# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022124
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## Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

 1 Beijing Computational Science Research Center, Beijing 100193, China 2 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

* Corresponding author: Zhimin Zhang

Received  February 2022 Revised  May 2022 Early access June 2022

Fund Project: This work is supported in part by the National Natural Science Foundation of China via grants NSFC 11871092, 12131005 and NSAF U1930402

We analyze superconvergence property of the lowest order curl-curl conforming finite element method based on polynomial preserving recovery (PPR) for the two-dimensional quad-curl problem on triangular meshes. We observe that the linear interpolation of $\nabla \times \boldsymbol u_h$ ($\boldsymbol u_h$ is the numerical solution) can be written as a linear combination of solutions of two discrete Poisson equations obtained by the usual linear finite element method. Therefore, the superconvergence analysis of the quad-curl problem can be attributed to the analysis of the Poisson equation. Then, with the help of the existing superconvergence results for the Poisson equation, we prove that recovered $\nabla \times \nabla \times \boldsymbol u_h$ (by applying PPR to $\nabla \times \boldsymbol u_h$) is superconvergent to $\nabla \times \nabla \times \boldsymbol u$. Based on this superconvergent result, we derive an asymptotically exact a posteriori error estimator. Numerical tests are provided to demonstrate effectiveness of the proposed method and confirm our theoretical findings.

Citation: Baiju Zhang, Zhimin Zhang. Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022124
##### References:
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show all references

##### References:
 [1] S. C. Brenner, J. Sun and L.-Y. Sung, Hodge decomposition methods for a quad-curl problem on planar domains, J. Sci. Comput., 73 (2017), 495-513.  doi: 10.1007/s10915-017-0449-0. [2] F. Cakoni and H. Haddar, A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media, Inverse Probl. Imaging, 1 (2007), 443-456.  doi: 10.3934/ipi.2007.1.443. [3] G. Chen, J. Cui and L. Xu, A hybridizable discontinuous Galerkin method for the quad-curl problem, J. Sci. Comput., 87 (2021), Paper No. 16, 23 pp. doi: 10.1007/s10915-021-01420-3. [4] L. Chen, Ifem: An innovative finite element methods package in matlab, Preprint, University of Maryland. [5] L. Chen and J. Xu, A posteriori error estimator by post-processing, in Adaptive Computations: Theory and Algorithms (eds. T. Tang and J. Xu), no. 6 in Mathematics Monograph Series, Science Press, 2007, chapter 2, 34-67. [6] W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054. [7] H. Guo, Z. Zhang, R. Zhao and Q. Zou, Polynomial preserving recovery on boundary, J. Comput. Appl. Math., 307 (2016), 119-133.  doi: 10.1016/j.cam.2016.03.003. [8] Q. Hong, J. Hu, S. Shu and J. Xu, A discontinuous Galerkin method for the fourth-order curl problem, J. Comput. Math., 30 (2012), 565-578.  