Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

Baiju Zhang; Zhimin Zhang

doi:10.3934/dcdsb.2022124

Article Contents

2023, Volume 28, Issue 2: 1323-1341. Doi: 10.3934/dcdsb.2022124

This issue Previous Article Periodic solutions in distribution of stochastic lattice differential equations Next Article Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential

Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

Baiju Zhang^1, and
Zhimin Zhang^1,2, ,

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

^* Corresponding author: Zhimin Zhang
^* Corresponding author: Zhimin Zhang

Received: January 31, 2022

Revised: April 30, 2022

Early access: June 2022

Published: February 2023

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 65N50, 65N30, 65N15, 41A25.

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Figure 1. Notation in the patch $ \Omega_e $

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Figure 2. Four types of uniform meshes (a) Regular pattern; (b) Chevron pattern; (c) Criss-cross pattern; (d)Union-Jack pattern

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Figure 3. 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)

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Figure 4. 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)

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Figure 5. The initial mesh (left) and the adaptively refined mesh (right) after 20 adaptive iterations (the lowest order case)

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Figure 6. 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)

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Figure 7. The initial mesh (left) and the adaptively refined mesh (right) after 20 adaptive (the higher order case) iterations

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Figure 8. 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)

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Table 1. 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

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Table 2. 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

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