Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials

In this paper, we develop a nodal discontinuous Galerkin method for solving the time-dependent Maxwell?s equations when metamaterials are involved. Both semi- and fully-discrete schemes are constructed. Numerical stability and error estimate are proved for both schemes. Numerical results are presented to demonstrate that the method is not only efficient but also very effective in solving metamaterial Maxwell?s equations.

[1]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[2]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[3]  B. Rivière,et al.  A Combined Mixed Finite Element and Discontinuous Galerkin Method for Miscible Displacement Problem in Porous Media , 2002 .

[4]  Chi-Wang Shu,et al.  A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes , 2006, J. Comput. Phys..

[5]  Daniel Weile,et al.  Electromagnetic Metamaterials: Physics and Engineering Explorations (Engheta, N. and Ziolkowski, R.W.; 2006) [Book Review] , 2007, IEEE Antennas and Propagation Magazine.

[6]  Pingwen Zhang,et al.  Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions , 2004 .

[7]  Ilaria Perugia,et al.  Interior penalty method for the indefinite time-harmonic Maxwell equations , 2005, Numerische Mathematik.

[8]  Wei Yang,et al.  Modeling Backward Wave Propagation in Metamaterials by the Finite Element Time-Domain Method , 2013, SIAM J. Sci. Comput..

[9]  Zhimin Zhang,et al.  Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media , 2010, J. Comput. Phys..

[10]  Jichun Li,et al.  Development of discontinuous Galerkin methods for Maxwell's equations in metamaterials and perfectly matched layers , 2011, J. Comput. Appl. Math..

[11]  Richard Ziolkowski,et al.  Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs. , 2003, Optics express.

[12]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[13]  Eric T. Chung,et al.  Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids , 2013, J. Comput. Phys..

[14]  Hailiang Liu,et al.  The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems , 2008, SIAM J. Numer. Anal..

[15]  Jiajia Waters,et al.  An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwell’s equations in metamaterials , 2012 .

[16]  L. Fezoui,et al.  Convergence and stability of a discontinuous galerkin time-domain method for the 3D heterogeneous maxwell equations on unstructured meshes , 2005 .

[17]  Stéphane Lanteri,et al.  Locally implicit discontinuous Galerkin method for time domain electromagnetics , 2010, J. Comput. Phys..

[18]  D. Schötzau,et al.  Interior penalty discontinuous Galerkin method for Maxwell's equations , 2007 .

[19]  Chi-Wang Shu,et al.  Locally divergence-free discontinuous Galerkin methods for the Maxwell equations , 2004, Journal of Computational Physics.

[20]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[21]  E. Montseny,et al.  Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell's equations , 2008, J. Comput. Phys..

[22]  David R. Smith,et al.  Metamaterials: Theory, Design, and Applications , 2009 .

[23]  Willie J Padilla,et al.  Composite medium with simultaneously negative permeability and permittivity , 2000, Physical review letters.

[24]  Jan S. Hesthaven,et al.  Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations , 2002 .

[25]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[26]  S. Lanteri,et al.  Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell's equations in dispersive media. , 2013 .

[27]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[28]  Jichun Li.  Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials , 2009 .

[29]  Richard W. Ziolkowski,et al.  Maxwellian material based absorbing boundary conditions , 1999 .

[30]  Jan S. Hesthaven.  Time-domain finite element methods for Maxwell"s equations in metamaterials , 2014 .