Error analysis of finite element approximations of diffusion coefficient identification for elliptic and parabolic problems

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 H1(Ω) 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 a priori weighted L2(Ω ) estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Further, these estimates also allow deriving standard L2(Ω ) error estimates under a positivity condition that can be verified for certain problem data. Numerical experiments are provided to complement the error analysis.