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Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with \({\textit{EQ}}_1^{rot}\) Nonconforming Finite Element

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Abstract

Nonlinear parabolic equation is studied with a linearized Galerkin finite element method. First of all, a time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, a rigorous analysis for the regularity of the time-discrete system is presented based on the proof of the temporal error skillfully. On the other hand, the spatial error is derived \(\tau \)-independently with the above achievements. Then, the superclose result of order \(O(h^2+\tau ^2)\) in broken \(H^1\)-norm is deduced without any restriction of \(\tau \). The two typical characters of the \({\textit{EQ}}_1^{rot}\) nonconforming FE (see Lemma 1 below) play an important role in the procedure of proof. At last, numerical results are provided in the last section to confirm the theoretical analysis. Here, h is the subdivision parameter, and \(\tau \), the time step.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11271340).

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  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, China

    Dongyang Shi, Junjun Wang & Fengna Yan

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  1. Dongyang Shi
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  2. Junjun Wang
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  3. Fengna Yan
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Correspondence to Dongyang Shi.

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Shi, D., Wang, J. & Yan, F. Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with \({\textit{EQ}}_1^{rot}\) Nonconforming Finite Element. J Sci Comput 70, 85–111 (2017). https://doi.org/10.1007/s10915-016-0243-4

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  • DOI: https://doi.org/10.1007/s10915-016-0243-4

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