Numerical analysis of nonlocal fracture models in Hölder space
Provides rigorous numerical analysis for nonlocal fracture models, which is important for computational mechanics but is an incremental theoretical extension.
This work derives convergence rates for finite difference approximations of nonlocal fracture models with double-well potentials, showing that the rate depends on Hölder regularity and scales as \(C_s h^\gamma / \varepsilon^2\). The Hölder continuous solutions converge to brittle fracture in the vanishing nonlocality limit.
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 $γ\in (0,1]$. The rate of convergence of the finite difference approximation depends on the factor $C_s h^γ/ε^2$ where $ε$ 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.