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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 |
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. |
show all references
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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