American Institute of Mathematical Sciences

September  2014, 13(5): 2115-2126. doi: 10.3934/cpaa.2014.13.2115

On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach

 1 Department of Mathematics, Texas A\&M University, College Station, TX 77845, United States 2 University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204 - 3008

Received  September 2013 Revised  January 2014 Published  June 2014

In this article we investigate, via numerical computations, the intersection properties of the nodal set of the eigenfunctions of the Laplace-Beltrami operator for smooth surfaces in $R^3$ (the nodal set of a continuous function is the set of those points at which the function vanishes). First, we briefly discuss the numerical solution of the eigenvalue/eigenfunction problem for the Laplace-Beltrami operator on bounded surfaces of $R^3$, and then consider some specific surfaces and visualize how the nodal lines intersect (or not) depending of the symmetries verified by the surface. After validating our computational methodology with the surface of a ring torus, we will investigate a simple surface without symmetry and observe that in that case the nodal set of the computed eigenfunctions consists of non intersecting lines, suggesting some conjecture. We observe also that for the above symmetry-free surface, the number of connected components of the nodal set varies non-monotonically with the rank of the associated eigenvalue (assuming that the eigenvalues are ordered by increasing value).
Citation: Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115
References:
 [1] W. Bangerth, R. Hartmann and G. Kanschat, deal.II-a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 24/1-24/27. doi: 10.1145/1268776.1268779. [2] A. Bonito, J. M. Cascón, P. Morin and R. H. Nochetto, AFEM for geometric PDE: The Laplace-Beltrami operator, in Analysis and Numerics of Partial Differential Equations, volume 4, pages 257-306. Springer INdAM Series, 2013. doi: 10.1007/978-88-470-2592-9_15. [3] A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), 734-771. doi: 10.1137/08072838X. [4] A. Bonito and J. E. Pasciak, Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator, Math. Comp., 81 (2012), 1263-1288. doi: 10.1090/S0025-5718-2011-02551-2. [5] A. Bonito and M. S. Pauletti, The step-38 tutorial progam: The laplace-beltrami operator, http://www.dealii.org. [6] S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. [7] G. Dziuk, An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603-611. doi: 10.1007/BF01385643. [8] F. J. Foss, II, R. Glowinski and R. H. W. Hoppe, On the numerical solution of a semilinear elliptic eigenproblem of Lane-Emden type. I. Problem formulation and description of the algorithms, J. Numer. Math., 15 (2007), 181-208. doi: 10.1515/jnma.2007.009. [9] F. J. Foss, II, R. Glowinski and R. H. W. Hoppe, On the numerical solution of a semilinear elliptic eigenproblem of Lane-Emden type. II. Numerical experiments, J. Numer. Math., 15 (2007), 277-298. doi: 10.1515/jnum.2007.013. [10] R. Glowinski and D. C. Sorensen, Computing the eigenvalues of the Laplace-Beltrami operator on the surface of a torus: a numerical approach, in Partial Differential Equations, volume 16 of Comput. Methods Appl. Sci., pages 225-232. Springer, Dordrecht, 2008. doi: 10.1007/978-1-4020-8758-5_12. [11] V. Hernandez, J. E. Roman and V. Vicente, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31 (2005), 351-362. doi: 10.1145/1089014.1089019. [12] R. Lai, Y. Shi, I. Dinov, T. F. Chan and A. W. Toga, Laplace-Beltrami nodal counts: a new signature for 3D shape analysis, in 2009 IEEE International symposium on biomedical imaging: From nano to macro, Vols 1 and 2, pages 694-697, 2009. IEEE Internaional Symposium on Biomedical Imaging-From Nano to Macro, Boston, MA, JUN 28-JUL 01, 2009. [13] R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide, volume 6 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. doi: 10.1137/1.9780898719628. [14] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078.

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References:
 [1] W. Bangerth, R. Hartmann and G. Kanschat, deal.II-a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 24/1-24/27. doi: 10.1145/1268776.1268779. [2] A. Bonito, J. M. Cascón, P. Morin and R. H. Nochetto, AFEM for geometric PDE: The Laplace-Beltrami operator, in Analysis and Numerics of Partial Differential Equations, volume 4, pages 257-306. Springer INdAM Series, 2013. doi: 10.1007/978-88-470-2592-9_15. [3] A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), 734-771. doi: 10.1137/08072838X. [4] A. Bonito and J. E. Pasciak, Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator, Math. Comp., 81 (2012), 1263-1288. doi: 10.1090/S0025-5718-2011-02551-2. [5] A. Bonito and M. S. Pauletti, The step-38 tutorial progam: The laplace-beltrami operator, http://www.dealii.org. [6] S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. [7] G. Dziuk, An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), 603-611. doi: 10.1007/BF01385643. [8] F. J. Foss, II, R. Glowinski and R. H. W. Hoppe, On the numerical solution of a semilinear elliptic eigenproblem of Lane-Emden type. I. Problem formulation and description of the algorithms, J. Numer. Math., 15 (2007), 181-208. doi: 10.1515/jnma.2007.009. [9] F. J. Foss, II, R. Glowinski and R. H. W. Hoppe, On the numerical solution of a semilinear elliptic eigenproblem of Lane-Emden type. II. Numerical experiments, J. Numer. Math., 15 (2007), 277-298. doi: 10.1515/jnum.2007.013. [10] R. Glowinski and D. C. Sorensen, Computing the eigenvalues of the Laplace-Beltrami operator on the surface of a torus: a numerical approach, in Partial Differential Equations, volume 16 of Comput. Methods Appl. Sci., pages 225-232. Springer, Dordrecht, 2008. doi: 10.1007/978-1-4020-8758-5_12. [11] V. Hernandez, J. E. Roman and V. Vicente, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31 (2005), 351-362. doi: 10.1145/1089014.1089019. [12] R. Lai, Y. Shi, I. Dinov, T. F. Chan and A. W. Toga, Laplace-Beltrami nodal counts: a new signature for 3D shape analysis, in 2009 IEEE International symposium on biomedical imaging: From nano to macro, Vols 1 and 2, pages 694-697, 2009. IEEE Internaional Symposium on Biomedical Imaging-From Nano to Macro, Boston, MA, JUN 28-JUL 01, 2009. [13] R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide, volume 6 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. doi: 10.1137/1.9780898719628. [14] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078.
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