Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 184786, 15 pages
http://dx.doi.org/10.1155/2015/184786
Research Article

A Fourier Continuation Method for the Solution of Elliptic Eigenvalue Problems in General Domains

Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

Received 2 July 2015; Accepted 25 October 2015

Academic Editor: Salvatore Alfonzetti

Copyright © 2015 Oscar P. Bruno et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. O. P. Bruno and M. Lyon, “High-order unconditionally stable FC-AD solvers for general smooth domains. I. Basic elements,” Journal of Computational Physics, vol. 229, no. 6, pp. 2009–2033, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. L. N. Trefethen and T. Betcke, “Computed eigenmodes of planar regions,” in Recent Advances in Differential Equations and Mathematical Physics, vol. 412 of Contemporary Mathematics, pp. 297–314, American Mathematical Society, Providence, RI, USA, 2006. View at Google Scholar
  3. F. Scheben and I. G. Graham, “Iterative methods for neutron transport eigenvalue problems,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2785–2804, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. L. Fox, P. Henrici, and C. Moler, “Approximations and bounds for eigenvalues of elliptic operators,” SIAM Journal on Numerical Analysis, vol. 4, no. 1, pp. 89–102, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  5. T. Betcke and L. N. Trefethen, “Reviving the method of particular solutions,” SIAM Review, vol. 47, no. 3, pp. 469–491, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. P. Amore, “Solving the Helmholtz equation for membranes of arbitrary shape: numerical results,” Journal of Physics A: Mathematical and Theoretical, vol. 41, no. 26, Article ID 265206, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. G. Sathej and R. Adhikari, “The eigenspectra of Indian musical drums,” Journal of the Acoustical Society of America, vol. 125, no. 2, pp. 831–838, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. L. Banjai, “Eigenfrequencies of fractal drums,” Journal of Computational and Applied Mathematics, vol. 198, no. 1, pp. 1–18, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. R. H. Hoppe, H. Wu, and Z. Zhang, “Adaptive finite element methods for the Laplace eigenvalue problem,” Journal of Numerical Mathematics, vol. 18, no. 4, pp. 281–302, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. S. Min and D. Gottlieb, “Domain decomposition spectral approximations for an eigenvalue problem with a piecewise constant coefficient,” SIAM Journal on Numerical Analysis, vol. 43, no. 2, pp. 502–520, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. R. Rannacher, A. Westenberger, and W. Wollner, “Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error,” Journal of Numerical Mathematics, vol. 18, no. 4, pp. 303–327, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. T. A. Driscoll, “Eigenmodes of isospectral drums,” SIAM Review, vol. 39, no. 1, pp. 1–17, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Boffi, “Finite element approximation of eigenvalue problems,” Acta Numerica, vol. 19, pp. 1–120, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. Berland, C. Bogey, O. Marsden, and C. Bailly, “High-order, low dispersive and low dissipative explicit schemes for multiple-scale and boundary problems,” Journal of Computational Physics, vol. 224, no. 2, pp. 637–662, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. W. D. Henshaw and D. W. Schwendeman, “Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement,” Journal of Computational Physics, vol. 227, no. 16, pp. 7469–7502, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. N. Albin and O. P. Bruno, “A spectral FC solver for the compressible Navier-Stokes equations in general domains I: explicit time-stepping,” Journal of Computational Physics, vol. 230, no. 16, pp. 6248–6270, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proceedings of the IEEE, vol. 93, no. 2, pp. 216–231, 2005. View at Publisher · View at Google Scholar
  18. M. Lyon and O. P. Bruno, “High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations,” Journal of Computational Physics, vol. 229, no. 9, pp. 3358–3381, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. F. M. Gomes and D. C. Sorenson, “ARPACK++: A C++ implementation of ARPACK eigenvalue package,” Tech. Rep. CRPC-TR97729, Center for Research on Parallel Computation, 1997. View at Google Scholar
  20. P. Amore, J. P. Boyd, F. M. Fernandez, and B. Rösler, “High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson Extrapolation of second order finite differences,” http://arxiv.org/abs/1509.02795.
  21. R. Blikberg and T. Sørevik, “Load balancing and OpenMP implementation of nested parallelism,” Parallel Computing, vol. 31, no. 10-12, pp. 984–998, 2005. View at Publisher · View at Google Scholar · View at Scopus