doi: 10.4208/jcm.1206-m3572. [9] K. Hu, Q. Zhang and Z. Zhang, Simple curl-curl-conforming finite elements in two dimensions, SIAM J. Sci. Comput., 42 (2020), A3859–3877. (arXiv: 2004.12507v2, 2021) doi: 10.1137/20M1333390. [10] P. Monk and J. Sun, Finite element methods for Maxwell's transmission eigenvalues, SIAM J. Sci. Comput., 34 (2012), B247–B264. doi: 10.1137/110839990. [11] A. Naga and Z. Zhang, The polynomial-preserving recovery for higher order finite element methods in 2D and 3D, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 769-798.  doi: 10.3934/dcdsb.2005.5.769. [12] A. Naga and Z. Zhang, A posteriori error estimates based on the polynomial preserving recovery, SIAM J. Numer. Anal., 42 (2004), 1780-1800.  doi: 10.1137/S0036142903413002. [13] J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874.  doi: 10.1137/100785478. [14] J. Sun, A mixed FEM for the quad-curl eigenvalue problem, Numer. Math., 132 (2016), 185-200.  doi: 10.1007/s00211-015-0708-7. [15] Z. Sun, F. Gao, C. Wang and Y. Zhang, A quadratic $C^0$ interior penalty method for the quad-curl problem, Math. Model. Anal., 25 (2020), 208-225.  doi: 10.3846/mma.2020.9796. [16] R. Verführt, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner, Stuttgart., 1996. [17] L. Wang, Q. Zhang, J. Sun and Z. Zhang, A priori and a posteriori error estimates for the quad-curl eigenvalue problem, ESAIM Math. Model. Numer. Anal., 56 (2022), 1027-1051.  doi: 10.1051/m2an/2022027. [18] H. Wu and Z. Zhang, Can we have superconvergent gradient recovery under adaptive meshes?, SIAM J. Numer. Anal., 45 (2007), 1701-1722.  doi: 10.1137/060661430. [19] J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp., 73 (2004), 1139-1152.  doi: 10.1090/S0025-5718-03-01600-4. [20] Q. Zhang, L. Wang and Z. Zhang, $H$(curl$^2)$-conforming finite elements in 2 dimensions and applications to the quad-curl problem, SIAM J. Sci. Comput., 41 (2019), A1527–A1547. doi: 10.1137/18M1199988. [21] Z. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM J. Sci. Comput., 26 (2005), 1192-1213.  doi: 10.1137/S1064827503402837. [22] B. Zheng, Q. Hu and J. Xu, A nonconforming finite element method for fourth order curl equations in $\mathbb R^3$, Math. Comp., 80 (2011), 1871-1886.  doi: 10.1090/S0025-5718-2011-02480-4. [23] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg., 24 (1987), 337-357.  doi: 10.1002/nme.1620240206. [24] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702. [25] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. II. Error estimates and adaptivity, Internat. J. Numer. Methods Engrg., 33 (1992), 1365-1382.  doi: 10.1002/nme.1620330703.
Notation in the patch $\Omega_e$
Four types of uniform meshes (a) Regular pattern; (b) Chevron pattern; (c) Criss-cross pattern; (d)Union-Jack pattern
The errors of $\left\|(\nabla \times)^{2}\left(\boldsymbol{u}-\boldsymbol{u}_{h}\right)\right\|_{0} / \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0}$ and $\left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{0} / \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0}$ for (a) Regular pattern; (b) Chevron pattern; (c) Criss-cross pattern; (d) Union-Jack pattern (the lowest order case)
