A Coupling Method of Mixed and Lagrange Finite Elements for Linear Elasticity Problem

TL;DR

This paper introduces a coupled finite element method combining mixed and Lagrange elements to accurately and efficiently model stress concentrations in elasticity problems.

Contribution

It develops a novel coupling approach that ensures well-posedness and optimal error estimates, balancing accuracy and computational cost.

Findings

The method achieves optimal convergence rates.

Numerical experiments confirm theoretical predictions.

The approach effectively captures stress concentrations.

Abstract

This paper proposes a finite element method that couples mixed and Lagrange finite elements to efficiently capture stress concentrations in elasticity problems. The method employs conforming mixed finite elements in regions with stress concentration, while standard Lagrange elements are used elsewhere, achieving a balance between stress accuracy and computational efficiency. The well-posedness of the coupled formulation and optimal a priori error estimates are established, even when the size of the mixed finite element subregion is $O(h)$. Numerical experiments are presented to verify the theoretical convergence rates and to demonstrate the effectiveness and efficiency of the proposed method.