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Convergence Analysis of the Fully Discrete Hybridizable Discontinuous Galerkin Method for the Allen–Cahn Equation Based on the Invariant Energy Quadratization Approach

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Abstract

In this paper, we develop a fully discrete scheme to solve the well-known Allen–Cahn equation, where space is discretized by the hybridizable discontinuous Galerkin method, and the time discretization is based on the newly developed Invariant Energy Quadratization approach. At each time step, the scheme results in a linear and uniquely solvable algebraic system. The scheme is shown to be unconditionally energy stable, and the optimal error estimates are rigorously established. Some numerical examples are presented to illustrate the temporal and spatial order of accuracy.

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Acknowledgements

The authors would like to thank the anonymous referees for the valuable comments and constructive suggestions to improve this paper.

Funding

The work of J. Wang was supported by the National Natural Science Foundation of China (Grant No. 11801171). The work of K. Pan was supported by the National Natural Science Foundation of China (Grant No. 41874086) and Science Challenge Project (Grant No. TZ2016002). The work of X. Yang was partially supported by National Science Foundation of USA with grant number DMS-2012490.

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Authors and Affiliations

  1. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, China

    Jiangxing Wang

  2. HNP-LAMA, School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Kejia Pan

  3. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

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  1. Jiangxing Wang
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  2. Kejia Pan
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  3. Xiaofeng Yang
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Correspondence to Xiaofeng Yang.

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Wang, J., Pan, K. & Yang, X. Convergence Analysis of the Fully Discrete Hybridizable Discontinuous Galerkin Method for the Allen–Cahn Equation Based on the Invariant Energy Quadratization Approach. J Sci Comput 91, 49 (2022). https://doi.org/10.1007/s10915-022-01822-x

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