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A High-Order Two-Grid Difference Method for Nonlinear Time-Fractional Biharmonic Problems and Its Unconditional \(\alpha \)-Robust Error Estimates

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Abstract

In this work, we propose and analyze a high-order mapping operator between two grids to construct a high-order two-grid difference algorithm for nonlinear partial differential equations. This algorithm is then applied to solve a nonlinear time-fractional biharmonic equation for illustration, in which the cut-off technique and the auxiliary scheme approach are used to reduce the requirement on the nonlinear term to the local Lipschitz continuous condition, and the energy estimates are performed to avoid the usage of the inverse estimates and thus the time-step conditions. To treat the fractional operator, the Alikhanov’s scheme on the graded mesh is applied to deal with the weak singularity at initial time, while the compact difference method based on the order reduction is employed for high-order spatial discretization. The above methods and improvements as well as the properties of the high-order mapping operator are integrated with the analysis of the two-grid method to prove the unique solvability and the unconditionally robust error estimates of the proposed schemes under different norms. The developed techniques are further extended to discretize and analyze the two-grid method of two-dimensional problems. Numerical examples are provided to verify the effectiveness and efficiency of the two-grid algorithms.

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Funding

This work is supported in part by funds from the National Natural Science Foundation of China Nos. 11971482 and 12131014, the Fundamental Research Funds for the Central Universities No. 202264006, the OUC Scientific Research Program for Young Talented Professionals.

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Authors and Affiliations

  1. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China

    Hongfei Fu & Bingyin Zhang

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Xiangcheng Zheng

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  1. Hongfei Fu
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  2. Bingyin Zhang
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  3. Xiangcheng Zheng
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Correspondence to Xiangcheng Zheng.

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Fu, H., Zhang, B. & Zheng, X. A High-Order Two-Grid Difference Method for Nonlinear Time-Fractional Biharmonic Problems and Its Unconditional \(\alpha \)-Robust Error Estimates. J Sci Comput 96, 54 (2023). https://doi.org/10.1007/s10915-023-02282-7

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  • DOI: https://doi.org/10.1007/s10915-023-02282-7

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