OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS *

In this paper, we obtain optimal error estimates in both L 2 -norm and H(curl)-norm for the Nedelec edge flnite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L 2 error estimates into the L 2 estimate of a discrete divergence-free function which belongs to the edge flnite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

[1]  V. Girault,et al.  Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in ³ , 1988 .

[2]  Daniele Boffi,et al.  EDGE FINITE ELEMENTS FOR THE APPROXIMATION OF MAXWELL RESOLVENT OPERATOR , 2002 .

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

[4]  Leszek Demkowicz,et al.  Finite Element Methods for Maxwell Equations , 2007 .

[5]  P. Monk.  A Simple Proof of Convergence for an Edge Element Discretization of Maxwell’s Equations , 2003 .

[6]  Annalisa Buffa,et al.  Remarks on the Discretization of Some Noncoercive Operator with Applications to Heterogeneous Maxwell Equations , 2005, SIAM J. Numer. Anal..

[7]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[8]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

[9]  Peter Monk,et al.  Discrete compactness and the approximation of Maxwell's equations in R3 , 2001, Math. Comput..

[10]  Douglas N. Arnold,et al.  Multigrid in H (div) and H (curl) , 2000, Numerische Mathematik.

[11]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

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

[13]  Alberto Valli,et al.  An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations , 1999, Math. Comput..

[14]  Long Chen.  FINITE ELEMENT METHODS FOR MAXWELL EQUATIONS , 2010 .

[15]  Peter Monk,et al.  A finite element method for approximating the time-harmonic Maxwell equations , 1992 .

[16]  G. Wittum,et al.  Adaptive local multigrid methods for solving time-harmonic eddy-current problems , 2006, IEEE Transactions on Magnetics.

[17]  Gabriel Wittum,et al.  Adaptive Local Multigrid Methods for the Solution of Time Harmonic Eddy Current Problems , 2005 .

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

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

[20]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .