Adaptive Numerical Solution of MFS Systems

The linear systems arising from MFS calculations share certain numerical effects with other systems involving radial basis functions. These effects concern approximation error and stability, which are closely related, and they can already be studied for simple interpolation problems without PDEs. In MFS calculations, they crucially depend on the position and density of the source points and the collocation points. In turn, the choice of these points must depend on the smoothness and possible singularities of the solution. This contribution provides an adaptive method which chooses good source points automatically. A series of examples shows that the adaptive choice of source points follows the theoretical predictions quite well.