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

Jin, B; Zhou, Z; (2021) Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems. SIAM Journal on Numerical Analysis , 59 (1) pp. 119-142. 10.1137/20m134383x. Green open access

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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 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.

Type: Article
Title: Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems
Open access status: An open access version is available from UCL Discovery
DOI: 10.1137/20m134383x
Publisher version: http://dx.doi.org/10.1137/20m134383x
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10118804
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