A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems

TL;DR

This paper introduces a domain decomposition method using a FETI-like approach for meshfree discretizations of nonlocal problems, enabling scalable and efficient large-scale simulations.

Contribution

It presents the first rigorous multi-domain numerical method for 2D nonlocal operators with finite horizon, combining meshfree discretization with a Lagrange multiplier approach.

Findings

The proposed algorithm demonstrates strong and weak scalability.

It outperforms standard distributed solution methods.

The method is validated through several 2D numerical tests.

Abstract

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 illustrate the strong and 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.