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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.

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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