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 $\underlineλ \to \infty$. The results validate the effectiveness of Mini element and proposed augmented subspace algorithm for reliable and efficient computation in the nearly incompressible regime.