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A posteriori-driven adaptive strategy for solving inverse Cauchy problems in diffusion-reaction models

Author

Listed:
  • Hamdi, Hafida
  • Nachaoui, Mourad
  • Bergam, Amal
  • Nachaoui, Abdeljalil

Abstract

This work addresses the inverse Cauchy problem for the modified Helmholtz equation using an alternating iterative approach. The central contribution lies in the design of novel local error indicators based on a posteriori analysis, which simultaneously assess the accuracy of the spatial discretization and the convergence behavior of the iterative algorithm. Unlike standard methods, our strategy leverages a comparative assessment of these indicators to drive an adaptive mesh refinement process. This adaptive framework ensures a more balanced distribution of computational resources, significantly reducing the numerical cost while maintaining high solution accuracy. The proposed methodology is validated through a series of synthetic and application-driven numerical experiments, demonstrating both its effectiveness and robustness in reconstructing inaccessible boundary data.

Suggested Citation

  • Hamdi, Hafida & Nachaoui, Mourad & Bergam, Amal & Nachaoui, Abdeljalil, 2026. "A posteriori-driven adaptive strategy for solving inverse Cauchy problems in diffusion-reaction models," Applied Mathematics and Computation, Elsevier, vol. 518(C).
    • Handle: RePEc:eee:apmaco:v:518:y:2026:i:c:s0096300325006277
    DOI: 10.1016/j.amc.2025.129902
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    References listed on IDEAS

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    1. Benedict Barnes & Isaac Addai & Francis Ohene Boateng & Ishmael Takyi, 2022. "Solving the Helmholtz Equation Together with the Cauchy Boundary Conditions by a Modified Quasi‐Reversibility Regularization Method," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    2. Zhang, Bei & Chen, Shaochun & Zhao, Jikun, 2014. "A posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1062-1075.
    3. Benedict Barnes & Isaac Addai & Francis Ohene Boateng & Ishmael Takyi & Ram Jiwari, 2022. "Solving the Helmholtz Equation Together with the Cauchy Boundary Conditions by a Modified Quasi-Reversibility Regularization Method," Journal of Mathematics, Hindawi, vol. 2022, pages 1-15, December.
    4. Li, Yixin & Hu, Xianliang, 2022. "Artificial neural network approximations of Cauchy inverse problem for linear PDEs," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    5. Ellabib, A. & Nachaoui, A., 2008. "An iterative approach to the solution of an inverse problem in linear elasticity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(2), pages 189-201.
    6. Bergam, A. & Chakib, A. & Nachaoui, A. & Nachaoui, M., 2019. "Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 865-878.
    7. Xiong, Xiangtuan & Cao, Xiaoxiao & He, Shumei & Wen, Jin, 2016. "A modified regularization method for a Cauchy problem for heat equation on a two-layer sphere domain," Applied Mathematics and Computation, Elsevier, vol. 290(C), pages 240-249.
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