Curl recovery for the lowest order rectangular edge element
Introduction
Finite element recovery methods are post-processing methods, which can reconstruct new numerical approximations to achieve better results with finite element solutions. Firstly, the gradient recovery can provide high accuracy approximate gradients, and, secondly, it can offer an asymptotically exact posteriori error estimators. There are a variety of researches and applications about gradient recovery methods. However, most of them apply averaging methods [1], [13] and global or local projections[2], [9]. The real effective breakthrough in the postprocessing technique is made by Zienkiewicz and Zhu, through presenting the superconvergent patch recovery (SPR) technique [24], [25], [26], [27]. The SPR technique is based on a least-squares fitting of derivatives at the known superconvergent points. It generates superconvergent stress (gradient) values in the whole domain [14], [28], [31], [32]. Subsequently, many improved SPR techniques have emerged [4], [17]. Zhang and Naga developed the polynomial preserving recovery (PPR) method [20], [29], [30]. Huang and Yi [12] proposed a novel gradient recovery method, i.e., superconvergent cluster recovery (SCR) method.
Unfortunately, these recovery methods are the Crouzeix–Raviart element [8] or C0 finite element methods [5], [6], [7]. According to our best knowledge, there are few works on curl recovery for the edge element. Recently Wang and her collaborators [21] employed hierarchical basis constructing a global recovery superconvergent result by a least squares method for Maxwell’s equations.
In present work, taking the 2-D time-harmonic Maxwell’s equation as an example, we propose and analyze two curl recovery methods for the lowest order rectangular edge element, and numerical examples show that the recovery methods can obtain superconvergent curl approximations. One approach is to define a local patch of element which has four element centers, which fits a Q1,1 polynomial. Another approach is to define a local patch of element which has six edge centers, which fits Q1,2 × Q2,1 polynomials. The raised approach is bounded, and proved to preserve polynomial of Q1,2 × Q2,1. The curl recovery approaches for the edge element are easy to implement and independent of the problem, same as SPR [26] and PPR [30] on C0 finite element methods. These results can be applied to many researches [22], [23].
The rest of the article is arranged as follows: In Sect. 2, we introduce preliminaries on time-harmonic Maxwell’s equations. In Sect. 3, we present two approaches on curl recovery and analyze the related properties. Sect. 4 presents the proof of our superconvergent results. In Sect. 5, numerical examples are presented to support our theoretical findings. Finally, some conclusions are drawn in Sect.
Section snippets
Notation and the lowest order rectangular edge element for time-harmonic Maxwell’s equations
In this section, we introduce some notation and the lowest order rectangular edge element for 2-D time-harmonic Maxwell’s equations.
Let Ω be a bounded domain with Lipschitz continuous boundary in a two-dimensional space. In the whole article, we will use the standard notations for the classical Sobolev spaces, which is the same as in [3]. For a subdomain D, Wk,p(D) denotes the classical Sobolev space with norm ‖ · ‖k,p,D, and the seminorm | · |k,p,D. When . In the present
Curl Recovery Method for the lowest order rectangular edge element
In this section, we will give two curl recovery techniques which is based on Zienkiewicz-Zhu’s Superconvergence Patch Recovery (SPR) [24], [25], [26], [27] and Zhang’s Polynomial Preserving Recovery (PPR) [20], [30] for the lowest order rectangular edge element, respectively.
Superconvergence analysis
In this section, we first give some general results for both SPR and PPR, and then we utilize these results and the supercloseness between the curl of the finite element solution and the curl of the interpolation to prove the superconvergence property of our curl recovery operators.
Numerical results
To confirm our theoretical analysis, in this section, some numerical examples are given to show that the theoretical results can be realized in numerical experiments. For convenience, we choose the physical domain with analytical solutionsand the parameter and piecewise constant .
From Table 1, Table 3 and Table 5, we see that the superconvergence order are O(h2) for and on uniform rectangular
Conclusion
In this paper, we present the recovered curl results of the edge element for 2−D time−harmonic Maxwell’s equation. We believe that our superconvergence conclusions can be extended to 3−D problems. The recovered curl results for triangular element will be studied in the future.
Acknowledgement
This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (11901197, 41974133 and 11671157).
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