A meshless method based on the method of fundamental solution for solving the steady-state heat conduction problems

Motivated by the incompleteness of the method of fundamental solution (MFS) for the interior problem, we give another meshless method both theoretically and numerically for recovering the temperature and the heat flux based on the normal derivative of the fundamental solution in this paper. Although the problems under investigation are well-posed, we should note that the method presented here results in an ill-conditioned system and this is a feature of the numerical method employed in the present approach. The ill-posedness of this system is given by the potential function. In order to overcome the ill-posedness of the system, the Tikhonov regularization method, as well as Morozov's discrepancy principle for selecting an appropriate regularization parameter, are used to increase the stability of this method. Then three kinds of boundary value problems are presented to show the effectiveness of this method with some examples, whilst the comparisons with the MFS is presented. The numerical convergence and stability of this method are also analyzed.