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An adaptive edge finite element method for the Maxwell's equations in metamaterials
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In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.
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Keywords:
- Maxwell's equations,
- wave source terms,
- a-posteriori error estimator,
- adaptive edge finite element method,
- metamaterial media
Citation: Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials[J]. Electronic Research Archive, 2020, 28(2): 961-976. doi: 10.3934/era.2020051
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Abstract
In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.
References
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Figure 1. Example 4.2: The first line and the second line are the real values of
$ E_1 $ and the meshes, respectively. From left to right:$ 8510 $ Ndof (for the initial mesh),$ 133620 $ Ndof (by using$ \eta^{r0}_{K_l} $ ) after$ 14 $ refinements,$ 139743 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) and$ 132334 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) with the same times of 12 refinements -
Figure 2. Example 4.2: Snapshots of numerical solution and adaptive meshes for the real values of
$ E_1 $ after$ 10 $ refinements. First two columns:$ 142833 $ Ndof (by using$ \eta^{r1}_{K_l} $ ); The last two columns:$ 138064 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) -
Figure 3. Example 4.3: The first is the initial mesh with
$ 6090 $ Ndof and the last three are the real values of$ E_1 $ based on the initial mesh -
Figure 4. Example 4.3: The first line and the second line are the real values of
$ E_1 $ and the meshes with$ (x_0, y_0) = (1, 1.45) $ and$ m_p = 2 $ , respectively. From left to right:$ 299395 $ Ndof (using$ \eta^{r0}_{K_l} $ ) after$ 18 $ refinements,$ 315182 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) and$ 273473 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) with the same times of$ 13 $ refinements -
Figure 5. Example 4.3: Snapshots of numerical solution and adaptive meshes for the real values of
$ E_1 $ after$ 13 $ refinements with$ (x_0, y_0) = (1, 1.45) $ and$ m_p = 2 $ . First two columns:$ 303142 $ Ndof (by using$ \eta^{r1}_{K_l} $ ); The last two columns:$ 278572 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) -
Figure 6. Example 4.3: The first line and the second line are the real values of
$ E_1 $ and the meshes with$ (x_0, y_0) = (-1, 1.45) $ and$ m_p = -2 $ , respectively. From left to right:$ 265615 $ Ndof (by using$ \eta^{r0}_{K_l} $ ) after$ 21 $ refinements,$ 282153 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) and$ 237085 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) with the same times of$ 14 $ refinements -
Figure 7. Example 4.3: Snapshots of numerical solution and adaptive meshes for the real values of
$ E_1 $ after$ 14 $ refinements with$ (x_0, y_0) = (-1, 1.45) $ and$ m_p = -2 $ . First two columns:$ 282422 $ Ndof (by using$ \eta^{r1}_{K_l} $ ); The last two columns:$ 245356 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) -
Figure 8. Example 4.3: The first line and the second line are the real values of
$ E_1 $ and the meshes, respectively. From left to right:$ 323340 $ Ndof (by using$ \eta^{r0}_{K_l} $ ) after$ 21 $ refinements,$ 306934 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) and$ 280265 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) with the same times of$ 13 $ refinements -
Figure 9. Example 4.3: Snapshots of numerical solution and adaptive meshes for the real values of
$ E_1 $ after$ 13 $ refinements. First two columns:$ 284570 $ Ndof (by using$ \eta^{r1}_{K_l} $ ); The last two columns:$ 271643 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) -
Figure 10. Example 4.4: The first line, the second line and the third line are the real values of
$ E_1 $ , the real values of$ E_2 $ and the meshes, respectively. From left to right:$ 344405 $ Ndof (by using$ \eta^{r0}_{K_l} $ ) after$ 3 $ refinements,$ 344620 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) and$ 332141 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) with the same times of$ 17 $ refinements -
Figure 11. Example 4.4: The first column, the second column and the third column are snapshots of numerical solution for the real values of
$ E_1 $ ,$ E_2 $ and adaptive meshes, respectively. The first line:$ 393423 $ Ndof after$ 23 $ refinements (by using$ \eta^{r1}_{K_l} $ ); The second line:$ 416438 $ Ndof after$ 21 $ refinements (by using$ \eta^{r2}_{K_l} $ ) -
Figure 12. Example 4.5: The first line and the second line are the real values of
$ E_2 $ and the meshes, respectively. From left to right:$ 794533 $ Ndof (by using$ \eta^{r0}_{K_l} $ ) after$ 4 $ refinements,$ 506294 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) after$ 23 $ refinements and$ 505234 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) after$ 17 $ refinements -
Figure 13. Example 4.5: Snapshots of numerical solution and adaptive meshes for the real values of
$ E_2 $ . First two columns:$ 468957 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) after$ 28 $ refinements; The last two columns:$ 656397 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) after$ 25 $ refinements - Figure 14. Example 4.6: The computational domain for the cloak simulation
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Figure 15. Example 4.6: The first line and the second line are the real values of
$ E_2 $ and the meshes, respectively. From left to right:$ 445224 $ Ndof (by using$ \eta^{r0}_{K_l} $ ) after$ 126 $ refinements,$ 323420 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) after$ 10 $ refinements,$ 120272 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) after$ 30 $ refinements -
Figure 16. Example 4.6: Snapshots of numerical solution and adaptive meshes for the real values of
$ E_2 $ . First two columns:$ 291690 $ Ndof (by using$ \eta^{r1}_{K_l} $ ) after$ 10 $ refinements; The last two columns:$ 78497 $ Ndof (by using$ \eta^{r2}_{K_l} $ ) after$ 34 $ refinements