An improved a priori error analysis for finite element approximations of Signorini's problem

International audienceThe present paper is concerned with the unilateral contact model in linear elastostatics, the so-called Signorini problem (our results can also be applied to the scalar Signorini problem). A standard continuous linear finite element approximation is first chosen to approach the twodimensional problem. We develop a new error analysis in the $H^1$-norm using estimates on Poincare constants with respect to the size of the areas of the noncontact sets. In particular we do not assume any additional hypothesis on the finiteness of the set of transition points between contact and noncontact. This approach allows us to establish better error bounds under sole $H^{\tau}$ assumptions on the solution: if $3/2 < \tau < 2$ we improve the existing rate by a factor $h^{(\tau−3/2)^2}$ and if $\tau= 2$ the existing rate $(h^{3/4})$ is improved by a new rate of $h \sqrt{\vert\ln(h)\vert}$. Using the same finite element spaces as previously we then consider another discrete approximation of the (nonlinear) contact condition in which the same kind of analysis leads to the same convergence rates as for the first approximation