Numerical analysis of nonlocal fracture models in H\"older space

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 a evolving displacement field in H\"{o}lder space with H\"{o}lder 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 H\"older 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 semi-discrete approximation. The H\"older continuous evolutions are seen to converge to a brittle fracture evolution in the limit of vanishing nonlocality.