TY - JOUR
T1 - Error analysis of finite element approximations of diffusion coefficient identification for elliptic and parabolic problems
AU - Jin, Bangti
AU - Zhou, Zhi
N1 - Funding Information:
\ast Received by the editors June 8, 2020; accepted for publication (in revised form) October 5, 2020; published electronically January 7, 2021. https://doi.org/10.1137/20M134383X \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The work of the first author was supported by the UK EPSRC grant EP/T000864/1. The work of the second author was supported by the Hong Kong Research Grants Council grant 15304420. \dagger Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK ([email protected], [email protected]). \ddagger Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong ([email protected]).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
PY - 2021/2
Y1 - 2021/2
N2 - 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.
AB - 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.
KW - Error estimate
KW - Finite element approximation
KW - Parameter identification
KW - Tikhonov regularization
UR - http://www.scopus.com/inward/record.url?scp=85103760092&partnerID=8YFLogxK
U2 - 10.1137/20M134383X
DO - 10.1137/20M134383X
M3 - Journal article
SN - 1095-7170
VL - 59
SP - 119
EP - 142
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 1
ER -