Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time-dependent Maxwell's equations modeling wave propagation in metamaterials. One-order higher global superclose results in the L2 norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011

[1]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[2]  Jan Brandts,et al.  Superconvergence of mixed finite element semi-discretizations of two time-dependent problems , 1999 .

[3]  Qun Lin,et al.  Global superconvergence for Maxwell's equations , 2000, Math. Comput..

[4]  Maciej Paszyński,et al.  Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications , 2007 .

[5]  Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part II: L∞‐error estimates , 1991 .

[6]  Peter Monk,et al.  Superconvergence of finite element approximations to Maxwell's equations , 1994 .

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

[8]  Ningning Yan,et al.  Superconvergent derivative recovery for the intermediate finite element family of the second type , 2001 .

[9]  Superconvergence of recovered gradients of discrete time/piecewise linear Galerkin approximations for linear and nonlinear parabolic problems , 1994 .

[10]  Jun Zou,et al.  Fully discrete finite element approaches for time-dependent Maxwell's equations , 1999, Numerische Mathematik.

[11]  Yunqing Huang,et al.  Recent Advances in Time-Domain Maxwell's Equations in Metamaterials , 2009, HPCA.

[12]  Aihua Wood,et al.  Finite Element Analysis for Wave Propagation in Double Negative Metamaterials , 2007, J. Sci. Comput..

[13]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[14]  Costas M. Soukoulis,et al.  Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials , 2008 .

[15]  Pekka Neittaanmäki,et al.  On superconvergence techniques , 1987 .

[16]  Jinchao Xu,et al.  OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS * , 2009 .

[17]  L. Wahlbin.  Superconvergence in Galerkin Finite Element Methods , 1995 .

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

[19]  Valery Shklover,et al.  Negative Refractive Index Materials , 2006 .

[20]  M. Raffetto,et al.  Existence, uniqueness and finite element approximation of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials , 2005 .

[21]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[22]  Zhiming Chen,et al.  Finite Element Methods with Matching and Nonmatching Meshes for Maxwell Equations with Discontinuous Coefficients , 2000, SIAM J. Numer. Anal..

[23]  Weiying Zheng,et al.  An Adaptive Finite Element Method for the H- ψ Formulation of Time-dependent Eddy Current Problems , 2006, Numerische Mathematik.

[24]  L. Demkowicz.  One and two dimensional elliptic and Maxwell problems , 2006 .

[25]  M. Raffetto,et al.  WELL-POSEDNESS AND FINITE ELEMENT APPROXIMABILITY OF TIME-HARMONIC ELECTROMAGNETIC BOUNDARY VALUE PROBLEMS INVOLVING BIANISOTROPIC MATERIALS AND METAMATERIALS , 2009 .

[26]  Ivo Babuška,et al.  Computer‐based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations , 1996 .

[27]  Vidar Thomée,et al.  Discretization with variable time steps of an evolution equation with a positive-type memory term , 1996 .

[28]  K. Sohlberg.  A Special Issue on Theoretical and Computational Studies of Interlocked Molecules and Molecular Devices , 2006 .

[29]  Qun Lin,et al.  Superconvergence analysis for Maxwell's equations in dispersive media , 2007, Math. Comput..