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 | References | Similar Articles | Additional Information

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