In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞(Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.
Keywords: Integro-differential equations, nonlocal diffusion, Neumann-type boundary condition, meshless, asymptotic compatibility
@article{M2AN_2021__55_S1_S811_0,
author = {You, Huaiqian and Lu, Xin Yang and Trask, Nathaniel and Yu, Yue},
title = {An asymptotically compatible approach for {Neumann-type} boundary condition on nonlocal problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S811--S851},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020058},
mrnumber = {4221328},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020058/}
}
TY - JOUR AU - You, Huaiqian AU - Lu, Xin Yang AU - Trask, Nathaniel AU - Yu, Yue TI - An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S811 EP - S851 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020058/ DO - 10.1051/m2an/2020058 LA - en ID - M2AN_2021__55_S1_S811_0 ER -
%0 Journal Article %A You, Huaiqian %A Lu, Xin Yang %A Trask, Nathaniel %A Yu, Yue %T An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S811-S851 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020058/ %R 10.1051/m2an/2020058 %G en %F M2AN_2021__55_S1_S811_0
You, Huaiqian; Lu, Xin Yang; Trask, Nathaniel; Yu, Yue. An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S811-S851. doi: 10.1051/m2an/2020058
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