Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Superconvergence Analysis of High-Order Rectangular Edge Elements for Time-Harmonic Maxwell’s Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, high-order rectangular edge elements are used to solve the two dimensional time-harmonic Maxwell’s equations. Superconvergence for the Nédélec interpolation at the Gauss points is proved for both the second and third order edge elements. Using this fact, we obtain the superconvergence results for the electric field \(\mathbf {E}\), magnetic field H and \(curl\mathbf {E}\) in the discrete \(l^2\) norm when the Maxwell’s equations are solved by both elements. Extensive numerical results are presented to justify our theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Ainsworth, M., Coyle, J.: Hierarchic hp-edge element families for Maxwell’ equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Eng. 190, 6709–6733 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babus̆ka, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K.: Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace’s, Poisson’s, and the elasticity equations. Numer. Methods Partial Differ. Equ. 12(3), 347–392 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part II: general unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2004)

    Article  MATH  Google Scholar 

  4. Boffi, D., Costabel, M., Dauge, M., Demkowicz, L.: Discrete compactness for the hp version of rectangular edge finite elements. SIAM J. Numer. Anal. 44, 979–1004 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brandts, J.H.: Superconvergence of mixed finite element semi-discretizations of two time-dependent problems. Appl. Math. 44(1), 43–53 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, W., Shu, C.-W., Yang, Y., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations. SIAM J. Numer. Anal. 53(4), 1651–1671 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao, W., Zhang, Z., Zou, Q.: Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 52(2), 2555–2573 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Celiker, F., Cockburn, B.: Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection–diffusion problems in one space dimension. Math. Comput. 76(257), 67–96 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection–diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044–4072 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chung, E.T., Ciarlet, P., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cockburn, B., Dong, B., Guzmán, J.: A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40, 141–187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Qiu, W., Shi, K.: Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comput. 81, 1327–1353 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Franz, S., Roos, H.-G.: Superconvergence of a Galerkin FEM for higher-order elements in convection–diffusion problems. Numer. Math. Theory Methods Appl. 7(3), 356–373 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Huang, Y., Li, J., Lin, Q.: Superconvergence analysis for time-dependent Maxwell’s equations in metamaterials. Numer. Methods Partial Differ. Equ. 28(6), 1794–1816 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, Y., Li, J., Wu, C.: Averaging for superconvergence: verification and application of 2D edge elements to Maxwell’s equations in metamaterials. Comput. Methods Appl. Mech. Eng. 255, 121–132 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, Y., Li, J., Wu, C., Yang, W.: Superconvergence analysis for linear tetrahedral edge elements. J. Sci. Comput. 62(1), 122–145 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, Y., Li, J., Yang, W., Sun, S.: Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials. J. Comput. Phys. 230(22), 8275–8289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, J., Huang, Y.: Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer Series in Computational Mathematics 43. Springer, New York (2013)

  19. Li, J., Shields, S.: Superconvergence analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes. Numer. Math. 134(4), 741–781 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lin, Q., Lin, J.F.: High accuracy approximation of mixed finite element for 2-D Maxwell equations. Acta Math. Sci. Ser. A Chin. 23, 499–503 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Lin, Q., Yan, N.: Global superconvergence for Maxwell’s equations. Math. Comput. 69(229), 159–176 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin, Q., Yan, N.: The construction and analysis of high accurate finite element methods. Hebei University Press, Hebei (1996)

    Google Scholar 

  23. Monk, P.: Superconvergence of finite element approximations to Maxwell’s equations. Numer. Methods Partial Differ. Equ. 10(6), 793–812 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Qiao, Z., Yao, C., Jia, S.: Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-harmonic Maxwell’s equations. J. Sci. Comput. 46(1), 1–19 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Santos, J.E., Sheen, D.: On the existence and uniqueness of solutions to Maxwell’s equations in bounded domains with application to magnetotellurics. Math. Models Methods Appl. Sci. 10(4), 615–628 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wheeler, M.F., Whiteman, J.R.: Superconvergence of recovered gradients of discrete time/piecewise linear Galerkin approximations for linear and nonlinear parabolic problems. Numer. Methods Partial Differ. Equ. 10(3), 271–294 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xie, Z., Zhang, Z.: Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D. Math. Comput. 79(269), 35–45 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, Z.: Finite element superconvergence approximation for one-dimensional singularly perturbed problems. Numer. Methods Partial Differ. Equ. 18(3), 374–395 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-D convection–diffusion problems. Math. Comput. 72(243), 1147–1177 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, Z., Yan, N., Sun, T.: Superconvergent derivative recovery for the intermediate finite element family of the second type. IMA J. Numer. Anal. 21(3), 643–665 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

    Ming Sun

  2. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    Jichun Li

  3. School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, People’s Republic of China

    Peizhen Wang

  4. Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Zhimin Zhang

  5. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

Authors
  1. Ming Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jichun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peizhen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhimin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jichun Li.

