A Galerkin Radial Basis Function Method for Nonlocal Diffusion

Bond, Stephen D.; Lehoucq, Richard B.; Rowe, Stephen T.

doi:10.1007/978-3-319-06898-5_1

Cited by 6 publications

(6 citation statements)

References 15 publications

(34 reference statements)

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Further, we comment that in the DG formulation, it is not essential to require that the discrete approximations are made of piecewise polynomials. In fact, more general discrete function spaces such as those represented by reproducing kernel spaces, radial basis functions, partition of unity, and other generalized/extended finite element basis [6,7,9,12,34,49] can also be used. We note that the key is to have the basis functions satisfying the volumetric constraints so that they conform with the modified energy spaces.…”

Section: Dg Approximationmentioning

confidence: 99%

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Tian¹,

Du²

2015

SIAM J. Numer. Anal.

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We present a nonconforming discontinuous Galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints. The nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions. The key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin. It works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions. A convergence theory is established for two types of finite dimensional spaces, with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown to be conditionally convergent. The analysis is based on the recently developed framework of asymptotically compatible schemes for parametrized variational problems. It also relies crucially on a successful extension of a compactness result of Bourgain, Brezis, and Mironescu to function sequences associated with a sequence of kernels that is assumed to be convergent to a more general limiting kernel. Numerical experiments are carried out to offer additional observations on the performance of our proposed method.1. Introduction. This paper serves as a first attempt to develop discontinuous and nonconforming Galerkin finite element discretization of nonlocal diffusion (ND) and nonlocal peridynamic models defined on a bounded spatial domain in any given space dimension. Since first introduced by Silling in [45], peridynamics (PD), which is an integral-type nonlocal continuum theory, has demonstrated effectiveness in modeling material singularities [4,32,47,48]. As pointed out in [21], linear scalar PD operators also share similarities with ND operators, thus making the study of PD relevant to the study of ND models, as well as fractional diffusion models, in various applications [2,11,13,19,22,27,29,30,31,33,36,40]. Along with research efforts on the mathematical analysis of PD/ND models, a variety of numerical methods have been studied and implemented, including finite difference, finite element, quadrature, and particle based methods [1,8,14,24,25,32,35,42,44,46,50,51,52,53,54,56,57]. In terms of Galerkin type approximations, most of the existing analyses have focused on the conforming ones and very few studies have been carried out for nonconforming discontinuous Galerkin (DG) finite element discretizations.

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Section: Dg Approximationmentioning

confidence: 99%

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Tian¹,

Du²

2015

SIAM J. Numer. Anal.

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“…The aim of this work is to develop a unified meshless pseudospectral method to solve both classical (α = 2) and fractional (α < 2) PDEs. Its unique feature -compatibility with the classical Laplacian -makes our method distinguish from other numerical methods (e.g., in [13,14,3,2,7,1,6,7]) for the fractional Laplacian. Moreover, our method takes great advantage of the Laplacian (−∆) α 2 (for both α = 2 and α < 2) of generalized inverse multiquadric functions so as to bypass numerical approximations to the hypersingular integral of fractional Laplacian in (1.4).…”

Section: Introductionmentioning

confidence: 99%

A universal solution scheme for fractional and classical PDEs

Wu¹,

Zhang²

2021

Preprint

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We propose a unified meshless method to solve classical and fractional PDE problems with (−∆) α 2 for α ∈ (0, 2]. The classical (α = 2) and fractional (α < 2) Laplacians, one local and the other nonlocal, have distinct properties. Therefore, their numerical methods and computer implementations are usually incompatible. We notice that for any α ≥ 0, the Laplacian (−∆) α 2 of generalized inverse multiquadric (GIMQ) functions can be analytically written by the Gauss hypergeometric function, and thus propose a GIMQ-based method. Our method unifies the discretization of classical and fractional Laplacians and also bypasses numerical approximation to the hypersingular integral of fractional Laplacian. These two merits distinguish our method from other existing methods for the fractional Laplacian. Extensive numerical experiments are carried out to test the performance of our method. Compared to other methods, our method can achieve high accuracy with fewer number of unknowns, which effectively reduces the storage and computational requirements in simulations of fractional PDEs. Moreover, the meshfree nature makes it free of geometric constraints and enables simple implementation for any dimension d ≥ 1. Additionally, two approaches of selecting shape parameters, including condition numberindicated method and random-perturbed method, are studied to avoid the ill-conditioning issues when large number of points.

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“…The essence of the projection-based ROM is to project the full-order governing partial differential equations (PDE), e.g., 3-D Navier-Stokes (NS) equations, onto a reduced subspace spanned by a group of basis functions, which can either be data-based basis such as proper orthogonal decomposition (POD) modes [38] or dictionary-based basis including polynomials [39], wavelets [40], and radial basis functions [41]. It is expected that the reduced system after projection can be solved more efficiently.…”

Section: Introductionmentioning

confidence: 99%

A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations

Gao

Zhu

Wang

2020

Computer Methods in Applied Mechanics and Engineering

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Image-based computational fluid dynamics (CFD) modeling enables derivation of hemodynamic information (e.g., flow field, wall shear stress, and pressure distribution), which has become a paradigm in cardiovascular research and healthcare. Nonetheless, the predictive accuracy largely depends on precisely specified boundary conditions and model parameters, which, however, are usually uncertain (or unknown) in most patient-specific cases. Quantifying the uncertainties in model predictions due to input randomness can provide predictive confidence and is critical to promote the transition of CFD modeling in clinical applications.In the meantime, forward propagation of input uncertainties often involves numerous expensive CFD simulations, which is computationally prohibitive in most practical scenarios. This paper presents an efficient bi-fidelity surrogate modeling framework for uncertainty quantification (UQ) in cardiovascular simulations, by leveraging the accuracy of high-fidelity models and efficiency of low-fidelity models. Contrary to most data-fit surrogate models with several scalar quantities of interest, this work aims to provide high-resolution, full-field predictions (e.g., velocity and pressure fields). Moreover, a novel empirical error bound estimation approach is introduced to evaluate the performance of the surrogate a priori.The proposed framework is tested on a number of vascular flows with both standardized and patient-specific vessel geometries, and different combinations of high-and low-fidelity models are investigated. The results show that the bi-fidelity approach can achieve high * Corresponding authors. jwang33@nd.edu (J.-X. Wang), xueyu-zhu@uiowa.edu (X. Zhu) predictive accuracy with a significant reduction of computational cost, exhibiting its merit and effectiveness. Particularly, the uncertainties from a high-dimensional input space can be accurately propagated to clinically relevant quantities of interest (e.g., wall shear stress) in the patient-specific case using only a limited number of high-fidelity simulations, suggesting a good potential in practical clinical applications.

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A Galerkin Radial Basis Function Method for Nonlocal Diffusion

Cited by 6 publications

References 15 publications

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

A universal solution scheme for fractional and classical PDEs

A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations

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