The errors of $\left\|(\nabla \times)^{2}\left(\boldsymbol{u}-\boldsymbol{u}_{h}\right)\right\|_{0} / \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0}$, $\left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{0} / \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0}$ and $\left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{\infty, N}/\left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{\infty, N}$ for (a) Regular pattern; (b) Chevron pattern; (c) Criss-cross pattern; (d) Union-Jack pattern (the higher order case)
The initial mesh (left) and the adaptively refined mesh (right) after 20 adaptive iterations (the lowest order case)
Numerical result for Example 2: (a) Numerical errors; (b) Effective index (c) $\left\|p-p_{h}\right\|_{0}$ and $\left\|p-\bar{p}_{h}\right\|_{0}$ (the lowest order case)
The initial mesh (left) and the adaptively refined mesh (right) after 20 adaptive (the higher order case) iterations
Numerical result for Example 2: (a) Numerical errors; (b) Effective index (c) $\left\|p-p_{h}\right\|_{0}$ and $\left\|p-\bar{p}_{h}\right\|_{0}$ (the higher order case)
Example 1: Numerical results by the lowest-order curl-curl conforming element on different mesh patterns
 Mesh $N$ $\frac{ \left\|\boldsymbol{u}-\boldsymbol{u}_{h}\right\|_{0} }{\|\boldsymbol{u}\|_{0}}$ rate $\frac{ \left\|(\nabla \times)^{2}\left(\boldsymbol{u}-\boldsymbol{u}_{h}\right)\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\frac{ \left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\left\|p-p_{h}\right\|_{0}$ 81 5.585e-01 6.557e-01 8.166e-01 3.722e-07 289 2.615e-01 0.60 3.883e-01 0.41 3.932e-01 0.57 1.040e-10 Regular 1089 1.279e-01 0.54 2.041e-01 0.48 1.297e-01 0.84 8.063e-11 4225 6.352e-02 0.52 1.034e-01 0.50 3.544e-02 0.96 5.228e-10 16641 3.170e-02 0.51 5.188e-02 0.50 9.108e-03 0.99 4.948e-09 66049 1.585e-02 0.50 2.596e-02 0.50 2.298e-03 1.00 5.293e-08 81 4.980e-01 6.597e-01 8.234e-01 5.743e-05 289 2.469e-01 0.55 3.843e-01 0.42 3.390e-01 0.70 6.862e-08 Chevron 1089 1.258e-01 0.51 2.032e-01 0.48 1.044e-01 0.89 8.795e-11 4225 6.325e-02 0.51 1.033e-01 0.50 2.786e-02 0.97 9.234e-10 16641 3.167e-02 0.50 5.186e-02 0.50 7.103e-03 1.00 7.202e-09 66049 1.584e-02 0.50 2.596e-02 0.50 1.787e-03 1.00 6.719e-08 145 3.586e-01 4.975e-01 6.558e-01 3.776e-08 545 1.777e-01 0.53 2.556e-01 0.50 2.232e-01 0.81 3.451e-11 Criss-cross 2113 9.066e-02 0.50 1.286e-01 0.51 6.367e-02 0.93 3.579e-10 8321 4.560e-02 0.50 6.442e-02 0.50 1.668e-02 0.98 2.953e-09 33025 2.284e-02 0.50 3.222e-02 0.50 4.246e-03 0.99 3.739e-08 131585 1.142e-02 0.50 1.611e-02 0.50 1.069e-03 1.00 2.450e-07 81 4.148e-01 5.476e-01 8.964e-01 1.276e-11 289 2.443e-01 0.42 3.695e-01 0.31 4.428e-01 0.55 3.624e-10 Union-Jack 1089 1.252e-01 0.50 1.902e-01 0.50 1.431e-01 0.85 8.954e-11 4225 6.316e-02 0.50 9.580e-02 0.51 3.869e-02 0.97 8.687e-10 16641 3.166e-02 0.50 4.799e-02 0.50 9.899e-03 0.99 7.013e-09 66049 1.584e-02 0.50 2.401e-02 0.50 2.493e-03 1.00 7.267e-08
