We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share “nonlocal” interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. This work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.

我们提出了一种有效模拟非局部问题的域分解方法。我们的方法基于非局部扩散问题的多域表述，其中子域共享非局部视界大小的“非局部”界面。该非局部方程系统首先根据非局部能量的最小化进行重写，然后使用无网格近似进行离散化，最后通过拉格朗日乘子方法以类似于有限元撕裂和互连方法的方式求解。具体来说，我们提出了一种用于求解拉格朗日乘子系统的分布式投影梯度算法，其未知数决定了非局部界面条件子域之间。对多达 1.91 亿个未知数的问题进行的几次二维数值测试说明了我们算法的强和弱可扩展性，它优于问题的分布式数值解的标准方法。这项工作是在二维多域设置中对具有有限范围的非局部算子进行的第一次严格数值研究，因此，它是在大规模模拟中增加非局部模型使用的基本步骤。