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Convergence analysis and error estimates of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time

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Abstract

This paper is presented for the convergence analysis of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time, which arises from the theory of viscoelastic materials with edge friction. First, the existence and uniqueness of the solutions for the evolutionary variational inequality of the second-order in time are proved, which are mainly based on the fixed point theorem. Second, the convergence analysis of the interpolating element-free Galerkin method is presented for them. The error estimates show that the convergence order depends not only on the number of basis functions in the interpolating moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.

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Acknowledgements

We would like to express our gratitude to Mr. Xiaoyue Lu for his contributions to the numerical examples of this paper. We are also grateful to the editors and reviewers for their valuable comments and constructive suggestions on our paper.

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Authors and Affiliations

  1. School of Rail Transportation, Soochow University, Suzhou, 215131, China

    Quan Shen

  2. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Rui Ding & Zhengcheng Zhu

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  1. Quan Shen
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  2. Rui Ding
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  3. Zhengcheng Zhu
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Correspondence to Rui Ding.

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Communicated by Abimael Loula.

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This work was supported by the National Natural Science Foundation of China (No. 11401416 and No. 11771319).

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Shen, Q., Ding, R. & Zhu, Z. Convergence analysis and error estimates of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time. Comp. Appl. Math. 39, 130 (2020). https://doi.org/10.1007/s40314-020-01154-2

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  • DOI: https://doi.org/10.1007/s40314-020-01154-2

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