Optimal Convergence for Discrete Variational Inequalities Modelling Signorini Contact in 2D and 3D without Additional Assumptions on the Unknown Contact Set

Abstract

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 two-dimensional 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 (e.g., the three-dimensional problem, quadratic finite elements), additional assumptions on the exact solution (in particular on the unknown contact set, see [Z. Belhachmi and F. Ben Belgacem, Math. Comp., 72 (2003), pp. 83--104; S. Hüeber and B. Wohlmuth, SIAM J. Numer. Anal., 43 (2005), pp. 156--173; B. Wohlmuth, A. Popp, M. Gee, and W. Wall, Comput. Mech. 49 (2012), pp. 735--747] had to be used. In this paper, we consider finite element approximations of the two-dimensional and three-dimensional Signorini problems with linear and quadratic finite elements. In the analysis, we remove all the additional assumptions and prove optimal $H^1$-error estimates with only the 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 noncontact set.