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arXiv · 2105.00144

Taylor-Hood like finite elements for nearly incompressible strain gradient elasticity problems

Yulei Liao·Pingbing Ming·Yun Xu
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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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Yulei Liao, Pingbing Ming, Yun Xu. 2023-01-26. Taylor-Hood like finite elements for nearly incompressible strain gradient elasticity problems. https://doi.org/10.1007/s10915-023-02135-3

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