Identification of Potential in Diffusion Equations from Terminal Observation: Analysis and Discrete Approximation
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DOI10.1137/21M1446708zbMATH Open1506.65144arXiv2201.01563OpenAlexW4221151204MaRDI QIDQ5043045
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Publication date: 27 October 2022
Published in: (Search for Journal in Brave)
Abstract: The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived extsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.
Full work available at URL: https://arxiv.org/abs/2201.01563
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