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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[1] Assous, F.; Degond, P.; Heintze, E.; Raviart, P. A.; Segre, J., On a finite element method for solving the three dimensional Maxwell equations, Journal of Computational Physics, 109, 222-237 (1993) · Zbl 0795.65087
[2] Baker, G. A.; Jureidini, W. N.; Karakashian, O. A., Piecewise solenoidal vector fields and the Stokes problem, SIAM Journal on Numerical Analysis, 27, 1466-1485 (1990) · Zbl 0719.76047
[3] Balsara, D. S.; Spicer, D. S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, Journal of Computational Physics, 149, 270-292 (1999) · Zbl 0936.76051
[4] Bramble, J.; Sun, T., A negative-norm least squares method for Reissner-Mindlin plates, Mathematics of Computation, 67, 901-916 (1998) · Zbl 0899.73544
[5] Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, 47, 217-235 (1985) · Zbl 0599.65072
[6] Ciarlet, P., The Finite Element Methods for Elliptic Problems (1975), North-Holland: North-Holland Amsterdam
[7] Cockburn, B., Discontinuous Galerkin methods for convection-dominated problems, (Barth, T. J.; Deconinck, H., High-Order Methods for Computational Physics. High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering, vol. 9 (1999), Springer: Springer Berlin), 69-224 · Zbl 0937.76049
[8] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Mathematics of Computation, 54, 545-581 (1990) · Zbl 0695.65066
[9] Cockburn, B.; Karniadakis, G.; Shu, C.-W., The development of discontinuous Galerkin methods, (Cockburn, B.; Karniadakis, G.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications. Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11 (2000), Springer: Springer Berlin), 3-50, Part I: Overview · Zbl 0989.76045
[10] Cockburn, B.; Luskin, M.; Shu, C.-W.; Süli, E., Enhanced accuracy by post-processing for finite element methods for hyperbolic equations, Mathematics of Computation, 72, 577-606 (2003) · Zbl 1015.65049
[11] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics, 141, 199-224 (1998) · Zbl 0920.65059
[12] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35, 2440-2463 (1998) · Zbl 0927.65118
[13] Cockburn, B.; Shu, C.-W., Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16, 173-261 (2001) · Zbl 1065.76135
[14] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability preserving high order time discretization methods, SIAM Review, 43, 89-112 (2001) · Zbl 0967.65098
[15] Hesthaven, J. S.; Warburton, T., Nodal high-order methods on unstructured grids. I. time-domain solution of Maxwell’s equations, Journal of Computational Physics, 181, 186-221 (2002) · Zbl 1014.78016
[16] Jiang, B.-N.; Wu, J.; Povinelli, L. A., The origin of spurious solutions in computational electromagnetics, Journal of Computational Physics, 125, 104-123 (1996) · Zbl 0848.65086
[17] Jiang, G.-S.; Shu, C.-W., On cell entropy inequality for discontinuous Galerkin methods, Mathematics of Computation, 62, 531-538 (1994) · Zbl 0801.65098
[18] Karakashian, O. A.; Jureidini, W. N., A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 35, 93-120 (1998) · Zbl 0933.76047
[19] Munz, C.-D.; Omnes, P.; Schneider, R.; Sonnendrücker, E.; Voß, U., Divergence correction techniques for Maxwell solvers based on a hyperbolic model, Journal of Computational Physics, 161, 484-511 (2000) · Zbl 0970.78010
[20] Nicolaides, R. A.; Wang, D.-Q., Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions, Mathematics of Computation, 67, 947-963 (1998) · Zbl 0907.65116
[22] Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Transactions on Antenna Propagation AP, 14, 302-307 (1966) · Zbl 1155.78304
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