In this paper,we are concerned with the fast solvers for higher order edge finite element discretizations of Maxwell's equations.We present the preconditioners for the first family and second family of higher order N′ed′elec element equations,respectively.By combining the stable decompositions of two kinds of edge finite element spaces with the abstract theory of auxiliary space preconditioning,we prove that the corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform grids.We also present some numerical experiments to demonstrate the theoretical results.
ZHONG LiuQiang1,2,SHU Shi1,2,SUN DuDu3&TAN Lin4 1School of Mathematical and Computational Sciences,Xiangtan University,Xiangtan 411105,China 2Hunan Key Laboratory for Computation and Simulation in Science and Engineering,Xiangtan 411105,China 3Institute of Computational Mathematics and Scientific/Engineering Computing,Academy of Mathematicsand Systems Science,Graduate University of Chinese Academy of Sciences,Chinese Academy Sciences,P.O.Box 2719,Beijing 100190,China 4Department of Math-Physics,Nanhua University,Hengyang 421001,China
In this paper, we obtain optimal error estimates in both L^2-norm and H（curl）-norm for the Nedelec edge finite 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 finite 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.
