Abstract
We present a novel adaptive higher-order finite element (hp-FEM) algorithm to solve non-symmetric elliptic eigenvalue problems. This is an extension of our prior work on symmetric elliptic eigenvalue problems. The method only needs to make one call to a generalized eigensolver on the coarse mesh, and then it employs Newton’s or Picard’s methods to resolve adaptively a selected eigenvalue–eigenvector pair. The fact that the method does not need to make repeated calls to a generalized eigensolver not only makes it very efficient, but it also eliminates problems that pose great complications to adaptive algorithms, such as eigenvalue reordering or returning arbitrary linear combinations of eigenvectors associated with the same eigenvalue. New theoretical and numerical results for the non-symmetric case are presented.
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References
Babuška I, Osborn J (1987) Eigenvalue Problems, Technical note BN. Institute for Physical Science and Technology
Berini P (2000) Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures. Phys Rev B 61(15):10484–10503
Cliffe A, Hall E, Houston P (2010) Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. SIAM J Sci Comput 31(6):4607–4632
Giani S, Graham IG (2012) Adaptive finite element methods for computing band gaps in photonic crystals. Numerische Mathematik 121(1):31–64
Hackbusch W (1992) Elliptic differential equations. Springer, Berlin
Joannopoulos JD, Meade RD, Winn JN (1995) Photonic crystals. Molding the flow of light. Princeton University Press, Princeton
Lalor N, Priebsch H-H (2007) The prediction of low- and mid-frequency internal road vehicle noise: a literature survey. Proc IMechE D J Automob Eng 221(3):245–269
Lehoucq RB, Sorensen DC, Yang C (1998) ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia
Solin P, Cerveny J, Dubcova L, Andrs D (2010) Monolithic discretization of linear thermoelasticity problems via adaptive multimesh \(hp\)-FEM. J Comput Appl Math 234:2350–2357
Solin P, Giani S (2012) An iterative finite element method for elliptic eigenvalue problems. J Comput Appl Math 236(18):4582–4599
Solin P, Segeth K, Dolezel I (2003) Higher-order finite element methods. Chapman & Hall, CRC Press, London
Strang G, Fix GJ (1973) An analysis of the finite element method. Prentice–Hall, Englewood Cliffs
Hislop PD, Sigal IM (1996) Introduction to spectral theory. Springer, Berlin
Brenner SC, Scott LR (1994) The mathematical theory of finite element methods. Springer, Berlin
Acknowledgments
The first author was supported by the Subcontract No. 00089911 of Battelle Energy Alliance (U.S. Department of Energy intermediary) as well as by the Grant No. P105/10/1682 of the Grant Agency of the Czech Republic.
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Solin, P., Giani, S. An iterative adaptive hp-FEM method for non-symmetric elliptic eigenvalue problems. Computing 95 (Suppl 1), 183–213 (2013). https://doi.org/10.1007/s00607-012-0251-7
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DOI: https://doi.org/10.1007/s00607-012-0251-7
Keywords
- Partial differential equation
- Non-symmetric eigenvalue problem
- Iterative method
- Adaptive higher-order finite element method
- hp-FEM