## Locally divergence-free discontinuous Galerkin methods for the Maxwell equations.(English)Zbl 1049.78019

This paper is devoted to the study of the role of discontinuous Galerkin methods in the numerical analysis of the two-dimensional Maxwell system $\frac{\partial H_x}{\partial t}=-\frac{\partial E_z}{\partial y},\quad \frac{\partial H_y}{\partial t}=\frac{\partial E_z}{\partial x},\quad \frac{\partial E_z}{\partial t}=\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}.$ First the locally divergence-free space is introduced and a numerical formulation of the algorithm is given. Next, it is described a way to measure the divergence of a piecewise smooth function, and the $$L^2$$-projection is defined from the space of locally divergence-free discontinuous piecewise polynomials to a subspace that contains the globally divergence-free piecewise polynomials. An abstract result on the $$L^2$$-stability and several error estimates are also given.

### MSC:

 78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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### References:

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