[Paper Review] A quasinonlocal coupling method for nonlocal and local diffusion models

Qiang Du, Xingjie Helen Li+2 more|arXiv (Cornell University)|Apr 2, 2017

Numerical methods in engineering35 references4 citations

TL;DR

This paper introduces a quasinonlocal coupling method that seamlessly merges nonlocal and local diffusion models in one dimension using geometric reconstruction, ensuring flux balance, energy conservation, and the maximum principle. The method eliminates interfacial inconsistencies and boundary artifacts, with rigorous well-posedness, stability, and modeling error estimates established for the first time in nonlocal-to-local coupling.

ABSTRACT

In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new coupling framework removes interfacial inconsistency, ensures the flux balance, and satisfies energy conservation as well as the maximum principle, whereas none of existing coupling methods for nonlocal-to-local coupling satisfies all of these properties. We establish the well-posedness and provide the stability analysis of the coupling method. We investigate the difference to the local limiting problem in terms of the nonlocal interaction range. Furthermore, we propose a first order finite difference numerical discretization and perform several numerical tests to confirm the theoretical findings. In particular, we show that the resulting numerical result is free of artifacts near the boundary of the domain where a classical local boundary condition is used, together with a coupled fully nonlocal model in the interior of the domain.

Motivation & Objective

• To address interfacial inconsistencies and boundary layer artifacts in nonlocal-to-local coupling of diffusion models.
• To develop a coupling framework that preserves key physical properties: flux balance, energy conservation, and the maximum principle.
• To establish well-posedness and modeling error estimates for the coupled nonlocal-local problem.
• To provide a numerical discretization that avoids spurious artifacts near boundaries when using classical Dirichlet conditions.
• To extend the geometric reconstruction idea from atomistic-to-continuum methods to nonlocal diffusion models.

Proposed method

• The method employs geometric reconstruction to couple a nonlocal diffusion operator with a classical local PDE in one dimension, ensuring consistency at the interface.
• The coupling is formulated via a quasinonlocal operator that preserves linear consistency and satisfies the maximum principle.
• Well-posedness is proven using a quasinonlocal version of the Poincaré inequality, ensuring existence and uniqueness of solutions.
• Modeling error is estimated using the maximum principle, showing convergence to the local limit as the horizon parameter δ → 0.
• A first-order finite difference scheme is proposed for numerical discretization, preserving the physical properties of the continuous model.
• The method replaces volumetric boundary conditions with classical Dirichlet conditions, eliminating non-physical boundary layers in the nonlocal region.

Experimental results

Research questions

• RQ1 Can a nonlocal-to-local coupling method preserve flux balance, energy conservation, and the maximum principle simultaneously?
• RQ2 How can interfacial inconsistencies and ghost forces be eliminated in nonlocal-local coupling of diffusion models?
• RQ3 What is the modeling error between the coupled nonlocal-local solution and the local limit as δ → 0?
• RQ4 Can the geometric reconstruction framework be adapted to nonlocal diffusion models to ensure physical consistency?
• RQ5 How does the proposed coupling perform numerically when classical Dirichlet boundary conditions are used?

Key findings

• The proposed quasinonlocal coupling method ensures flux balance, energy conservation, and the maximum principle—properties not satisfied by existing coupling methods.
• Well-posedness of the coupled problem is rigorously proven using a quasinonlocal Poincaré inequality.
• Modeling error between the coupled solution and the local limit is estimated via the maximum principle, showing convergence as δ → 0.
• Numerical tests confirm first-order convergence in both solution and gradient under the L∞ norm, with orders of 0.99–1.00.
• The method successfully removes artificial boundary layers caused by volumetric constraints in fully nonlocal models when using classical Dirichlet boundary conditions.
• The local-nonlocal-local coupling formulation accurately captures singular behavior in the source term while matching local solutions on both sides of the nonlocal region.