Additional information

The first and third authors were partially supported by the China Scholarship Council. The first author was partially supported by National Natural Science Foundation of China (NSFC) Grant 11671233. The second author was partially supported by NSF Grants DMS-1416742, and NSFC 11671340. The fourth author is also supported in part by Grants NSFC 11471031, NSFC 91430216, and NSAF U1530401, and the US NSF Grant DMS-1419040.

Appendices

Appendix 1: The Degrees of Freedom and Basis Functions When k=2

The degrees of freedom \(c_i\) are [18, p.75]:

$$\begin{aligned}&c_i=\int _{l_i}\mathbf {E}(x,y)\cdot \mathbf {\tau }_idl,\ \ \ \ i=1,\ldots ,4,\\&c_5=\int _{l_1}(x-x_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_1dl,\quad \quad \quad c_6=\int _{l_2}(y-y_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_2dl,\\&c_7=\int _{l_3}(x-x_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_3dl,\quad \quad \quad c_8=\int _{l_4}(y-y_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_4dl,\\&c_9=\int _{K}\mathbf {E}(x,y)\cdot (1,0)'dxdy,\quad \quad \quad c_{10}=\int _{K}\mathbf {E}(x,y)\cdot (0,1)'dxdy,\\&c_{11}=\int _{K}\mathbf {E}(x,y)\cdot (x-x_c,0)'dxdy,\ \ c_{12}=\int _{K}\mathbf {E}(x,y)\cdot (0,y-y_c)'dxdy, \end{aligned}$$

and the basis function \(\varvec{N}_i\) can be expressed as:

$$\begin{aligned}&\varvec{N}_1=\left( \begin{array}{c} \frac{[3(y-y_c)+h_y][(y-y_c)-h_y]}{8h_xh_y^2}\\ 0 \end{array}\right) , \quad \varvec{N}_2=\left( \begin{array}{c} 0\\ \frac{[3(x-x_c)-h_x][(x-x_c)+h_x]}{8h^2_xh_y} \end{array}\right) ,\\&\varvec{N}_3=\left( \begin{array}{c} \frac{[3(y-y_c)-h_y][(y-y_c)+h_y]}{8h_xh_y^2}\\ 0 \end{array}\right) , \quad \varvec{N}_4=\left( \begin{array}{c} 0\\ \frac{[3(x-x_c)+h_x][(x-x_c)-h_x]}{8h^2_xh_y} \end{array}\right) ,\\&\varvec{N}_5=\left( \begin{array}{c} \frac{(x-x_c)[3(y-y_c)-3h_y][3(y-y_c)+h_y]}{8h^3_xh_y^2}\\ 0 \end{array}\right) , \quad \varvec{N}_6=\left( \begin{array}{c} 0\\ \frac{(y-y_c)[3(x-x_c)+3h_x][3(x-x_c)-h_x]}{8h^2_xh^3_y} \end{array}\right) ,\\&\varvec{N}_7=\left( \begin{array}{c} \frac{(x-x_c)[3(y-y_c)+3h_y][3(y-y_c)-h_y]}{8h^3_xh_y^2}\\ 0 \end{array}\right) , \quad \varvec{N}_8=\left( \begin{array}{c} 0\\ \frac{(y-y_c)[3(x-x_c)-3h_x][3(x-x_c)+h_x]}{8h^2_xh^3_y} \end{array}\right) ,\\&\varvec{N}_9=\left( \begin{array}{c} \frac{-3[(y-y_c)-h_y][(y-y_c)+h_y]}{8h_xh_y^3}\\ 0 \end{array}\right) , \quad \varvec{N}_{10}=\left( \begin{array}{c} 0\\ \frac{-3[(x-x_c)-h_x][(x-x_c)+h_x]}{8h^3_xh_y} \end{array}\right) ,\\&\varvec{N}_{11}=\left( \begin{array}{c} \frac{-9(x-x_c)[(y-y_c)-h_y][(y-y_c)+h_y]}{8h^3_xh_y^3}\\ 0 \end{array}\right) , \quad \varvec{N}_{12}=\left( \begin{array}{c} 0\\ \frac{-9(y-y_c)[(x-x_c)+h_x][(x-x_c)-h_x]}{8h^3_xh^3_y} \end{array}\right) . \end{aligned}$$

