A coupled complex boundary method for parameter identification in elliptic problems

ABSTRACT

In this paper, we study a parameter identification problem for elliptic partial differential equations. We reconstruct the coefficient with additional boundary measurements, including both Dirichlet and Neumann boundary conditions. To solve the problem, the coupled complex boundary method (CCBM), originally proposed in Cheng et al. [A novel coupled complex boundary method for solving inverse source problems, Inverse Probl. 30 (2014), p. 055002] is used. With CCBM, a complex boundary problem is introduced in such a way that the boundary conditions are coupled in a complex Robin boundary condition with a parameter τ. Using Tikhonov regularization, the coefficient is sought such that the imaginary part of the solution of the forward Robin boundary value problem vanishes in the problem domain, which brings advantages on robustness in reconstruction. Besides, the reconstruction is feasible even for very small regularization parameter through choosing the values of τ properly. Some theoretical analyses are given. Moreover, noise model is analysed and the finite element method is used for discretization. Numerical examples show the feasibility and stability of the proposed method.

Keywords:

• Coupled complex method
• parameter identification problem
• Tikhonov regularization
• optimization method
• finite element method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M32
• 65M60
• 35Q93

Acknowledgments

We thank the two anonymous referees for their careful review on our manuscript and for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China, the Natural Science Foundation of Human Province and Human Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering [11526194, 2016JJ4122, 2017TP1017].