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Preprint: Finite element approximation of nonlinear nonlocal models

Prashant K. Jha's picture

With Robert Lipton. Submitted to Mathematical Models and Methods in Applied Sciences. arXiv link: https://arxiv.org/abs/1710.07661

Abstract: We consider nonlocal nonlinear potentials and compute the rate of convergence of the finite element approximation to the peridynamic equation of motion. The present work is a continuation and extension of the work in [\refcite{CMPer-JhaLipton}] where the finite difference approximation of peridynamics equation was shown to converge at the rate $O(h^\gamma/\epsilon)$ in H\"older space $C^{0,\gamma}$. Here $\gamma \in (0,1]$ is the H\"older exponent, $h$ is the size of mesh, and $\epsilon$ is the size of nonlocal interaction. In this work the existence of $H^2$ solutions is shown. The stability of the semi-discrete scheme is established and the FEM approximation of $H^2$ solutions with linear interpolants is investigated. The FEM approximation is shown to converge at the rate $O(h^2/\epsilon^2)$. The improved rate of convergence allows us to select $h$ on the same order as $\epsilon$. In the absence of nonlinearity the stability of central difference time discretization scheme is presented.

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