Nonconforming Finite Element Methods for Wave Propagation in Metamaterials

Abstract

In this paper, nonconforming mixed finite element method is proposed to simulate the wave propagation in metamaterials. The error estimate of the semi-discrete scheme is given by convergence order O(h2), which is less than 40 percent of the computational costs comparing with the same effect by using Nédélec-Raviart element. A Crank-Nicolson full discrete scheme is also presented with O(τ2 + h2) by traditional discrete formula without using penalty method. Numerical examples of 2D TE, TM cases and a famous re-focusing phenomena are shown to verify our theories.

High-efficiency approximation of non-local dispersion model for light interaction with metallic nanostructures by modified two-grid method

“…For the high order elements, we believe that the same results can be obtained. For the other elements, for example, the nonconforming elements [19,21,22] can also lead to the same results as that of our MTGM. The partition can also be triangular or tetrahedral edge elements, the similar global post-processing interpolation operator exists [23].…”

“…be the discrete solutions of equations ( 19)- (22). Then for any 1 ≤ k ≤ N, the energy estimates hold…”

“…Now we carry out the superclose [8,9,10] error estimates for the scheme ( 19)- (22). In fact, Multiplying ( 1)-( 4) by test functions ψ ∈ L 2 (Ω), ϕ ∈ H 0 (curl; Ω), χ ∈ H 0 (div; Ω), ξ ∈ L 2 (Ω), respectively, integrating over space Ω, and integrating with respect to time from t = t k−1 to t k , we have…”

“…On the other hand, we also compare the CPU cost of the FEM and the MTGM in Table 3 with the same partition (h = 1/36) on the different time level. Here the CPU cost of the FEM is running by the discrete scheme ( 19)- (22) with 13 the partition h = 1/36. We can see that the MTGM take almost half as much CPU time as the FEM at the finial time t n = 0.1, 0.5, 0.6, 1.…”

New Energy Analysis of Yee Scheme for Metamaterial Maxwell's Equations on Non-Uniform Rectangular Meshes

“…Matical side there has recently been increased interest in the understanding of the mathematical properties of metamaterial Maxwell's equations relevant to numerical analysis. For example, finite-difference time-domain (FDTD) methods [10,[15][16][17], finite element methods [11,13,29,30], and the monograph [12].…”