The local tangential lifting method for moving interface problems on surfaces with applications

Abstract

In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces. One novelty of the new method is the local tangential lifting method to construct discrete delta functions on surfaces. The idea of the local tangential lifting method is to transform a local surface problem to a local two dimensional one on the tangent planes of surfaces at some selected surface nodes. Moreover, a surface version of the front tracking method is developed to track moving interfaces on surfaces. Strategies have been developed for computing geodesic curvatures of interfaces on surfaces. Various numerical examples are presented to demonstrate the accuracy of the new methods. It is also interesting to see the comparison of the dendritic solidification processes in two dimensional spaces and on surfaces.