Numerical analysis of nonlocal fracture models in hölder 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 Hölder space with Hölder exponent γ ∈ (0, 1]. The rate of convergence of the finite difference approximation depends on the factor Cshγ /2 where gives the length scale of nonlocal interaction, h is the discretization length, and Cs 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.
Publication Source (Journal or Book title)
SIAM Journal on Numerical Analysis
Jha, P., & Lipton, R. (2018). Numerical analysis of nonlocal fracture models in hölder space. SIAM Journal on Numerical Analysis, 56 (2), 906-941. https://doi.org/10.1137/17M1112236