Discrete Conservation Laws on Curved Surfaces II: A Dual Approach

Abstract

In this paper we discuss discrete conservation laws for diffusion equations over triangular surface meshes from the viewpoint of duality. Conservation laws are very important for us to model physical phenomenon on curved spaces. The key idea of our method is to use the concept of local dual meshes, which are determined by the centroid points of the triangles in the original meshes. The Green formula will provide us a natural way to give a discrete approximation of the Laplace--Beltrami operators on functions over curved regular surfaces. Then, we shall show that discrete conservation laws are fulfilled on the dual meshes for our algorithm. Moreover, our discrete Laplace--Beltrami operators have the local property: it only involves the information on the 1-ring of neighboring vertices. Note that this approach is quite different from the finite element/difference method, used very often in the filed of numerical analysis. Some convergence problems will also be discussed. Numerical simulations are given to s...