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Taylor-Hood Like Finite Elements for Nearly Incompressible Strain Gradient Elasticity Problems

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Abstract

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73–100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with 1/2 order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.

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Funding

The work of Liao and Ming were supported by National Natural Science Foundation of China through Grant No. 11971467. The work of Xu was supported by National Natural Science Foundation of China through Grant No. 11772067.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, No. 55 East Road Zhong-Guan-Cun, Beijing, 100190, China

    Yulei Liao & Pingbing Ming

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yulei Liao & Pingbing Ming

  3. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Yun Xu

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  1. Yulei Liao
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  2. Pingbing Ming
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  3. Yun Xu
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Correspondence to Pingbing Ming.

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Liao, Y., Ming, P. & Xu, Y. Taylor-Hood Like Finite Elements for Nearly Incompressible Strain Gradient Elasticity Problems. J Sci Comput 95, 4 (2023). https://doi.org/10.1007/s10915-023-02135-3

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