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Convergence and Superconvergence Analysis of a Nonconforming Finite Element Variable-Time-Step BDF2 Implicit Scheme for Linear Reaction-Diffusion Equations

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Abstract

In this paper, an effective fully-discrete implicit scheme for solving linear reaction-diffusion equations is constructed by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the nonconforming finite element methods in space. By introducing a modified energy projection operator, a discrete Laplace operator, the discrete orthogonal convolution kernels, we obtain the optimal and sharp error estimates of order \(O(h^2+\tau ^2)\) in \(L^2\)-norm and \(O(h+\tau ^2)\) in \(H^1\)-norm under a mild restriction \(0<r_k< r_{\max }\approx 4.8645\) for the ratio of adjacent time steps \(r_k\). Furthermore, with the help of a modified discrete Grönwall inequality and the combination technique of interpolation and projection operators, we achieved the superclose result between the interpolation function \(I_hu\) and finite element solution \(u_h\) in \(H^1\)-norm of order \(O(h^2+\tau ^2)\), which together with the interpolation postprocessing operator \(\Pi _{2h}\) leads to the global superconvergence result about \(u-\Pi _{2h}u_h\) in \(H^1\)-norm of order \(O(h^2+\tau ^2)\). Finally, numerical tests are provided to verify the theoretical analysis.

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Funding

This work is supported in part by the National Natural Science Foundation of China under grants Nos. 12171376, the Fundamental Research Funds for the Central Universities (No. 2042021kf0050) and WHU-2022-SYJS-0002. The numerical simulations in this work have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Lifang Pei & Chaofeng Zhang

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Yifan Wei

  3. School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Jiwei Zhang

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  1. Lifang Pei
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Correspondence to Jiwei Zhang.

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Pei, L., Wei, Y., Zhang, C. et al. Convergence and Superconvergence Analysis of a Nonconforming Finite Element Variable-Time-Step BDF2 Implicit Scheme for Linear Reaction-Diffusion Equations. J Sci Comput 98, 67 (2024). https://doi.org/10.1007/s10915-024-02456-x

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