Numerical Analysis of Nonlocal Fracture Models in Hölder Space

Abstract

In this work, we calculate the convergence rate of the finite difference approximation for a class of nonlocal fracture models. We consider two point force interactions characterized by a double well potential. We show the existence of an evolving displacement field in Hölder space with Hölder exponent $\gamma \in (0, 1]$. The rate of convergence of the finite difference approximation depends on the factor $C_s h^\gamma/\epsilon^2$ where $\epsilon$ gives the length scale of nonlocal interaction, $h$ is the discretization length, and $C_s$ is the maximum of Hölder norm of the solution and its second derivatives during the evolution. It is shown that the rate of convergence holds for both the forward Euler scheme as well as general single step implicit schemes. A stability result is established for the semidiscrete approximation. The Hölder continuous evolutions are seen to converge to a brittle fracture evolution in the limit of vanishing nonlocality.