For the interpolation \(I_{K} H\in Q_{1,1}\), the degrees of freedom \(d_i\) and the basis functions \(\psi _i\) are:

$$\begin{aligned}&d_1=\int _{K}H(x,y)dxdy, \quad \quad \quad \quad \quad d_2=\int _{K}H(x,y)(x-x_c)dxdy,\\&d_3=\int _{K}H(x,y)(y-y_c)dxdy, \quad \ d_4=\int _{K}H(x,y)(x-x_c)(y-y_c)dxdy, \end{aligned}$$

and the basis functions \(\psi _i\) are:

$$\begin{aligned}&\psi _1(x,y)=\frac{1}{4h_xh_y},\quad \quad \quad \psi _2(x,y)=\frac{3(x-x_c)}{4h^3_xh_y},\\&\psi _3(x,y)=\frac{3(y-y_c)}{4h_xh^3_y}, \quad \ \psi _4(x,y)=\frac{9(x-x_c)(y-y_c)}{4h^3_xh^3_y}. \end{aligned}$$

Appendix 2: The Degrees of Freedom and Basis Functions When k=3

The degrees of freedom \(c_i\) are

$$\begin{aligned}&c_i=\int _{l_i}\mathbf {E}(x,y)\cdot \mathbf {\tau }_idl,\ \ \ \ i=1,\ldots ,4,\\&c_5=\int _{l_1}(x-x_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_1dl,\quad \quad \quad c_6=\int _{l_2}(y-y_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_2dl,\\&c_7=\int _{l_3}(x-x_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_3dl,\quad \quad \quad c_8=\int _{l_4}(y-y_c)\mathbf {E}(x,y)\cdot \mathbf {\tau }_4dl,\\&c_9=\int _{l_1}(x-x_c)^2\mathbf {E}(x,y)\cdot \mathbf {\tau }_1dl,\quad \quad \quad c_{10}=\int _{l_2}(y-y_c)^2\mathbf {E}(x,y)\cdot \mathbf {\tau }_2dl,\\&c_{11}=\int _{l_3}(x-x_c)^2\mathbf {E}(x,y)\cdot \mathbf {\tau }_3dl,\quad \quad \ \ c_{12}=\int _{l_4}(y-y_c)^2\mathbf {E}(x,y)\cdot \mathbf {\tau }_4dl,\\&c_{13}=\int _{K}\mathbf {E}(x,y)\cdot (1,0)'dxdy,\quad \quad \quad \ c_{14}=\int _{K}\mathbf {E}(x,y)\cdot (0,1)'dxdy,\\&c_{15}=\int _{K}\mathbf {E}(x,y)\cdot (x-x_c,0)'dxdy,\quad \ c_{16}=\int _{K}\mathbf {E}(x,y)\cdot (0,y-y_c)'dxdy,\\&c_{17}=\int _{K}\mathbf {E}(x,y)\cdot (y-y_c,0)'dxdy,\quad \ c_{18}=\int _{K}\mathbf {E}(x,y)\cdot (0,x-x_c)'dxdy,\\&c_{19}=\int _{K}\mathbf {E}(x,y)\cdot \big ((x-x_c)(y-y_c),0\big )'dxdy,\\&c_{20}=\int _{K}\mathbf {E}(x,y)\cdot \big (0,(x-x_c)(y-y_c)\big )'dxdy,\\&c_{21}=\int _{K}\mathbf {E}(x,y)\cdot \big ((x-x_c)^2,0\big ) 'dxdy,\quad \quad \quad \ c_{22}=\int _{K}\mathbf {E}(x,y)\cdot \big (0,(y-y_c)^2\big )'dxdy,\\&c_{23}=\int _{K}\mathbf {E}(x,y)\cdot \big ((x-x_c)^2(y-y_c),0\big )'dxdy,\\&c_{24}=\int _{K}\mathbf {E}(x,y)\cdot \big (0,(x-x_c)(y-y_c)^2\big )'dxdy, \end{aligned}$$

and the basis function \(\varvec{N}_i\) can be expressed as:

