Analysis of an adaptive finite element method for recovering the Robin coefficient. (English) Zbl 1355.65148
Summary: 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.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 35R30 | Inverse problems for PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35R25 | Ill-posed problems for PDEs |
| 65N20 | Numerical methods for ill-posed problems for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
Robin inverse problem; a posteriori error estimates; adaptive finite element method; diffusion system; ill-posed nonlinear inverse problem; convergence; algorithm; Tikhonov regularization; numerical resultsReferences:
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