Finite element error analysis for elliptic parameter identification with power-type nonlinearity

TL;DR

This paper develops finite element error analysis for an elliptic parameter identification problem with power-type nonlinearity, establishing stability and error estimates that extend previous linear case results.

Contribution

It introduces conditional stability estimates and derives a priori error bounds for finite element approximations in a nonlinear elliptic setting, under weaker regularity assumptions.

Findings

Conditional stability estimates at the continuous level

A priori error estimates depending on mesh size, regularization, noise, and nonlinearity

Extension and sharpening of linear case error results

Abstract

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on piecewise linear finite elements. As one of our main novelties, we establish conditional stability estimates at the continuous level, which form the theoretical foundation of the present finite element analysis. Our stability analysis relies on tailored analytical tools, including Hardy-type inequalities, fractional Gagliardo-Nirenberg inequalities, and weighted spaces with singular distance weights. By invoking the achieved conditional stability together with the Carstensen quasi-interpolation operator and associated estimates in negative Sobolev spaces, we derive a priori error estimates for the proposed finite element approximation in terms of the mesh size, the regularization parameter, the noise level, and the nonlinearity exponent. Our results extend the recent stability and error estimates for the linear case by Jin et al. \cite{jin2022convergence} and sharpen their error estimates and convergence order under weaker regularity assumptions.