A level set immersed finite element method for parabolic problems on surfaces with moving interfaces

Abstract

This paper addresses the challenge of solving parabolic moving interface problems on surfaces. These problems have diverse applications, including the Stefan problem, solidification of dendrites on solid surfaces, and flow patterns on soap bubbles. The main difficulties lie in accurately discretizing complex surfaces, efficiently processing interface jump conditions, and tracking the moving interface. Existing numerical methods for interface problems on surfaces have limitations, such as handling only homogeneous jump conditions, having first-order accuracy, or requiring body-fitted nodes. To overcome these limitations, this paper proposes a second-order accurate immersed finite element method (IFEM) for solving parabolic moving interface problems on surfaces. The method is extended to handle non-homogeneous flux jump conditions by enriching the basis functions on interface elements. Furthermore, a novel computational framework is proposed by combining the IFEM with the level set method to track the moving interface. This framework simulates the heat conduction process involving moving interfaces in different velocity fields. The innovation of this paper lies in its ability to handle moving interface problems on surfaces with improved accuracy, efficiency, and versatility compared to existing methods. Verified through numerical simulation, the proposed method and computational framework enable the simulation of a wider range of heat conduction with moving interfaces on surfaces.