Pointwise Error Estimates for Finite Element Solutions of the Stokes Problem

Abstract

In this paper pointwise error estimates for general finite element approximations of the Stokes problem are established on quasi-uniform grids in R N . The results obtained in this paper improve and extend the existing error estimates in the maximum norm for the Stokes problem. The new pointwise error estimates exhibit a more local dependence of the errors on the true solution and as a by-product provide logarithm-free bounds for all errors except the error of the velocity approximation of the lowest order. Introduction.This paper is devoted to new pointwise error estimates of finite element approximations of the Stokes equations on general quasi-uniform meshes in R N . The results in this paper represent an improvement on and extension of the existing maximum norm error estimates found in Durán, Nochetto, and Wang [6], which were obtained for two-dimensional problems. Our analysis is based on the technique developed recently by Schatz [15,16] for the finite element method for second order elliptic problems (see also Schatz and Wahlbin [17]). In contrast to the traditional approach for proving error estimates in the maximum norm with the weighted function method (Scott [19], Natterer [10], Rannacher and Scott [14], etc.), the new method relies on the availability of local error estimate in energy norm for the underlying finite element discretization. The results in [15] indicate a more localized dependence of the errors on the derivatives of the true solution. As a consequence of these estimates error expansion inequalities have been derived and applied to superconvergence and extrapolation and a posteriori estimates (see [16,8]). The aim of this paper is to extend the new technique from elliptic problems to the Stokes problem, which has a saddle point nature and requires a more careful investigation.It is well known that the conforming finite element approximation (u h , p h ) to the true solution (u, p) of the Stokes problem admits the following optimal error estimates in energy and L 2 norms (Brezzi and Fortin [4], Girault and Raviart [7]):

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

“…Chen makes use of techniques originally developed by Schatz [21] to prove pointwise estimates for the Laplace equation. In this paper we also use the techniques found in [21] and our results are very similar to the results contained in [3]. However, in order to prove pointwise estimates Chen assumed local error estimates for subdomains that touch ∂Ω which are not contained in [2].…”

“…Recently, Girault et al [16] removed the logarithmic factor and extended the results to three dimensions. In this paper and in [3] the logarithmic factor is also not present for higher order elements. The proof in [16] uses techniques for maximum-norm estimates for finite element approximations of the Laplace equation [27], whereas in this paper and in [3] techniques from [21] were used.…”

“…In this paper and in [3] the logarithmic factor is also not present for higher order elements. The proof in [16] uses techniques for maximum-norm estimates for finite element approximations of the Laplace equation [27], whereas in this paper and in [3] techniques from [21] were used. This allows us to establish a more local dependence of the error on the exact solution as compared to the results in [16].…”

“…These pointwise estimates are optimal and describe how the error at a point x depends on the behavior of the exact solution in regions away from x. Recently, Chen [3] used the local estimates derived in [2] to prove pointwise estimates of conforming mixed methods for (1.1) on a domain Ω with a smooth boundary. Chen makes use of techniques originally developed by Schatz [21] to prove pointwise estimates for the Laplace equation.…”

Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

“…Recently, Burman et al [3] used weighted norms for a continuous interior penalty method and obtained a local error estimate for singularly perturbed problems. Also, Chen [4] and Guzman [7] provided pointwise error estimates of a conforming mixed method and discontinuous Galerkin method for the Stokes equation using weighted norm respectively. For other weighted norm error estimates, we refer to [6,11,[13][14][15] and references therein.…”