Finite Element Approximation of a Strain-Limiting Elastic Model

TL;DR

This paper develops a finite element method for a strain-limiting elastic model, proving strong convergence to the solution, establishing convergence rates under regularity assumptions, and introducing an efficient iterative algorithm with numerical validation.

Contribution

It introduces a novel finite element approximation and an iterative solution algorithm for a nonlinear strain-limiting elastic model, with proven convergence and computational efficiency.

Findings

Strong convergence of finite element approximations to the model's solution.

Convergence rates established under regularity assumptions.

Numerical experiments validate the theoretical results.

Abstract

We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. Assuming that the material parameters featuring in the model are Lipschitz-continuous, and assuming that the weak solution has additional regularity, the sequence of finite element approximations is shown to converge with a rate. An iterative algorithm is constructed for the solution of the system of nonlinear algebraic equations that arises from the finite element approximation. An appealing feature of the iterative algorithm is that it decouples the monotone and linear elastic parts of the nonlinearity in the model. In particular, our choice of piecewise constant approximation for the stress tensor (and continuous piecewise linear approximation for the displacement) allows us to compute the monotone part of the nonlinearity by solving an algebraic system with $d(d+1)/2$ unknowns independently on each element in the subdivision of the computational domain. The theoretical results are illustrated by numerical experiments.