Quasi-optimal adaptive hybridized mixed finite element methods for linear elasticity

TL;DR

This paper proves the quasi-optimal convergence of an adaptive hybridized mixed finite element method for solving the planar Navier--Lamé equations in linear elasticity, with improved applicability and implementation simplicity.

Contribution

It establishes uniform quasi-optimal convergence for an adaptive hybridized mixed finite element method, extending its applicability to traction boundary conditions.

Findings

Proves quasi-optimal convergence of the adaptive method.

Provides a discrete a posteriori upper bound and quasi-orthogonality result.

Method is directly applicable to traction boundary conditions.

Abstract

For the planar Navier--Lam\'e equation in mixed form with symmetric stress tensors, we prove the uniform quasi-optimal convergence of an adaptive method based on the hybridized mixed finite element proposed in [Gong, Wu, and Xu: Numer.~Math., 141 (2019), pp.~569--604]. The main ingredients in the analysis consist of a discrete a posteriori upper bound and a quasi-orthogonality result for the stress field under the mixed boundary condition. Compared with existing adaptive methods, the proposed adaptive algorithm could be directly applied to the traction boundary condition and be easily implemented.