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 uμ for every μ>0, its gap is positive almost everywhere, and |u−uμ|H1(Ω)≲μ1/2. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the L2 central-path boundary law. If the obstacle is locally the trace of an H2-function, we also prove a uniform local H2 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 uμ, with constants independent of s. If uμ is uniformly bounded in Hr(Ω), 3/2<r≤ k+1, then |u−uh|H1(Ω)≲ hr−1|uμ|Hr(Ω)+μ1/2; hence the sufficient balance s≲ h2r−1 preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with P1 and P2 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.
Related
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BibTeX
@misc{hansbo2026barrierregularized,
title = {{A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem}},
author = {Peter Hansbo and Mats G. Larson},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30470},
eprint = {2608.30470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30470},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem
AU - Peter Hansbo
AU - Mats G. Larson
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30470
UR - https://arxiv.org/abs/2608.30470
AB - 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)≲μ^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≤ k+1, then |u−u_h|_H^1(Ω)≲ h^r−1|u_μ|_H^r(Ω)+μ^1/2; hence the sufficient balance s≲ 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.
ER -