A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem

Abstract

We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $μ=γs$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_μ$ for every $μ>0$, its gap is positive almost everywhere, and $\|u-u_μ\|_{H^1(Ω)}\lesssimμ^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the \(L^2\) central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_μ$, with constants independent of $s$. If $u_μ$ is uniformly bounded in $H^r(Ω)$, $3/2<r\leq k+1$, then $\|u-u_h\|_{H^1(Ω)}\lesssim h^{r-1}\|u_μ\|_{H^r(Ω)}+μ^{1/2}$; hence the sufficient balance $s\lesssim h^{2r-1}$ preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with $P_1$ and $P_2$ elements on structured and unstructured meshes support the predicted rates and the local Newton theory, while showing that a rate-preserving smoothing may still resolve the contact set poorly.