Title: 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 a 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 semi-discrete approximation.
The Hölder continuous evolutions are seen to converge to a brittle fracture
evolution in the limit of vanishing nonlocality.