Trace and flux a priori error estimates in finite element approximations of Signorni-type problems

TL;DR

This paper derives new optimal a priori error estimates for the trace and flux in finite element solutions of Signorini-type variational inequalities, enhancing understanding of boundary approximation accuracy.

Contribution

It introduces novel a priori error estimates for boundary trace and flux in Signorini problems using Steklov-Poincaré operators and duality techniques.

Findings

Optimal error estimates for trace and flux achieved

Numerical results confirm theoretical convergence rates

Enhanced understanding of boundary approximation in variational inequalities

Abstract

Variational inequalities play in many applications an important role and are an active research area. Optimal a priori error estimates in the natural energy norm do exist but only very few results in other norms exist. Here we consider as prototype a simple Signorini problem and provide new optimal order a priori error estimates for the trace and the flux on the Signorini boundary. The a priori analysis is based on the exact and a mesh-dependent Steklov-Poincar\'e operator as well as on duality in Aubin-Nitsche type arguments. Numerical results illustrate the convergence rates of the finite element approach.