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

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