A comparative study of Domain Embedding Methods for regularized solutions of inverse Stefan problems

In this paper, various Domain Embedding Methods (DEMs) for an inverse Stefan problem are presented and compared. These DEMs extend the moving boundary domain to a larger, but simple and xed domain. The original unknown interface position is then replaced by a new unknown, which can be a boundary temperature or heat ux, or an internal heat source. In this way, the non-linear identi cation problem is transformed into a linear one in the enlarged domain. Using di erent physical quantities as the new unknown leads to di erent DEMs. They are analysed from various points of view (accuracy, e ciency, etc.) through two test problems, by a comparison with a common Front-Tracking Method (FTM). The rst test has a smooth temperature eld and the second one has some singularities. The advantage of the DEMs in solving the inverse problem and in computing the corresponding direct mapping is shown. In the direct problem, high-order accurate schemes could be obtained more easily with the DEMs than with the FTM. In the inverse problem, an iterative regularization and a Tikhonov regularization have been employed. For the FTM, the iterative regularization is not e cient|the solution oscillates when the data are noisy. As for the Tikhonov regularization, it requests special care to choose an adequate penalty term. In contrast, both the regularizations give good results with all the considered DEMs, except for the second test problem at the beginning (t =0+) when the value of the heat ux and the heat source tends to 1. Slightly di erent regularization e ects have been obtained when using di erent DEMs. Finally, an automatic choice of the optimal regularization parameter is also discussed, using data with di erent noise levels. We propose the use of the curve of the residual norm against the regularization parameter