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

Du, Qiang; Ju, Lili; Li, Tian; Zhou, Kun

doi:10.1090/s0025-5718-2013-02708-1

Cited by 72 publications

(55 citation statements)

References 39 publications

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“…This is the essential component of PIM. Similar integral approximation is also used in nonlocal diffusion and peridynamic models [1,[15][16][17]49].…”

Section: R(r)mentioning

confidence: 99%

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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The Laplace-Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral method (PIM), a novel numerical method for discretizing the Laplace-Beltrami operator on point clouds (Li et al. in Commun Comput Phys 22(1):228-258, 2017). In this paper, we analyze the convergence of PIM for Poisson equation with Neumann boundary condition on submanifolds that are isometrically embedded in Euclidean spaces. Keywords: Laplace-Beltrami operator, Neumann boundary, Point cloud, Point integral method, Convergence analysisMathematics Subject Classification: 65N12, 65N25, 65N75 BackgroundThe partial differential equations on manifolds arise in a wide variety of applications. In many problems, including material science [10,20], fluid flow [22,25], biology and biophysics [2,3,21,37], people need to study the physical process, for instance diffusion and convection, in curved surfaces which introduce different kinds of PDEs in surfaces. It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space. One fundamental assumption in the manifold model is that the point cloud samples a smooth manifold. Thus, the information of the manifold is very useful to understand the data or images. PDEs on the manifold, particularly the Laplace-Beltrami equation, encode several intrinsic information of the manifold, thus helping reveal the underlying structures in the data or images. To get the information encoded in PDEs, we need to solve them in the unstructured point cloud. Given that the point cloud is embedded in a high-dimensional space, the traditional methods for PDEs on 2D surfaces do not work.

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“…This is the essential component of PIM. Similar integral approximation is also used in nonlocal diffusion and peridynamic models [1,[15][16][17]49].…”

Section: R(r)mentioning

confidence: 99%

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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“…Since the interpolation operator Π k is not needed in that case, the term |v| s,ωT is replaced by v 0,T . Therefore, this implies also that in the reliability result given in [21], the constant C 3 is independent of δ under the conditionh k ≤ δ/2 (or say, T ⊂ B δ/2 (x) for any x ∈ T ). In addition, we note that if s ≥ 1/2, then Lu k 0,Ωs is undefined so that the error estimator (2.18) is no longer valid for this case, which is why we require s ∈ (0, 1/2) in this work.…”

Section: Reliability Of a Posteriori Error Estimatormentioning

confidence: 86%

“…Remark 3.10. For the case of bounded nonlocal operators with integrable kernel functions, one can similarly use the patch comparison technique in the last remark to show the reliability result (3.11) with a constant C 3 [21]. Since the interpolation operator Π k is not needed in that case, the term |v| s,ωT is replaced by v 0,T .…”

Section: Reliability Of a Posteriori Error Estimatormentioning

confidence: 99%

A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models

Du¹,

Li²,

Zhao³

2013

SIAM J. Numer. Anal.

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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. Introduction.In this work, we consider numerical approximations of some nonlocal diffusion models which arise in many fields such as image analysis [10,25,28], nonlocal diffusion [8,19], and continuum mechanics [37]. These models offer new alternatives to traditional PDE based models. For instance, the peridynamic (PD) theory proposed in [37] is an integral-type nonlocal continuum theory which incorporates the nonlocal nature of material interactions. It also connects continuum mechanics and molecular dynamics within a single framework [39]. Meanwhile, there has been much development in the mathematical theory of nonlocal models; see, for instance, an extensive treatment on nonlocal diffusion problems in [4]. In [18,26], a nonlocal vector calculus was developed to provide a more general variational setting for nonlocal models. More theoretical studies of related volume-constraint problems can be found in [19,30,20]. Within the context of PD-based nonlocal models, there have been a variety of numerical methods implemented for their approximations including finite difference, finite element, quadrature, and particle-based methods [3,9,15,24,27,29,35,38,42]. Given the ability of nonlocal PD models to simulate cracks or fractures, an adaptive method is a natural way to reduce the computational cost. Indeed, adaptive refinement for nonlocal PD-type models has been studied in [9] with meshless methods. Utilizing the nice variational structures of the volume-constrained problems associated with the linear nonlocal diffusion or PD operators and the strong connection to the variational PDE problems associated with elliptic operators, it is also natural to study finite element and adaptive finite element approximations of

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“…Studies of different meshing strategies, including comparisons between cubic meshes and centroidal Voronoi tessellations, appear in [29]. Discretizations of peridynamic and related nonlocal diffusion models using weak forms of governing equations appear, on the other hand, in [22,30,31], based on continuous and discontinuous Galerkin finite element methods. Related convergence studies for quadrature methods in peridynamics have been discussed in [32].…”

Section: Introductionmentioning

confidence: 99%

Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations

Seleson

2014

Computer Methods in Applied Mechanics and Engineering

147

107

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

Cited by 72 publications

References 39 publications

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models

Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations

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