Analysis of SDFEM on Shishkin Triangular Meshes and Hybrid Meshes for Problems with Characteristic Layers

Zhang, Jin; Liu, Xiaowei

doi:10.1007/s10915-016-0180-2

Cited by 23 publications

(5 citation statements)

References 16 publications

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“…For the purpose of estimating | ∫ d 0 a ′ (x)(u I − u)𝜒dx|, we use (15), (22), (23), and 𝜒(x) = ∫ x 0 𝜒 ′ (s)ds,…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…With the in-depth development of singular perturbation problems, there are many stabilization methods, the most typical of which is streamline diffusion finite element methods (SDFEMs) [10,12,[19][20][21][22][23][24]. Combining SDFEMs and layer adapted meshes, we can obtain a numerical solution with satisfactory stability and accuracy; see, for example, Liu and Zhang [22] and Babu and Ramanujam [25].…”

Section: Introductionmentioning

confidence: 99%

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Streamline diffusion finite element method for a singularly perturbed problem with an interior layer on Shishkin mesh

Zhang

2023

Math Methods in App Sciences

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With the continuous development of science and technology, singularly perturbed problems have attracted more and more attentions from computational fluid scholars. In this paper, we discuss a singularly perturbed convection–diffusion problem with a discontinuous convection. Generally speaking, due to this discontinuity, an interior layer appears in the solution. A streamline diffusion finite element method on Shishkin mesh is used to solve this problem, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to support the theoretical conclusion.

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“…For the purpose of estimating | ∫ d 0 a ′ (x)(u I − u)𝜒dx|, we use (15), (22), (23), and 𝜒(x) = ∫ x 0 𝜒 ′ (s)ds,…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Streamline diffusion finite element method for a singularly perturbed problem with an interior layer on Shishkin mesh

Zhang

2023

Math Methods in App Sciences

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“…Very often the size of diffusion is characterized by a parameter ε, which could be smaller by several orders of magnitude compared to the size of convection and/or reaction, resulting in narrow boundary or interior layers in which the solution changes extremely rapidly [31]. Classical numerical methods use layer-adapted meshes or introduce carefully designed artificial stability terms to solve these challenging problems [2,4,33,37,38].…”

Section: Introductionmentioning

confidence: 99%

Less Emphasis on Hard Regions: Curriculum Learning of PINNs for Singularly Perturbed Convection-Diffusion-Reaction Problems

Yufeng Wang,

Cong Xu,

Min Yang

et al. 2024

EAJAM

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Although physics-informed neural networks (PINNs) have been successfully applied in a wide variety of science and engineering fields, they can fail to accurately predict the underlying solution in slightly challenging convection-diffusion-reaction problems. In this paper, we investigate the reason of this failure from a domain distribution perspective, and identify that learning multi-scale fields simultaneously makes the network unable to advance its training and easily get stuck in poor local minima. We show that the widespread experience of sampling more collocation points in high-loss regions hardly help optimize and may even worsen the results. These findings motivate the development of a novel curriculum learning method that encourages neural networks to prioritize learning on easier non-layer regions while downplaying learning on harder regions. The proposed method helps PINNs automatically adjust the learning emphasis and thereby facilitates the optimization procedure. Numerical results on typical benchmark equations show that the proposed curriculum learning approach mitigates the failure modes of PINNs and can produce accurate results for very sharp boundary and interior layers. Our work reveals that for equations whose solutions have large scale differences, paying less attention to high-loss regions can be an effective strategy for learning them accurately.

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“…ere are many methods for solving singular perturbation problems. Recent convergence analysis of the finite element method is referred to [5][6][7][8][9][10][11][12][13][14]. Except the finite element method, the finite difference method is the most widely used one at present.…”

Section: Introductionmentioning

confidence: 99%

A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

Tian

Liu

2021

Mathematical Problems in Engineering

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In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.

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Analysis of SDFEM on Shishkin Triangular Meshes and Hybrid Meshes for Problems with Characteristic Layers

Cited by 23 publications

References 16 publications

Streamline diffusion finite element method for a singularly perturbed problem with an interior layer on Shishkin mesh

Streamline diffusion finite element method for a singularly perturbed problem with an interior layer on Shishkin mesh

Less Emphasis on Hard Regions: Curriculum Learning of PINNs for Singularly Perturbed Convection-Diffusion-Reaction Problems

A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

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