A CONVERGENT ADAPTIVE FINITE ELEMENT ALGORITHM FOR NONLOCAL DIFFUSION AND PERIDYNAMIC MODELS

In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions. The convergence of the adaptive finite element al...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 51; no. 2; pp. 1211 - 1234
Main Authors: DU, QIANG, TIAN, LI, ZHAO, XUYING
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Subjects:
A posteriori knowledge
Adaptive algorithms
Algorithms
Applied mathematics
Approximation
Computational mathematics
Convergence
Diffusion
Estimates
Estimators
Finite element method
Interpolation
Kernel functions
Line segments
Mathematical analysis
Mathematical functions
Mathematical models
Mathematical problems
Neighborhoods
Science
Sobolev spaces
Triangles
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions. The convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients, such as the upper bound of the estimator, the estimator reduction, and the orthogonality property. We also consider how the results are affected by the horizon parameter δ which characterizes the range of nonlocality. Numerical experiments are performed to verify our theoretical findings.
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ISSN:0036-1429
1095-7170
DOI:10.1137/120871638