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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]):

In this paper the Wilson nonconforming finite element is considered for solving a class of two-dimensional second-order elliptic boundary value problems. Superconvergence estimates and error expansions are obtained for both uniform and non-uniform rectangular meshes. A new lower bound of the error shows that the usual error estimates are optimal. Finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms.

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