Mini mixed finite element method for nearly incompressible linear elasticity problems
Abstract
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 → ∞. The results validate the effectiveness of Mini element and proposed augmented subspace algorithm for reliable and efficient computation in the nearly incompressible regime.
Related
- An interior penalty nonconforming finite element method for strain gradient elasticity
- A robust finite element method for incompressible magnetohydrodynamics using Whitney forms
- A Hybridized Staggered Discontinuous Galerkin--Mixed Finite Element Method for Strain Gradient Elasticity
- A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfvénic alignment in periodic domains
- A Modified Brinkman-Type Fictitious-Domain Finite Element Method for Incompressible Flow with Rigid Obstacles
BibTeX
@misc{guan2026mini,
title = {{Mini mixed finite element method for nearly incompressible linear elasticity problems}},
author = {Zhijin Guan and Yue Feng and Hehu Xie and Chenguang Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03780},
eprint = {2609.03780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03780},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Mini mixed finite element method for nearly incompressible linear elasticity problems
AU - Zhijin Guan
AU - Yue Feng
AU - Hehu Xie
AU - Chenguang Zhou
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03780
UR - https://arxiv.org/abs/2609.03780
AB - 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 → ∞. The results validate the effectiveness of Mini element and proposed augmented subspace algorithm for reliable and efficient computation in the nearly incompressible regime.
ER -