 Mesh $N$ $\frac{ \left\|\boldsymbol{u}-\boldsymbol{u}_{h}\right\|_{0} }{\|\boldsymbol{u}\|_{0}}$ rate $\frac{ \left\|(\nabla \times)^{2}\left(\boldsymbol{u}-\boldsymbol{u}_{h}\right)\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\frac{ \left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\left\|p-p_{h}\right\|_{0}$ 81 5.585e-01 6.557e-01 8.166e-01 3.722e-07 289 2.615e-01 0.60 3.883e-01 0.41 3.932e-01 0.57 1.040e-10 Regular 1089 1.279e-01 0.54 2.041e-01 0.48 1.297e-01 0.84 8.063e-11 4225 6.352e-02 0.52 1.034e-01 0.50 3.544e-02 0.96 5.228e-10 16641 3.170e-02 0.51 5.188e-02 0.50 9.108e-03 0.99 4.948e-09 66049 1.585e-02 0.50 2.596e-02 0.50 2.298e-03 1.00 5.293e-08 81 4.980e-01 6.597e-01 8.234e-01 5.743e-05 289 2.469e-01 0.55 3.843e-01 0.42 3.390e-01 0.70 6.862e-08 Chevron 1089 1.258e-01 0.51 2.032e-01 0.48 1.044e-01 0.89 8.795e-11 4225 6.325e-02 0.51 1.033e-01 0.50 2.786e-02 0.97 9.234e-10 16641 3.167e-02 0.50 5.186e-02 0.50 7.103e-03 1.00 7.202e-09 66049 1.584e-02 0.50 2.596e-02 0.50 1.787e-03 1.00 6.719e-08 145 3.586e-01 4.975e-01 6.558e-01 3.776e-08 545 1.777e-01 0.53 2.556e-01 0.50 2.232e-01 0.81 3.451e-11 Criss-cross 2113 9.066e-02 0.50 1.286e-01 0.51 6.367e-02 0.93 3.579e-10 8321 4.560e-02 0.50 6.442e-02 0.50 1.668e-02 0.98 2.953e-09 33025 2.284e-02 0.50 3.222e-02 0.50 4.246e-03 0.99 3.739e-08 131585 1.142e-02 0.50 1.611e-02 0.50 1.069e-03 1.00 2.450e-07 81 4.148e-01 5.476e-01 8.964e-01 1.276e-11 289 2.443e-01 0.42 3.695e-01 0.31 4.428e-01 0.55 3.624e-10 Union-Jack 1089 1.252e-01 0.50 1.902e-01 0.50 1.431e-01 0.85 8.954e-11 4225 6.316e-02 0.50 9.580e-02 0.51 3.869e-02 0.97 8.687e-10 16641 3.166e-02 0.50 4.799e-02 0.50 9.899e-03 0.99 7.013e-09 66049 1.584e-02 0.50 2.401e-02 0.50 2.493e-03 1.00 7.267e-08
Example 1: Numerical results by the higher order curl-curl conforming element on different mesh patterns
 Mesh $N$ $\frac{ \left\|\boldsymbol{u}-\boldsymbol{u}_{h}\right\|_{0} }{\|\boldsymbol{u}\|_{0}}$ rate $\frac{ \left\|(\nabla \times)^{2}\left(\boldsymbol{u}-\boldsymbol{u}_{h}\right)\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\frac{ \left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\frac{\left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{\infty, N}}{\left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{\infty, N}}$ rate $\left\|p-p_{h}\right\|_{0}$ 193 2.172e-01 2.330e-01 6.035e-01 5.858e-01 1.260e-06 705 6.330e-02 0.95 7.134e-02 0.91 1.860e-01 0.91 1.620e-01 0.99 2.586e-10 Regular 2689 1.642e-02 1.01 1.980e-02 0.96 2.021e-02 1.66 1.617e-02 1.72 2.655e-10 10497 4.135e-03 1.01 5.137e-03 0.99 1.612e-03 1.86 1.132e-03 1.95 3.119e-09 41473 1.035e-03 1.01 1.298e-03 1.00 1.294e-04 1.84 7.298e-05 2.00 5.176e-08 164865 2.587e-04 1.00 3.253e-04 1.00 1.107e-05 1.78 4.594e-06 2.00 8.128e-07 193 2.341e-01 2.372e-01 6.796e-01 6.794e-01 2.712e-06 705 6.517e-02 0.99 7.106e-02 0.93 1.743e-01 1.05 1.302e-01 1.28 7.263e-10 Chevron 2689 1.647e-02 1.03 1.903e-02 0.98 1.828e-02 1.68 1.948e-02 1.42 1.381e-10 10497 4.113e-03 1.02 4.855e-03 1.00 1.681e-03 1.75 2.543e-03 1.49 2.192e-09 41473 1.027e-03 1.01 1.220e-03 1.01 1.861e-04 1.60 3.182e-04 1.51 2.705e-08 164865 2.564e-04 1.01 3.054e-04 1.00 2.376e-05 1.49 3.974e-05 1.51 4.180e-07 353 1.229e-01 1.159e-01 3.876e-01 4.581e-01 7.219e-09 1345 3.307e-02 0.98 3.218e-02 0.96 5.325e-02 1.48 5.674e-02 1.56 3.201e-11 Criss-cross 5249 