Convergence results for finite element and finite difference approximation of nonlocal fracture

Abstract

We show apriori convergence of the nonlinear Peridynamic models. The model is shown to be well-posed in Hölder space and Sobolev H^2 space. Finite element approximation using linear conforming elements is shown to converge at the rate h^2 where h is the mesh size. Piecewise constant approximation (finite difference) on a uniform grid is shown to converge at the rate h. We numerically demonstrate the convergence for the Mode-I crack propagation problem. We conclude by discussing future works.