The basic H 1-finite element error estimate of order h with only H 2-regularity on the solution has not been yet established for the simplest 2D Signorini problem approximated by a discrete variational inequality (or the equivalent mixed method) and linear finite elements. To obtain an optimal error bound in this basic case and also when considering more general cases (three-dimensional problem, quadratic finite elements.. .), additional assumptions on the exact solution (in particular on the unknown contact set, see [5, 20, 35]) had to be used. In this paper we consider finite element approximations of the 2D and 3D Signorini problems with linear and quadratic finite elements. In the analysis, we remove all the additional assumptions and we prove optimal H 1-error estimates with the only standard Sobolev regularity. The main tools are local L 1 and L 2-estimates of the normal constraints and the normal displacements on the candidate contact area and error bounds depending both on the contact and on the non-contact set.

The present paper deals with linear and quadratic finite element approximations of the two and three-dimensional unilateral contact problems between two elastic bodies with nonmatching meshes. We propose a simple noninterpenetration condition (on the displacements) which is local (as the well known node-on-segment and node-on-face conditions) and accurate (like the mortar approach). This condition consists of averaging locally on a few elements the noninterpenetration. We achieve the corresponding optimal convergence analysis in 2D and 3D using various linear and quadratic elements and the computations of the Taylor patch test and Hertzian contact.