# 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 & 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,, \emph{ACM Trans. Math. Softw.}, 33 (2007), 1.  doi: 10.1145/1268776.1268779.  Google Scholar [2] A. Bonito, J. M. Cascón, P. Morin and R. H. Nochetto, AFEM for geometric PDE: The Laplace-Beltrami operator,, in \emph{Analysis and Numerics of Partial Differential Equations}, (2013), 257.  doi: 10.1007/978-88-470-2592-9_15.  Google Scholar [3] A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method,, \emph{SIAM J. Numer. Anal.}, 48 (2010), 734.  doi: 10.1137/08072838X.  Google Scholar [4] A. Bonito and J. E. Pasciak, Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator,, \emph{Math. Comp.}, 81 (2012), 1263.  doi: 10.1090/S0025-5718-2011-02551-2.  Google Scholar [5] A. Bonito and M. S. Pauletti, The step-38 tutorial progam: The laplace-beltrami operator,, \emph{http://www.dealii.org.}, ().   Google Scholar [6] S. Y. Cheng, Eigenfunctions and nodal sets,, \emph{Comment. Math. Helv.}, 51 (1976), 43.   Google Scholar [7] G. Dziuk, An algorithm for evolutionary surfaces,, \emph{Numer. Math.}, 58 (1991), 603.  doi: 10.1007/BF01385643.  Google Scholar [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,, \emph{J. Numer. Math.}, 15 (2007), 181.  doi: 10.1515/jnma.2007.009.  Google Scholar [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,, \emph{J. Numer. Math.}, 15 (2007), 277.  doi: 10.1515/jnum.2007.013.  Google Scholar [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 \emph{Partial Differential Equations}, (2008), 225.  doi: 10.1007/978-1-4020-8758-5_12.  Google Scholar [11] V. Hernandez, J. E. Roman and V. Vicente, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,, \emph{ACM Trans. Math. Software}, 31 (2005), 351.  doi: 10.1145/1089014.1089019.  Google Scholar [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 \emph{2009 IEEE International symposium on biomedical imaging: From nano to macro, (2009), 694.   Google Scholar [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), (1998).  doi: 10.1137/1.9780898719628.  Google Scholar [14] K. Uhlenbeck, Generic properties of eigenfunctions,, \emph{Amer. J. Math.}, 98 (1976), 1059.   Google Scholar

show all references

##### References:
 [1] W. Bangerth, R. Hartmann and G. Kanschat, deal.II-a general purpose object oriented finite element library,, \emph{ACM Trans. Math. Softw.}, 33 (2007), 1.  doi: 10.1145/1268776.1268779.  Google Scholar [2] A. Bonito, J. M. Cascón, P. Morin and R. H. Nochetto, AFEM for geometric PDE: The Laplace-Beltrami operator,, in \emph{Analysis and Numerics of Partial Differential Equations}, (2013), 257.  doi: 10.1007/978-88-470-2592-9_15.  Google Scholar [3] A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method,, \emph{SIAM J. Numer. Anal.}, 48 (2010), 734.  doi: 10.1137/08072838X.  Google Scholar [4] A. Bonito and J. E. Pasciak, Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator,, \emph{Math. Comp.}, 81 (2012), 1263.  doi: 10.1090/S0025-5718-2011-02551-2.  Google Scholar [5] A. Bonito and M. S. Pauletti, The step-38 tutorial progam: The laplace-beltrami operator,, \emph{http://www.dealii.org.}, ().   Google Scholar [6] S. Y. Cheng, Eigenfunctions and nodal sets,, \emph{Comment. Math. Helv.}, 51 (1976), 43.   Google Scholar [7] G. Dziuk, An algorithm for evolutionary surfaces,, \emph{Numer. Math.}, 58 (1991), 603.  doi: 10.1007/BF01385643.  Google Scholar [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,, \emph{J. Numer. Math.}, 15 (2007), 181.  doi: 10.1515/jnma.2007.009.  Google Scholar [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,, \emph{J. Numer. Math.}, 15 (2007), 277.  doi: 10.1515/jnum.2007.013.  Google Scholar [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 \emph{Partial Differential Equations}, (2008), 225.  doi: 10.1007/978-1-4020-8758-5_12.  Google Scholar [11] V. Hernandez, J. E. Roman and V. Vicente, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,, \emph{ACM Trans. Math. Software}, 31 (2005), 351.  doi: 10.1145/1089014.1089019.  Google Scholar [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 \emph{2009 IEEE International symposium on biomedical imaging: From nano to macro, (2009), 694.   Google Scholar [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), (1998).  doi: 10.1137/1.9780898719628.  Google Scholar [14] K. Uhlenbeck, Generic properties of eigenfunctions,, \emph{Amer. J. Math.}, 98 (1976), 1059.   Google Scholar
 [1] A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709 [2] Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078 [3] Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 [4] Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291 [5] Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295 [6] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [7] Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927 [8] Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034 [9] Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795 [10] Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689 [11] Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020096 [12] Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873 [13] A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769 [14] Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807 [15] Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211 [16] Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 [17] Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619 [18] Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965 [19] Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637 [20] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020178

2019 Impact Factor: 1.105