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

TL;DR

This paper introduces a family of mixed finite elements tailored for nearly incompressible strain gradient elasticity problems, demonstrating robustness, optimal convergence, and effectiveness through theoretical analysis and numerical validation.

Contribution

It develops a new finite element method that is robust for nearly incompressible strain gradient elasticity and proves its optimal convergence and robustness in the incompressible limit.

Findings

Optimal convergence rate achieved, robust in the incompressible limit

Method effectively handles boundary layer effects

Numerical results confirm theoretical predictions

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.