An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

You, Huaiqian; Lu, Xin Yang; Trask, Nathaniel Albert; Yu, Yue

doi:10.1051/m2an/2020058

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

Journal Article · Thu Feb 25 23:00:00 EST 2021 · Mathematical Modelling and Numerical Analysis

DOI:https://doi.org/10.1051/m2an/2020058· OSTI ID:1769929

You, Huaiqian ^[1]; Lu, Xin Yang ^[2]; Trask, Nathaniel Albert ^[3]; Yu, Yue ^[1]

Lehigh Univ., Bethlehem, PA (United States)
Lakehead Univ., Thunder Bay, ON (Canada)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research

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. Finally, 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.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1769929

Report Number(s):: SAND--2021-2178J; 694081

Journal Information:: Mathematical Modelling and Numerical Analysis, Journal Name: Mathematical Modelling and Numerical Analysis Vol. 55; ISSN 0764-583X

Publisher:: EDP SciencesCopyright Statement

Country of Publication:: United States

Language:: English

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