A domain decomposition scheme for couplings between local and nonlocal equations

TL;DR

This paper introduces a domain decomposition method for coupling local and nonlocal equations, demonstrating its convergence in both continuous and discrete frameworks, thereby advancing numerical solutions for hybrid PDE models.

Contribution

The paper develops and proves convergence of a Schwarz-type domain decomposition scheme specifically designed for local-nonlocal equation couplings, integrating Lion's framework.

Findings

Method converges in continuous setting

Method converges in discrete setting

Applicable to certain classes of local-nonlocal couplings

Abstract

We study a natural alternating method of Schwarz type (domain decomposition) for certain class of couplings between local and nonlocal operators. We show that our method fits into Lion's framework and prove, as a consequence, convergence in both, the continuous and the discrete settings.