Mixed discontinuous Galerkin approximation of the Maxwell operator.

*(English)*Zbl 1084.65115A discontinuous Galerkin discretization of the Maxwell operator in mixed form is introduced and analyzed. All the unknowns of the underlying system of the partial differential equations are approximated by discontinuous finite element spaces of the same order, so that the present approach can be applied to meshes with nonmatching interfaces and hanging nodes. A stabilization technique is also employed but the use of stabilization parameters is kept minimum.

The analysis is performed essentially in the case of the affine 2D and 3D finite elements. For piecewise constant coefficients, the method is shown to be stable and optimally convergent with respect to the mesh size. Numerical experiments for 2D quadrilateral elements are presented to highlight the performance of the proposed method for problems with both smooth and singular analytical solutions. Some numerical observations for the nonaffine element cases are also included, suggesting various interesting subjects to be analyzed.

The analysis is performed essentially in the case of the affine 2D and 3D finite elements. For piecewise constant coefficients, the method is shown to be stable and optimally convergent with respect to the mesh size. Numerical experiments for 2D quadrilateral elements are presented to highlight the performance of the proposed method for problems with both smooth and singular analytical solutions. Some numerical observations for the nonaffine element cases are also included, suggesting various interesting subjects to be analyzed.

Reviewer: Fumio Kikuchi (Tokyo)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35Q40 | PDEs in connection with quantum mechanics |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

78A25 | Electromagnetic theory, general |

78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |