A Splice Method for Local-to-Nonlocal Coupling of Weak Forms

TL;DR

This work introduces a splice method to couple nonlocal diffusion with a local model using weak formulations, enabling efficient finite element discretization while preserving desirable properties such as patch tests and asymptotic compatibility. The core idea is to construct a non-symmetric splicing matrix that combines local and nonlocal variational problems on overlapping subdomains, reducing the need for expensive full nonlocal assembly and avoiding intrusive kernel modifications. The method is shown to be well-posed in the discrete setting and is equivalent to a discretized optimization-based coupling, ensuring existence, uniqueness, and the ability to leverage optimization-based analysis. Numerical results in 1D and 2D demonstrate patch test satisfaction, convergence to local models as the horizon shrinks, and robustness in problems with discontinuities and time dependence. The work also discusses nontrivial considerations in choosing the local/nonlocal split and provides a pathway for extending the approach to time-dependent and multi-physics contexts.

Abstract

We propose a method to couple local and nonlocal diffusion models. By inheriting desirable properties such as patch tests, asymptotic compatibility and unintrusiveness from related splice and optimization-based coupling schemes, it enables the use of weak (or variational) formulations, is computationally efficient and straightforward to implement. We prove well-posedness of the coupling scheme and demonstrate its properties and effectiveness in a variety of numerical examples.

Figures (21)

• Figure 1: Sketch of geometric configurations of the subdomains: left-right splitting on the left and a nonlocal inclusion on the right; in top: 1D, bottom: 2D The red/blue highlighted boundaries are $\Gamma, \widetilde{\Gamma}$ respectively. Note that for simplicity, the domains were chosen such that the overlap $\Omega_{L}\cap\Omega_{N}$ is empty.
• Figure 2: Figure illustrating the degrees of freedom and the corresponding index sets. The color green corresponds to $\mathcal{I}_{L}$, $\mathcal{I}_{N}$, red corresponds to $\mathcal{I}_{\Gamma}$, $\mathcal{I}_{{N, \mathcal{I}}}$ and blue corresponds to $\mathcal{I}_{\widetilde{\Gamma}}$, $\mathcal{I}_{{N, \widetilde{\mathcal{I}}}}$ in the local/nonlocal case respectively. Note that $\mathcal{I}_{\Gamma}\subset \mathcal{I}_{N}$ and $\mathcal{I}_{{N, \mathcal{I}}}\subset \mathcal{I}_{L}$. We show two local-nonlocal splittings. The top row shows $\Omega=(0,2)\times(0,1)$ with $\Omega_L = (0, 1)^{2}$. The bottom row shows $\Omega=(-1,1)^{2}$ with $\Omega_L = \Omega\setminus[-.25, .25]^2$.
• Figure 3: Figure illustrating the splice matrix for a 1D example. The top half rows correspond to dofs in the local subdomain, resulting in a bandwidth of 3 while the bottom half are dofs on the nonlocal subdomain.
• Figure 4: Example meshes for the non-matching meshes coupling. See \ref{['rem:domain-nonmatching']} regarding the notation of the domains. In the 1D case, the domains are $\Omega_L = (-1, 0), \Omega_N = [0, 1)$ and for 2D they are $\Omega_L = (-1, 0) \times (-1, 1), \Omega_N = [0, 1) \times (-1, 1)$ and $\Omega_L = (-1, 1)^2 \setminus [-.25, .25]^2, \Omega_N = (-.25 , .25 )^2$.
• Figure 5: The domains of the optimization method. Left: matching $\mathbb{P}_{1}-\mathbb{P}_{1}$ discretizations. Right: non-matching $\mathbb{P}_{1}-\mathbb{P}_{0}$ discretizations. The blue color edges corresponds to the support of $\Theta_{L}$, the green area corresponds to the support of $\Theta_{N}$. Green and yellow area together constitute $\Omega_b$.

Theorems & Definitions (24)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 4.1
• Theorem 4.1
• Lemma A.1: Strong Local Maximum Principle
• proof
• Lemma A.2: Weak maximum principle for the nonlocal problem
• Remark A.1
• Remark A.2