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Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary Observations

A Correction to this article was published on 14 September 2020

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Abstract

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank–Nicolson Galerkin method is applied to the least squares functional with a quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and a corresponding convergence rate are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.

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Acknowledgements

The authors would like to thank the Referees and the Editor for their valuable comments and suggestions which helped to improve our paper. T.N.T. Quyen gratefully acknowledges support of the University of Goettingen, Germany. N.T. Son is supported in part by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), Grant 101.01-2017.319. The paper is completed when he is with the Université catholique de Louvain under the support of EOS Project No. 30468160 funded by FNRS and FWO.

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Hào, D.N., Quyen, T.N.T. & Son, N.T. Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary Observations. Appl Math Optim 84, 2289–2325 (2021). https://doi.org/10.1007/s00245-020-09710-2

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Keywords

  • Inverse source problem
  • Tikhonov regularization
  • Crank–Nicolson Galerkin method
  • Source condition
  • Convergence rates
  • Ill-posedness
  • Parabolic problem

Mathematics Subject Classification

  • 35R25
  • 35R30
  • 65J20
  • 65J22