### Abstract

In this paper, a theoretical framework with a completely new theory is presented for the general curlcurl-graddiv second-order elliptic eigenproblem when discretized by the recently developed local L-2 projected C-0 finite element method. The theoretical framework consists of two Fortin-type interpolations and an Inf-Sup inequality associated with a trilinear curl/div form. From this framework, error estimates are readily derived for the source problem as well as the eigenproblem. The new theory is verified for the local L-2 projected C-0 finite element method with elementwise linear element L-2 projections applied to div and curl operators and the C-0 linear elements enriched with some face-bubbles and element-bubbles for the singular solution. The challenging question is whether C-0 elements can give spectrally correct approximations when eigenfunctions are singular (not in (H-1(Omega))(d) space). It is shown that optimal error bounds can be obtained from this theoretical framework with O(h(2r)) for eigenvalues while O(h(r)) for singular eigenfunctions in (H-r(Omega))(d) (d = 2, 3), where r <1, and that the local L-2 projected C-0 finite element method is spectrally correct.

Original language | English |
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Pages (from-to) | 1678-1714 |

Number of pages | 37 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 51 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |

### Keywords

- SPECTRAL APPROXIMATION
- ELECTROMAGNETIC EIGENPROBLEMS
- L-2 projected C-0 finite element method
- FLUID-STRUCTURE INTERACTION
- Fortin-type interpolation
- Inf-Sup inequality
- singular solution
- EIGENVALUE PROBLEMS
- DISCONTINUOUS GALERKIN APPROXIMATION
- SINGULAR FIELD METHOD
- DISCRETE COMPACTNESS
- curlcurl-graddiv eigenproblem
- error estimates
- STRUCTURE SYSTEMS
- MAXWELL EQUATIONS
- spectral correct approximation
- POLYHEDRAL DOMAINS

## Cite this

^{2 }projected C

^{0 }finite element method.

*SIAM Journal on Numerical Analysis*,

*51*(3), 1678-1714. https://doi.org/10.1137/100812136