$$\begin{aligned}&\varvec{N}_1=\left( \begin{array}{c} \frac{3[5(x-x_c)^2-3h_x^2][6(y-y_c)^2-(y-y_c-h_y)^2][(y-y_c)-h_y]}{32h^3_xh_y^3}\\ 0 \end{array}\right) ,\\&\varvec{N}_2=\left( \begin{array}{c} 0\\ \frac{-3[5(y-y_c)^2-3h_y^2][6(x-x_c)^2-(x-x_c+h_x)^2][(x-x_c)+h_x]}{32h^3_xh^3_y} \end{array}\right) ,\\&\varvec{N}_3=\left( \begin{array}{c} \frac{-3[5(x-x_c)^2-3h_x^2][6(y-y_c)^2-(y-y_c+h_y)^2][(y-y_c)+h_y]}{32h^3_xh_y^3}\\ 0 \end{array}\right) ,\\&\varvec{N}_4=\left( \begin{array}{c} 0\\ \frac{3[5(y-y_c)^2-3h_y^2][6(x-x_c)^2-(x-x_c-h_x)^2][(x-x_c)-h_x]}{32h^3_xh^3_y} \end{array}\right) , \end{aligned}$$
$$\begin{aligned}&\varvec{N}_5=\left( \begin{array}{c} \frac{-3(x-x_c)[6(y-y_c)^2-(y-y_c-h_y)^2][(y-y_c)-h_y]}{8h^3_xh_y^3}\\ 0 \end{array}\right) ,\\&\varvec{N}_6=\left( \begin{array}{c} 0\\ \frac{3(y-y_c)[6(x-x_c)^2-(x-x_c+h_x)^2][(x-x_c)+h_x]}{8h^3_xh^3_y} \end{array}\right) ,\\&\varvec{N}_7=\left( \begin{array}{c} \frac{3(x-x_c)[6(y-y_c)^2-(y-y_c+h_y)^2][(y-y_c)+h_y]}{8h^3_xh_y^3}\\ 0 \end{array}\right) ,\\&\varvec{N}_8=\left( \begin{array}{c} 0\\ \frac{-3(y-y_c)[6(x-x_c)^2-(x-x_c-h_x)^2][(x-x_c)-h_x]}{8h^3_xh^3_y} \end{array}\right) ,\\&\varvec{N}_9=\left( \begin{array}{c} \frac{-15[3(x-x_c)^2-h_x^2][6(y-y_c)^2-(y-y_c-h_y)^2][(y-y_c)-h_y]}{32h^5_xh_y^3}\\ 0 \end{array}\right) ,\\&\varvec{N}_{10}=\left( \begin{array}{c} 0\\ \frac{15[3(y-y_c)^2-h_y^2][6(x-x_c)^2-(x-x_c+h_x)^2][(x-x_c)+h_x]}{32h^3_xh^5_y} \end{array}\right) ,\\&\varvec{N}_{11}=\left( \begin{array}{c} \frac{15[3(x-x_c)^2-h_x^2][6(y-y_c)^2-(y-y_c+h_y)^2][(y-y_c)+h_y]}{32h^5_xh_y^3}\\ 0 \end{array}\right) ,\\&\varvec{N}_{12}=\left( \begin{array}{c} 0\\ \frac{-15[3(y-y_c)^2-h_y^2][6(x-x_c)^2-(x-x_c-h_x)^2][(x-x_c)-h_x]}{32h^3_xh^5_y} \end{array}\right) ,\\ \end{aligned}$$
$$\begin{aligned}&\varvec{N}_{13}=\left( \begin{array}{c} \frac{9[5(x-x_c)^2-3h_x^2][(y-y_c)^2-h_y^2]}{32h^3_xh_y^3}\\ 0 \end{array}\right) , \quad \varvec{N}_{14}=\left( \begin{array}{c} 0\\ \frac{9[5(y-y_c)^2-3h_y^2][(x-x_c)^2-h_x^2]}{32h^3_xh^3_y} \end{array}\right) ,\\&\varvec{N}_{15}=\left( \begin{array}{c} \frac{-9(x-x_c)[(y-y_c)^2-h_y^2]}{8h^3_xh_y^3}\\ 0 \end{array}\right) , \quad \varvec{N}_{16}=\left( \begin{array}{c} 0\\ \frac{-9(y-y_c)[(x-x_c)^2-h_x^2]}{8h^3_xh^3_y} \end{array}\right) ,\\&\varvec{N}_{17}=\left( \begin{array}{c} \frac{45[5(x-x_c)^2-3h_x^2][(y-y_c)^2-h_y^2](y-y_c)}{32h^3_xh_y^5}\\ 0 \end{array}\right) , \quad \varvec{N}_{18}=\left( \begin{array}{c} 0\\ \frac{45[5(y-y_c)^2-3h_y^2][(x-x_c)^2-h_x^2](x-x_c)}{32h^5_xh^3_y} \end{array}\right) ,\\&\varvec{N}_{19}=\left( \begin{array}{c} \frac{-45(x-x_c)[(y-y_c)^2-h_y^2](y-y_c)}{8h^3_xh_y^5}\\ 0 \end{array}\right) , \quad \varvec{N}_{20}=\left( \begin{array}{c} 0\\ \frac{-45(y-y_c)[(x-x_c)^2-h_x^2](x-x_c)}{8h^5_xh^3_y} \end{array}\right) ,\\&\varvec{N}_{21}=\left( \begin{array}{c} \frac{-45[3(x-x_c)^2-h_x^2][(y-y_c)^2-h_y^2]}{32h^5_xh_y^3}\\ 0 \end{array}\right) , \quad \varvec{N}_{22}=\left( \begin{array}{c} 0\\ \frac{-45[3(y-y_c)^2-h_y^2][(x-x_c)^2-h_x^2]}{32h^3_xh^5_y} \end{array}\right) ,\\&\varvec{N}_{23}=\left( \begin{array}{c} \frac{-225[3(x-x_c)^2-h_x^2][(y-y_c)^2-h_y^2](y-y_c)}{32h^5_xh_y^5}\\ 0 \end{array}\right) , \quad \varvec{N}_{24}=\left( \begin{array}{c} 0\\ \frac{-225[3(y-y_c)^2-h_y^2][(x-x_c)^2-h_x^2](x-x_c)}{32h^5_xh^5_y} \end{array}\right) . \end{aligned}$$

