A cookbook for finite element methods for nonlocal problems, including quadrature rules and approximate Euclidean balls

D’Elia, Marta; Gunzburger, Max; Vollmann, Christian

doi:10.48550/arxiv.2005.10775

Cited by 8 publications

(27 citation statements)

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“…Even though we have found the explicit formula of the optimal test space norm, as shown in Lemma 3.6, for the nonlocal convection-dominated diffusion problem (16), it involves the computation of the inverse operator (−L δ ) −1 which is computationally prohibitive in practice. To alleviate the computational difficulties, we now limit ourselves to one dimension (d = 1) and discuss an approximation of the optimal test norm (27).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…We remark that the intersection of an element and horizon in one dimension, Ω j ∩ B δ (x p ), is an interval which is not hard to find. The intersecting geometry becomes more complicated in higher dimensions and related discussions on numerical integration for finite element implementations of nonlocal models can be found in [11,16]. To reduce the integration error, we have used p (p) + N over quadrature points.…”

Section: Nonlocal Kernelsmentioning

confidence: 99%

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A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

Leng,

Tian,

Demkowicz

et al. 2021

Preprint

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We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al. ( 2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform h-and p-refinements, and adaptive h-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Nonlocal Kernelsmentioning

confidence: 99%

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

Leng,

Tian,

Demkowicz

et al. 2021

Preprint

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“…In [9], Du et al consider a discontinuous Galerkin method. See the recent monographs by D'Elia et al for a review on finite element based methods [8] and other methods [7] that also apply to non-integrable spatial kernels.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of some nonlocal diffusion discretization schemes

Galiano¹

2021

Preprint

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We study two numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and on the Fourier transform. Apart from the usual time-space discretization, these algorithms also use the discretization of the range of the solution (quantization). We show that the discrete approximations converge to the continuous solution in suitable functional spaces, and provide error estimates. Finally, we present some numerical experiments illustrating the performance of the algorithms, specially focusing in the execution time.

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“…However, the utilization of nonlocal models in applications that could benefit from their improved predictive capabilities is hindered by several modeling and numerical challenges. These include the unresolved treatment of nonlocal interfaces [2,12], the nontrivial prescription of nonlocal volume constraints (the nonlocal counterpart of boundary conditions) [15,23], and the fact that computational costs attendant to the use of nonlocal problems may become prohibitive as the extent of the nonlocal interactions increases; see, e.g., [17,22] for variational methods and [13,Chapter 7] for mesh-free methods. Other critical challenges are related to the uncertain nature of model parameters; in fact, modeling parameters such as δ and those characterizing the kernel, applied forces, and/or sources can be non-measurable, sparse, and/or subject to noise.…”

mentioning

confidence: 99%

“…However, we address this issue in Section 4.1. A comprehensive discussion of how to effectively handle cut elements can be found in [22].…”

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confidence: 99%

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

Capodaglio¹,

D’Elia²,

Gunzburger³

et al. 2020

Preprint

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A rigorous mathematical framework is provided for a substructuring-based domaindecomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, singledomain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

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A cookbook for finite element methods for nonlocal problems, including quadrature rules and approximate Euclidean balls

Cited by 8 publications

References 0 publications

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

Error analysis of some nonlocal diffusion discretization schemes

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

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