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High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem.

J S Hesthaven1, T Warburton

  • 1Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA. jan.hesthaven@brown.edu

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|August 13, 2004
PubMed
Summary

This study introduces a novel discontinuous element scheme using high-order nodal elements to solve the challenging Maxwell eigenvalue problem. This method offers an efficient and robust alternative to traditional Galerkin finite-element methods.

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Area of Science:

  • Computational electromagnetics
  • Numerical analysis

Background:

  • The Maxwell eigenvalue problem presents significant numerical challenges, primarily due to its large null space.
  • Standard Galerkin finite-element methods (FEM) with curl-conforming elements are widely used but can be computationally intensive.

Purpose of the Study:

  • To propose and evaluate a high-order discontinuous element scheme as an alternative to conventional FEM for solving Maxwell's eigenvalue problem.
  • To investigate the applicability and robustness of this new scheme in both two- and three-dimensional scenarios.

Main Methods:

  • Implementation of a discontinuous element scheme utilizing high-order nodal elements.
  • Application to both 2D and 3D Maxwell eigenvalue problems.
  • Modification of the 3D scheme to address formulation challenges inherent in discontinuous methods.

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Main Results:

  • The proposed scheme demonstrates effectiveness and robustness for 2D Maxwell eigenvalue problems.
  • Initial challenges in the 3D formulation were successfully overcome with minor, scheme-specific modifications.
  • Numerical experiments validate the approach for general problems.

Conclusions:

  • Discontinuous element schemes offer a high-order accurate, efficient, and robust alternative for solving Maxwell's equations in both frequency and time domains.
  • The findings suggest a promising new direction for computational electromagnetics research and application.