An inverse geometric problem in steady state heat conduction - the solution and stability analysis

The paper addresses a numerical method for boundary identification in a problem governed by Laplace's equation. The proposed numerical procedure for discrete reconstruction of the unknown boundary from the given temperature data is based on the Trefftz method. In contrast to the procedures described in the reference papers, the present approach requires significantly less and easier computation. The paper undertakes analysis of the resistance of the solution to small perturbations of the prescribed temperature condition at the unknown part of the boundary. We define and then estimate a sensitivity factor which allows quantitative assessment of the relationship between temperature measurement errors and boundary identification errors, even if the exact solution is not known. The included numerical examples demonstrate the effectiveness of the proposed method for boundary reconstruction and present the analysis of numerical stability using a sensitivity factor.