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Based on a new a posteriori error estimator, an adaptive finite element method is proposed for recovering the Robin coefficient involved in a diffusion system from some boundary measurement. The a posteriori error estimator cannot be derived for this ill-posed nonlinear inverse problem as was done for the existing a posteriori error estimators for direct problems. Instead, we shall derive the a posteriori error estimator from our convergence analysis of the adaptive algorithm. We prove that the adaptive algorithm guarantees a convergent subsequence of discrete solutions in an energy norm to some exact triplet (the Robin coefficient, state and costate variables) determined by the optimality system of the least-squares formulation with Tikhonov regularization for the concerned inverse problem. Some numerical results are also reported to illustrate the performance of the algorithm.

Analysis of an Adaptive Finite Element Method for Recovering the Robin Coefficient