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Error reduction of the adaptive conforming and nonconforming finite element methods with red–green refinement

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Abstract

In this paper, we analyze the convergence of the adaptive conforming and nonconforming \(P_1\) finite element methods with red–green refinement based on standard Dörfler marking strategy. Since the mesh after refining is not nested into the one before, the usual Galerkin-orthogonality or quasi-orthogonality for newest vertex bisection does not hold for this case. To overcome such a difficulty, we develop some new quasi-orthogonality instead under certain condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive methods by establishing the reduction of some total errors. To weaken the condition on the initial mesh, we propose a modified red–green refinement and prove the convergence of the associated adaptive methods under a much weaker condition on the initial mesh (Condition B). Furthermore, we also develop an initial mesh generator which guarantee that all the interior triangles are equilateral triangles (satisfy Condition A) and the other triangles containing at least one vertex on the boundary satisfy Condition B.

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Notes

  1. We give up the symbol \(\nabla _h\) which was used in the previous literature (\(h\) seems corresponding to the current mesh \(\mathcal{T }_h\)). Because, for non-nested red–green refinement, there will be some triangles in our proofs which neither belongs to the current mesh \(\mathcal{T }_h\) nor the previous mesh \(\mathcal{T }_H\). This means the meaning of \(\nabla _h u_H\) will be puzzling. For example, see the four triangles in the parallelogram \(\overline{A_1A_2A_4A_6}\) in Fig.  4. Therefore, herein we use another symbol \(\tilde{\nabla }\) to stand for piecewise gradient.

  2. Since two level meshes are not nested. Some quantities like \(\tilde{\nabla } v|_T\) or \(\tilde{\nabla } v|_E\) in the proof of Lemmas and 5.3 may be piecewise constant on some triangle \(T\) or edge \(E\). Hence we define the edge jump for general piecewise linear functions not merely on nonconforming \(P_1\) functions. However there is no confusion since the edge jump is well-defined.

  3. The “PLTMG” package after the version 8.0 has changed data structures to use the longest-edge bisection while the previous versions is based on the red–green refinement.

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Acknowledgments

The authors are very thankful to Professor Qiang Du who made many valuable discussions in Algorithm 5, and to the Department of Mathematics, Pennsylvania State University for the hospitality during the second-named author’s visit, and also to the anonymous referees who made many valuable comments and corrections of the English and typesetting mistakes.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Zhongci Shi & Xuying Zhao

  2. Beijing Computational Science Research Center, Beijing, 100084, People’s Republic of China

    Xuying Zhao

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  1. Zhongci Shi
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  2. Xuying Zhao
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Correspondence to Xuying Zhao.

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Supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS. The second-named author was supported by the National Nature Science Foundation of China under Grant 11201462, and China Postdoctoral Science Foundation under Grant 20110490279.

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Shi, Z., Zhao, X. Error reduction of the adaptive conforming and nonconforming finite element methods with red–green refinement. Numer. Math. 123, 553–584 (2013). https://doi.org/10.1007/s00211-012-0495-3

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