High-order algorithms for solving eigenproblems over discrete surfaces

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays 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 (LTL) method with the configuration equation to develop a new effective and convergent algorithm to solve the eigenvalue problems of 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.

[1]  Colin B. Macdonald,et al.  Solving eigenvalue problems on curved surfaces using the Closest Point Method , 2011, J. Comput. Phys..

[2]  T. Banchoff,et al.  Differential Geometry of Curves and Surfaces , 2010 .

[3]  Xiangmin Jiao,et al.  High-Order Numerical Integration over Discrete Surfaces , 2012, SIAM J. Numer. Anal..

[4]  Sheng-Gwo Chen,et al.  Estimating normal vectors and curvatures by centroid weights , 2004, Comput. Aided Geom. Des..

[5]  Martin Rumpf,et al.  Anisotropic geometric diffusion in surface processing , 2000 .

[6]  Sheng-Gwo Chen,et al.  A new intrinsic numerical method for PDE on surfaces , 2009, Int. J. Comput. Math..

[7]  Sheng-Gwo Chen,et al.  A geometric interpretation of weighted normal vectors and its improvements , 2005, International Conference on Computer Graphics, Imaging and Visualization (CGIV'05).

[8]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[9]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[10]  Bart M. ter Haar Romeny,et al.  Geometry-Driven Diffusion in Computer Vision , 1994, Computational Imaging and Vision.

[11]  Sheng-Gwo Chen,et al.  Discrete Conservation Laws on Curved Surfaces , 2013, SIAM J. Sci. Comput..

[12]  Arthur W. Toga,et al.  Anisotropic Laplace-Beltrami eigenmaps: Bridging Reeb graphs and skeletons , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[13]  Arthur W. Toga,et al.  Harmonic Surface Mapping with Laplace-Beltrami Eigenmaps , 2008, MICCAI.

[14]  Mark Meyer,et al.  Implicit fairing of irregular meshes using diffusion and curvature flow , 1999, SIGGRAPH.

[15]  Niklas Peinecke,et al.  Laplace-spectra as fingerprints for shape matching , 2005, SPM '05.

[16]  Mei-Hsiu Chi,et al.  Derivatives estimation for 2D digital data points , 2006 .

[17]  Steven J. Ruuth,et al.  A simple embedding method for solving partial differential equations on surfaces , 2008, J. Comput. Phys..