We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nédélec edge elements, for three-dimensional indefinite time-harmonic Maxwell equations. We prove that the AE-FEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is the establishment of a quasi-orthogonality and a localized a posteriori error estimator.
In this paper, we obtain optimal error estimates in both L^-norm and iî(curl)-norm for the Nédélec 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 Û error estimates into the L^ 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 Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.Mathematics subject classification: 65N30, 35Q60.
Two new two-grid algorithms are proposed for solving the Maxwell eigenvalue problem. The new methods are based on the two-grid methodology recently proposed by Xu and Zhou [Math. Comp., 70 (2001), pp. 17–25] and further developed by Hu and Cheng [Math. Comp., 80 (2011), pp. 1287–1301] for elliptic eigenvalue problems. The new two-grid schemes reduce the solution of the Maxwell eigenvalue problem on a fine grid to one linear indefinite Maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid. The new schemes, therefore, save total computational cost. The error estimates reveals that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results.
SUMMARYThe standard adaptive edge finite element method (AEFEM), using first/second family Nédélec edge elements with any order, for the three-dimensional H(curl)-elliptic problems with variable coefficients is shown to be convergent for the sum of the energy error and the scaled error estimator. The special treatment of the data oscillation and the interior node property are removed from the proof. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi-optimal.
Using the plane wave expansion method we have studied the effect of symmetry on the complete acoustic band gaps in two-dimensional binary phononic crystals including five types of straight rod arranged in hexagonal lattices. For each type of rod, two cases are considered: water rods in mercury (low-density scatters in high-density background) and mercury rods in water (high-density scatters in low-density background). The crystals' symmetry can be changed by rotating the noncircular rods, and the results show that the extreme (both the maximum and minimum) gaps at any given filling factor appear under high symmetry. So the above maximum gaps can be obtained by adjusting the rods' orientation. And the peaks of those maximum gaps are relative to the rods' rotational symmetry, the relations of which are different for the two cases with different density contrast.
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