Finite element approximation of a strain-limiting elastic model

Abstract

AbstractWe 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. A rate of convergence for the sequence of finite element approximations is shown provided that the material parameters featuring in the model are Lipschitz continuous and that the exact solution possesses additional regularity. A rate of convergence for the sequence of finite element approximations is shown provided that the material parameters featuring in the model are Lipschitz continuous and that the exact solution possesses additional regularity. 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.