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A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models


Authors: Qiang Du, Lili Ju, Li Tian and Kun Zhou
Journal: Math. Comp. 82 (2013), 1889-1922
MSC (2010): Primary 65J15, 65R20, 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2013-02708-1
Published electronically: May 8, 2013
MathSciNet review: 3073185
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Abstract: In this paper, we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond-based peridynamic models. In particular, we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems. A posteriori error estimators are consequently prompted, the reliability and efficiency of the estimators are proved. Connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space. Numerical experiments (1D) are also given to test the theoretical conclusions.


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Additional Information

Qiang Du
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: qdu@math.psu.edu

Lili Ju
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: ju@math.sc.edu

Li Tian
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: tian@math.psu.edu

Kun Zhou
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: zhou@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02708-1
Keywords: Peridynamic models, nonlocal diffusion, a posteriori error estimate, finite element
Received by editor(s): April 27, 2011
Received by editor(s) in revised form: March 6, 2012
Published electronically: May 8, 2013
Additional Notes: This work was supported in part by the U.S. Department of Energy Office of Science under grant number DE-SC0005346 and by the U.S. National Science Foundation under grant number DMS-1016073.
Article copyright: © Copyright 2013 American Mathematical Society

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