A nonlocal coupled system: analysis and discretization

TL;DR

This paper studies a nonlocal coupled system involving regional fractional Laplacians, proving existence, regularity, and convergence of a finite element discretization with an efficient Schwarz-type solver, supported by numerical validation.

Contribution

It introduces a new analysis of a nonlocal coupled system with fractional Laplacians, along with a finite element discretization and an efficient Schwarz-type iterative method.

Findings

Existence and uniqueness of the energy minimizer are established.

A priori error estimates for the finite element discretization are derived.

The Schwarz-type method converges geometrically and is validated numerically.

Abstract

We analyze a nonlocal coupled system arising as the Euler--Lagrange equations of an energy functional involving regional fractional Laplacians of orders $s_1$ and $s_2$ ($ 0 < s_1,s_2 < 1$), each acting on a separate disjoint domain and coupled through a nonlocal interaction term depending on a kernel $J$. Under suitable assumptions on the domains and the kernel, we prove existence and uniqueness of the energy minimizer and derive regularity estimates in fractional Sobolev spaces. We introduce a finite element discretization and establish a priori error estimates. We develop an alternating Schwarz-type method for both the continuous and discrete problems and prove its geometric convergence. Numerical experiments validate the theoretical predictions and illustrate the performance of the method.