Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

TL;DR

This paper develops a new error analysis framework for finite element approximations of diffusion coefficient identification in elliptic and parabolic PDEs, providing weighted and standard error estimates with verified positivity conditions.

Contribution

It introduces a novel error analysis approach based on weighted $L^2$ estimates for finite element methods in diffusion coefficient recovery, including both elliptic and parabolic problems.

Findings

Weighted $L^2$ error estimates depend only on problem data

Standard $L^2$ error estimates are derived under positivity conditions

Numerical experiments confirm theoretical error bounds

Abstract

In this work, we present a novel error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem. It is based on the standard regularized output least-squares formulation with an $H^1(\Omega)$ seminorm penalty, and then discretized using the Galerkin finite element method with conforming piecewise linear finite elements for both state and coefficient, and backward Euler in time in the parabolic case. We derive \textit{a priori} weighted $L^2(\Omega)$ estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Further, these estimates also allow deriving standard $L^2(\Omega)$ error estimates, under a positivity condition that can be verified for certain problem data. Numerical experiments are provided to complement the error analysis.