8.442e-03 1.00 8.365e-03 0.99 4.282e-03 1.85 4.302e-03 1.89 3.095e-10 20737 2.122e-03 1.01 2.114e-03 1.00 3.315e-04 1.86 2.826e-04 1.98 3.293e-09 82433 5.312e-04 1.00 5.301e-04 1.00 3.029e-05 1.73 2.928e-05 1.64 3.004e-08 328705 1.329e-04 1.00 1.326e-04 1.00 3.320e-06 1.60 3.356e-06 1.57 2.800e-07 193 2.598e-01 2.609e-01 6.164e-01 6.926e-01 2.712e-06 705 6.387e-02 1.08 6.818e-02 1.04 1.577e-01 1.05 1.336e-01 1.27 7.470e-10 Union-Jack 2689 1.610e-02 1.03 1.854e-02 0.97 1.568e-02 1.72 1.327e-02 1.73 1.639e-10 10497 4.020e-03 1.02 4.767e-03 1.00 1.362e-03 1.79 1.370e-03 1.67 1.687e-09 41473 1.003e-03 1.01 1.201e-03 1.00 1.393e-04 1.66 1.580e-04 1.57 2.933e-08 164865 2.505e-04 1.01 3.008e-04 1.00 1.603e-05 1.57 1.918e-05 1.53 3.935e-07
 Mesh $N$ $\frac{ \left\|\boldsymbol{u}-\boldsymbol{u}_{h}\right\|_{0} }{\|\boldsymbol{u}\|_{0}}$ rate $\frac{ \left\|(\nabla \times)^{2}\left(\boldsymbol{u}-\boldsymbol{u}_{h}\right)\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\frac{ \left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{0} }{ \left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{0} }$ rate $\frac{\left\|(\nabla \times)^{2} \boldsymbol{u}-G_{h}^{\perp} I_{h} \nabla \times \boldsymbol{u}_{h}\right\|_{\infty, N}}{\left\|(\nabla \times)^{2} \boldsymbol{u}\right\|_{\infty, N}}$ rate $\left\|p-p_{h}\right\|_{0}$ 193 2.172e-01 2.330e-01 6.035e-01 5.858e-01 1.260e-06 705 6.330e-02 0.95 7.134e-02 0.91 1.860e-01 0.91 1.620e-01 0.99 2.586e-10 Regular 2689 1.642e-02 1.01 1.980e-02 0.96 2.021e-02 1.66 1.617e-02 1.72 2.655e-10 10497 4.135e-03 1.01 5.137e-03 0.99 1.612e-03 1.86 1.132e-03 1.95 3.119e-09 41473 1.035e-03 1.01 1.298e-03 1.00 1.294e-04 1.84 7.298e-05 2.00 5.176e-08 164865 2.587e-04 1.00 3.253e-04 1.00 1.107e-05 1.78 4.594e-06 2.00 8.128e-07 193 2.341e-01 2.372e-01 6.796e-01 6.794e-01 2.712e-06 705 6.517e-02 0.99 7.106e-02 0.93 1.743e-01 1.05 1.302e-01 1.28 7.263e-10 Chevron 2689 1.647e-02 1.03 1.903e-02 0.98 1.828e-02 1.68 1.948e-02 1.42 1.381e-10 10497 4.113e-03 1.02 4.855e-03 1.00 1.681e-03 1.75 2.543e-03 1.49 2.192e-09 41473 1.027e-03 1.01 1.220e-03 1.01 1.861e-04 1.60 3.182e-04 1.51 2.705e-08 164865 2.564e-04 1.01 3.054e-04 1.00 2.376e-05 1.49 3.974e-05 1.51 4.180e-07 353 1.229e-01 1.159e-01 3.876e-01 4.581e-01 7.219e-09 1345 3.307e-02 0.98 3.218e-02 0.96 5.325e-02 1.48 5.674e-02 1.56 3.201e-11 Criss-cross 5249 8.442e-03 1.00 8.365e-03 0.99 4.282e-03 1.85 4.302e-03 1.89 3.095e-10 20737 2.122e-03 1.01 2.114e-03 1.00 3.315e-04 1.86 2.826e-04 1.98 3.293e-09 82433 5.312e-04 1.00 5.301e-04 1.00 3.029e-05 1.73 2.928e-05 1.64 3.004e-08 328705 1.329e-04 1.00 1.326e-04 1.00 3.320e-06 1.60 3.356e-06 1.57 2.800e-07 193 2.598e-01 2.609e-01 6.164e-01 6.926e-01 2.712e-06 705 6.387e-02 1.08 6.818e-02 1.04 1.577e-01 1.05 1.336e-01 1.27 7.470e-10 Union-Jack 2689 1.610e-02 1.03 1.854e-02 0.97 1.568e-02 1.72 1.327e-02 1.73 1.639e-10 10497 4.020e-03 1.02 4.767e-03 1.00 1.362e-03 1.79 1.370e-03 1.67 1.687e-09 41473 1.003e-03 1.01 1.201e-03 1.00 1.393e-04 1.66 1.580e-04 1.57 2.933e-08 164865 2.505e-04 1.01 3.008e-04 1.00 1.603e-05 1.57 1.918e-05 1.53 3.935e-07
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