A face-based LTL method for solving diffusion equations and Cahn-Hilliard equations on stationary surfaces

Abstract

In this paper we shall develop a face-based local tangential lifting method to handle the problem of discrete conservation laws for diffusion equations and solve Cahn-Hilliard equations over curved surfaces. To do so, we need to give a new discrete approximation of the Laplace-Beltrami operators on functions over curved surfaces. Our method is a simple effective numerical method for solving diffusion equations on curved surfaces. The key idea of our face-based local tangential lifting method is to approximate locally the underlying surface and functions with respect to faces in the discrete model of the surface by using the centroid method. Indeed, our algorithm is a generalized finite difference method and an intrinsic geometry method. It gives a natural way to approximate directly the Laplace-Beltrami operators on a triangular mesh. Discrete conservation laws for diffusion equations on triangular meshes are also discussed.