteriori error estimator

Numerical experiments show optimal convergence with adaptive methods

Abstract

In this paper we analyze the finite element approximation of the Stokes equations with non-smooth 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 [Z. Cai and Y. Wang, Math. Comp., 78(266):771-787, 2009]. 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.