Finite element approximation of nonlocal fracture models

We consider nonlocal nonlinear potentials and estimate the rate of convergence of time stepping schemes to the peridynamic equation of motion. We begin by establishing the existence of $H^2$ solutions over any finite time interval. Here spatial approximation by finite element interpolations are considered. The energy stability of the associated semi-discrete time stepping scheme is established and the approximation of strong and weak formulations of the evolution using FE interpolations of $H^2$ solutions are investigated. The strong and weak form of approximations are shown to converge to the actual solution in the mean square norm at the rate $C_t\Delta t +C_s h^2/\epsilon^2$ where $h$ is the mesh size, $\epsilon$ is the size of nonlocal interaction and $\Delta t$ is the time step. The constants $C_t$ and $C_s$ are independent of $\Delta t$, and $h$. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed.