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

Abstract

We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain $\Omega$, subject to a regularized Tresca friction-type boundary condition on a subset $\Gamma$ of $\partial \Omega$. 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 $\omega \subset \Omega$ 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.