## Abstract

We present two hybridizable discontinuous Galerkin (HDG) methods for the numerical solution of the time-harmonic Maxwell's equations. The first HDG method explicitly enforces the divergence-free condition and thus necessitates the introduction of a Lagrange multiplier. It produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the Lagrange multiplier. The second HDG method does not explicitly enforce the divergence-free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only. For both HDG methods, the approximate vector field converges with the optimal order of k+1 in the L^{2}-norm, when polynomials of degree k are used to represent all the approximate variables. We propose elementwise postprocessing to obtain a new H^{curl}-conforming approximate vector field which converges with order k+1 in the H^{curl}-norm. We present extensive numerical examples to demonstrate and compare the performance of the HDG methods.

Original language | English (US) |
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Pages (from-to) | 7151-7175 |

Number of pages | 25 |

Journal | Journal of Computational Physics |

Volume | 230 |

Issue number | 19 |

DOIs | |

State | Published - Aug 10 2011 |

### Bibliographical note

Funding Information:We would like to thank Mr. Zhang Wujun of UMN for adaptively refined meshes. J. Peraire and N.C. Nguyen would like to acknowledge the Singapore-MIT Alliance and the AFOSR Grant FA9550-08-1-0350 for partially supporting this work. B. Cockburn would like to acknowledge the National Science Foundation for partially supporting this work through Grant DMS-0712955 .

## Keywords

- Computational electromagnetics
- Discontinuous Galerkin methods
- Finite element method
- Hybrid/mixed methods
- Maxwell's equations
- Postprocessing