For the interpolation \(I_{\hat{K}} H\in Q_{2,2}(\hat{K})\), the degrees of freedom \(d_i\) are:

$$\begin{aligned}&d_1=\int _{\hat{K}}H(\hat{x},\hat{y})d\hat{x}d\hat{y}, \quad \quad d_2=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{x}d\hat{x}d\hat{y}, \quad \ d_3=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{y}d\hat{x}d\hat{y},\\&d_4=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{x}\hat{y}d\hat{x}d\hat{y}, \quad d_5=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{x}^2d\hat{x}d\hat{y}, \quad d_6=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{y}^2d\hat{x}d\hat{y},\\&d_7=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{x}^2\hat{y}d\hat{x}d\hat{y}, \ \ d_8=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{x}\hat{y}^2d\hat{x}d\hat{y}, \ \ d_9=\int _{\hat{K}}H(\hat{x},\hat{y})\hat{x}^2\hat{y}^2d\hat{x}d\hat{y}, \end{aligned}$$

and the basis function \(\hat{\psi }_i\) are:

$$\begin{aligned}&\hat{\psi }_1(\hat{x},\hat{y})=\frac{81}{64}-\frac{135}{64}\hat{x}^2 -\frac{135}{64}\hat{y}^2+\frac{225}{64}\hat{x}^2\hat{y}^2,\\&\hat{\psi }_2(\hat{x},\hat{y})=\frac{27}{16}\hat{x}-\frac{45}{16}\hat{x}\hat{y}^2, \ \ \hat{\psi }_3(\hat{x},\hat{y})=\frac{27}{16}\hat{y}-\frac{45}{16}\hat{x}^2\hat{y}, \ \ \hat{\psi }_4(\hat{x},\hat{y})=\frac{9}{4}\hat{x}\hat{y},\\&\hat{\psi }_5(\hat{x},\hat{y})=-\,\frac{135}{64}+\frac{405}{64}\hat{x}^2+\frac{225}{64}\hat{y}^2-\frac{675}{64}\hat{x}^2\hat{y}^2,\\&\hat{\psi }_6(\hat{x},\hat{y})=-\,\frac{135}{64}+\frac{225}{64}\hat{x}^2+\frac{405}{64}\hat{y}^2-\frac{675}{64}\hat{x}^2\hat{y}^2,\\&\hat{\psi }_7(\hat{x},\hat{y})=-\,\frac{45}{16}\hat{y}+\frac{135}{16}\hat{x}^2\hat{y}, \ \ \hat{\psi }_8(\hat{x},\hat{y})=-\,\frac{45}{16}\hat{x}+\frac{135}{16}\hat{x}\hat{y}^2,\\&\hat{\psi }_9(\hat{x},\hat{y})=\frac{225}{64}-\frac{675}{64}\hat{x}^2-\frac{675}{64}\hat{y}^2+\frac{2025}{64}\hat{x}^2\hat{y}^2. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, M., Li, J., Wang, P. et al. Superconvergence Analysis of High-Order Rectangular Edge Elements for Time-Harmonic Maxwell’s Equations. J Sci Comput 75, 510–535 (2018). https://doi.org/10.1007/s10915-017-0544-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0544-2

Keywords

Search

Navigation