A nonlocal coupled system: analysis and discretization

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.