近不可压缩线性弹性问题的Mini混合有限元方法
Mini mixed finite element method for nearly incompressible linear elasticity problems
- School of Science, East China University of Technology（华东交通大学理学院）
- School of Mathematics, Jilin University（吉林大学数学学院）
- SKLMS, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
- Department of Mathematics, Beijing University of Technology（北京工业大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出基于Mini单元的混合有限元方法，解决近不可压缩线性弹性问题的锁现象，开发非嵌套增量子空间算法，经理论分析和数值实验验证其无锁性与收敛鲁棒性。
AI中文摘要：
本文研究近不可压缩线性弹性边值问题及其相关特征值问题的数值求解方法。针对近不可压缩极限下标准低阶单元存在的锁现象，提出一种基于Mini单元的混合有限元格式以规避该问题。对于边值问题，本文建立混合变分格式的适定性，并推导关于拉梅常数$\boldsymbol{\u03BB}$一致的先验误差估计，从而证明该方法无锁特性。对于特征值问题，本文在混合有限元框架内开发一种高效的非嵌套增量子空间算法。针对离散特征值近似和所提出的迭代求解器，本文提供全面的收敛性分析，证明其收敛速率与$\boldsymbol{\u03BB}$无关。对两类问题的数值实验均验证了理论结论，显示当$\boldsymbol{\u03BB} \to \u221e$时，方法具有最优收敛速率和鲁棒性。上述结果证实了Mini单元与所提增量子空间算法在近不可压缩领域可靠高效计算的有效性。
英文摘要：
This paper addresses the numerical solution of nearly incompressible linear elasticity boundary value problems and their associated eigenvalue problems. A mixed finite element formulation based on Mini element is proposed to circumvent the locking phenomenon that plagues standard low-order elements in the nearly incompressible limit. For the boundary value problem, we establish the well-posedness of mixed variational formulation and derive a priori error estimates that are uniform with respect to the Lamé constant $\underlineλ$, thereby proving the method's locking-free property. For the eigenvalue problem, we develop an efficient non-nested augmented subspace algorithm designed within the mixed finite element framework. A comprehensive convergence analysis is provided for the discrete eigenvalue approximation and the proposed iterative solver, demonstrating that the convergence rates remain independent of $\underlineλ$. Numerical experiments on both problems confirm the theoretical conclusions, showing optimal convergence rates and robustness as $\underlineλ \to \infty$. The results validate the effectiveness of Mini element and proposed augmented subspace algorithm for reliable and efficient computation in the nearly incompressible regime.
