Convergence Analysis for the Recovery of the Friction Threshold in a Scalar Tresca Model

TL;DR

This work addresses the inverse problem of recovering the friction threshold $a$ in a scalar elliptic model with a regularised Tresca boundary condition. It introduces a simple finite‑element, Newton‑type scheme that exploits the assumption that $a$ lies in a finite‑dimensional subspace, and proves local uniqueness and second‑order convergence of the discrete friction $a_h$ to the true $a$, along with quadratic convergence of Newton iterations for suitable starting points. A Keller‑type framework is developed to guarantee stability and convergence under noise, with an $O(h^2)$ bias and a data‑driven error term. The authors also establish well‑posedness and high‑order regularity for the nonlinear problem, derive finite‑element error estimates for both the nonlinear and linearized problems, and provide Lipschitz stability for the linearized inverse mapping. Numerical experiments on a 2D geometry corroborate the theory, showing the predicted convergence rates and robustness to measurement noise.

Abstract

We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain $Ω$, subject to a regularized Tresca friction-type boundary condition on a subset $Γ$ of $\partial Ω$. The friction threshold, a positive function appearing in this boundary condition, is assumed to be unknown and serves as the coefficient to be recovered in our inverse problem. Assuming that (i) the friction threshold lies in a finite dimensional space with known basis functions, (ii) the right hand sides of the partial differential equation are known, and (iii) the solution to the partial differential equation on some small open subset $ω\subset Ω$ is available, we develop an iterative computational method for the recovery of the friction threshold. This algorithm is simple to implement and is based on piecewise linear finite elements. We show that the proposed algorithm converges in second order to a function $a_h$ and, moreover, that $a_h$ converges in second order in the finite element's mesh size $h$ to the true (unknown) friction threshold. We highlight our theoretical results by simulations that confirm our rates numerically.

Figures (4)

• Figure 1: Sketch of the geometrical configuration. For given $q=u^{(\tilde{a})}|_\omega \in L^2(\Omega)$, $f \in H^1(\Omega)$ and $g \in H^{3/2}(\Gamma)$ the aim is to reconstruct the function $\tilde{a} \in V_J \subset C^2(\Gamma)$.
• Figure 2: Left: The exact solution on the domain $\Omega_h$. Right: Convergence of the finite element solution as a function of the mesh size $h$. The $H^1$-error decreases linearly, the $L^2$-error decreases quadratically, in accordance with Lemma \ref{['lem:H1']} and Lemma \ref{['lem:L2']}. The dashed lines have slope one and two respectively.
• Figure 3: Top left: Visualization of the exact solution on the domain $\Omega$. Top middle: Visualization of $q$ on the domain $\omega$. Top right to bottom right: The initial guess, the iterates 25, 100 and the final iteration 160.
• Figure 4: Double logarithmic plots showing the maximal mesh size $h$ against the relative error from \ref{['eq:errrel']}. The dashed line has slope 2. Left: Unperturbed data $q$ is given. Right: Noisy data $q^\delta$ for different noise levels $\sigma$ are given.

Theorems & Definitions (33)

• Theorem 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3