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Volume 15, Issue 4
A Dual-Horizon Nonlocal Diffusion Model and Its Finite Element Discretization

Mingchao Bi, Lili Ju & Hao Tian

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 990-1008.

Published online: 2022-10

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  • Abstract

In this paper, we present a dual-horizon nonlocal diffusion model, in which the influence area at each point consists of a standard sphere horizon and an irregular dual horizon and its geometry is determined by the distribution of the varying horizon parameter. We prove the mass conservation and maximum principle of the proposed nonlocal model, and establish its well-posedness and convergence to the classical diffusion model. Noticing that the dual horizon-related term in fact vanishes in the corresponding variational form of the model, we then propose a finite element discretization for its numerical solution, which avoids the difficulty of accurate calculations of integrals on irregular intersection regions between the mesh elements and the dual horizons. Various numerical experiments in two and three dimensions are also performed to illustrate the usage of the variable horizon and demonstrate the effectiveness of the numerical scheme.

  • AMS Subject Headings

65R20, 65N35, 74S05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-990, author = {Bi , MingchaoJu , Lili and Tian , Hao}, title = {A Dual-Horizon Nonlocal Diffusion Model and Its Finite Element Discretization}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {990--1008}, abstract = {

In this paper, we present a dual-horizon nonlocal diffusion model, in which the influence area at each point consists of a standard sphere horizon and an irregular dual horizon and its geometry is determined by the distribution of the varying horizon parameter. We prove the mass conservation and maximum principle of the proposed nonlocal model, and establish its well-posedness and convergence to the classical diffusion model. Noticing that the dual horizon-related term in fact vanishes in the corresponding variational form of the model, we then propose a finite element discretization for its numerical solution, which avoids the difficulty of accurate calculations of integrals on irregular intersection regions between the mesh elements and the dual horizons. Various numerical experiments in two and three dimensions are also performed to illustrate the usage of the variable horizon and demonstrate the effectiveness of the numerical scheme.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0004s }, url = {http://global-sci.org/intro/article_detail/nmtma/21087.html} }
TY - JOUR T1 - A Dual-Horizon Nonlocal Diffusion Model and Its Finite Element Discretization AU - Bi , Mingchao AU - Ju , Lili AU - Tian , Hao JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 990 EP - 1008 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0004s UR - https://global-sci.org/intro/article_detail/nmtma/21087.html KW - Nonlocal diffusion, dual-horizon, maximum principle, finite element discretization, asymptotically compatible. AB -

In this paper, we present a dual-horizon nonlocal diffusion model, in which the influence area at each point consists of a standard sphere horizon and an irregular dual horizon and its geometry is determined by the distribution of the varying horizon parameter. We prove the mass conservation and maximum principle of the proposed nonlocal model, and establish its well-posedness and convergence to the classical diffusion model. Noticing that the dual horizon-related term in fact vanishes in the corresponding variational form of the model, we then propose a finite element discretization for its numerical solution, which avoids the difficulty of accurate calculations of integrals on irregular intersection regions between the mesh elements and the dual horizons. Various numerical experiments in two and three dimensions are also performed to illustrate the usage of the variable horizon and demonstrate the effectiveness of the numerical scheme.

Mingchao Bi, Lili Ju & Hao Tian. (2022). A Dual-Horizon Nonlocal Diffusion Model and Its Finite Element Discretization. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 990-1008. doi:10.4208/nmtma.OA-2022-0004s
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