Communications on Applied Mathematics and Computation >

2020 , Vol. 2 >Issue 2: 261 - 303

DOI: https://doi.org/10.1007/s42967-019-00051-8

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

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  • 1 CCDC Army Research Laboratory, Attn: FCDD-RLW-MB, Aberdeen Proving Ground, Aberdeen, MD 21005, USA;
    2 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Received date: 2019-05-22

  Revised date: 2019-10-01

  Online published: 2020-02-19

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Abstract

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We fnd that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible. We verify these fndings through extensive numerical experiments.

Key words： Nonlocal operator; Inhomogeneous local boundary condition; Nonlocal difusion; Asymptotic compatibility; Collocation method; Peridynamics; Functional calculus

Cite this article

Burak Aksoylu, Fatih Celiker, George A. Gazonas . Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility[J]. Communications on Applied Mathematics and Computation, 2020 , 2(2) : 261 -303 . DOI: 10.1007/s42967-019-00051-8

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