Discrete Conservation Law on Curved Surfaces

TL;DR

This paper introduces an intrinsic numerical method for computing differential operators on curved surfaces, ensuring conservation laws and divergence theorem are satisfied on triangular meshes.

Contribution

The paper presents a novel, unified approach to directly compute derivatives on discretized surfaces that preserves fundamental geometric properties.

Findings

The method accurately computes derivatives on curved surfaces.

It ensures divergence theorem and conservation laws hold on meshes.

The approach is effective for scalar and vector functions.

Abstract

In this paper we shall introduce a simple, effective numerical method for finding differential operators for scalar and vector-valued functions on surfaces. The key idea of our algorithm is to develop an intrinsic and unified way to compute directly the partial derivatives of functions defined on triangular meshes which are the discretization of regular surfaces under consideration. Most importantly, the divergence theorem and conservation laws on triangular meshes are fulfilled.