Abstract In this paper, we derive a priori error estimates for variational inequalities of the first kind in an abstract framework. This is done by combining the first Strang Lemma and the Falk Theorem. The main application consists of the derivation of a priori error estimates for Galerkin methods, in which “variational crimes” may perturb the underlying variational inequality. Different types of perturbations are incorporated into the abstract framework and are discussed in various examples. For instance, the perturbation caused by an inexact quadrature is examined in detail for the Laplacian obstacle problem. For this problem, guaranteed rates for the approximation error resulting from the use of a higher-order finite element method are derived. In numerical experiments, the influence of the number of quadrature points on the approximation error and on the quadrature-related error itself is studied for several discretization methods.

The Scott–Vogelius element is a popular finite element for the discretization of the Stokes equations which enjoys inf-sup stability and gives divergence-free velocity approximations. However, it is well known that the convergence rates for the discrete pressure deteriorate in the presence of certain critical vertices in a triangulation of the domain. Modifications of the Scott–Vogelius element such as the recently introduced pressure-wired Stokes element also suffer from this effect. In this paper we introduce a simple modification strategy for these pressure spaces that preserves the inf-sup stability while the pressure converges at an optimal rate.