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Chen, Sheng-Gwo; Chi, Mei-Hsiu; Wu, Jyh-Yang

High-order algorithms for Laplace-Beltrami operators and geometric invariants over curved surfaces. (English) Zbl 1329.65263

J. Sci. Comput. 65, No. 2, 839-865 (2015).
Summary: The Laplace-Beltrami operators on curved surfaces play an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting method with the configuration equation to develop a new effective and convergent generalized finite difference method to approximate the Laplace-Beltrami operators acting on functions over discrete surfaces. The convergence rates of our algorithms of discrete Laplace-Beltrami operators over surfaces is \(O(r^n)\), \(n\geq 1\), where \(r\) represents the size of the mesh of discretization of the surface. The problem of high-order accuracies will also be discussed and used to compute geometric invariants of the underlying surfaces. Some convergence tests and eigenvalue computations on the sphere, tori and a dumbbell are presented.

Cited in 4 Documents

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
58J05 Elliptic equations on manifolds, general theory
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

configuration equation; discrete Laplace-Beltrami operator; numerical example; eigenvalue problem; convergence; finite difference method; algorithm; eigenvalue
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