Finite Differences and Finite Elements in Nonlocal Fracture Modeling: A Priori Convergence Rates

Abstract

In this chapter we present a rigorous convergence analysis of finite difference and finite element approximation of nonlinear nonlocal models. In the previous chapter, we considered a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175–209, 2000). There we showed, for a fixed horizon of nonlocal interaction 𝜖, that well-posed formulations of the model can be developed over Hölder spaces and Sobolev spaces. In this chapter we apply these formulations to show a priori convergence for the discrete finite difference and finite element methods. We show that the error made using the forward Euler in time and a finite difference (i.e., piecewise constant) discretization in space with time step Δt and spatial discretization h is of the order of O( Δt + h∕𝜖 ²). For a central difference approximation in time and piecewise linear finite element approximation in space, the approximation error is of the order of O( Δt + h ²∕𝜖 ²). We point out these are the first such error estimates for nonlinear nonlocal fracture formulations and are reported in Jha and Lipton (2017b Numerical analysis of nonlocal fracture models models in holder space. arXiv preprint arXiv:1701.02818. To appear in SIAM Journal on Numerical Analysis 2018) and Jha and Lipton (2017a, Finite element approximation of nonlocal fracture models. arXiv preprint arXiv:1710.07661). We then go on to prove the stability of the semi-discrete approximation and show that the energy of the discrete approximation is bounded in terms of work done by the body force and initial energy put into the system. We look forward to improvements and development of a posteriori error estimation in the coming years.