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ADI Galerkin finite element scheme for the two-dimensional semilinear partial intergro-differential equation with a weakly singular kernel

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Abstract

In the work, the two-dimensional semilinear partial intergro-differential equations with a weakly singular kernel is considered. We construct a fully-discrete scheme via investigating a backward Euler method and first-order convolution quadrature rule for the temporal discretization, in combination with finite elements for the spatial discretization. Then, an alternating direction implicit algorithm is employed to reduce computing costs, and then we linearize the semi-linear term. After that, we utilize the energy method to derive the unconditional stability and convergence in \(L^2\) norm with convergence orders \(O(k+h^{r+1})\), where k and h are parameters for time and space step sizes, respectively, and r is the degree of continuous piecewise polynomial. Numerical experiments are carried out to confirm the accuracy and efficiency of our method.

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Acknowledgements

The project was supported by National Natural Science Foundation of China (11701168, 11601144), the Scientific Research Fund of Hunan Provincial Education Department (18B304, 18C0531). Besides, the authors are very grateful for the helpful comments and suggestions provided via the reviewers.

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  1. Biao Yang and Haixiang Zhang are co-first authors.

Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, Hunan, People’s Republic of China

    Biao Yang, Haixiang Zhang, Xuehua Yang & Liang Tang

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  1. Biao Yang
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  2. Haixiang Zhang
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  3. Xuehua Yang
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Correspondence to Xuehua Yang.

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Yang, B., Zhang, H., Yang, X. et al. ADI Galerkin finite element scheme for the two-dimensional semilinear partial intergro-differential equation with a weakly singular kernel. J. Appl. Math. Comput. 68, 2471–2491 (2022). https://doi.org/10.1007/s12190-021-01609-7

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