Nonlocal $H$-convergence

TL;DR

This paper introduces the concept of nonlocal H-convergence using operator complex theory, establishing its uniqueness, characterisation, and applications to Maxwell's equations, nonlocal response models, and homogenisation problems.

Contribution

It defines nonlocal H-convergence, proves its fundamental properties, and demonstrates its applicability across various complex physical and mathematical models.

Findings

Nonlocal H-convergence is well-defined and unique.

Local and nonlocal H-convergence coincide for multiplication operators.

Applications include Maxwell's equations, nonlocal response theory, and homogenisation on manifolds.

Abstract

We introduce the concept of nonlocal $H$-convergence. For this we employ the theory of abstract closed complexes of operators in Hilbert spaces. We show uniqueness of the nonlocal $H$-limit as well as a corresponding compactness result. Moreover, we provide a characterisation of the introduced concept, which implies that local and nonlocal $H$-convergence coincide for multiplication operators. We provide applications to both nonlocal and nonperiodic fully time-dependent 3D Maxwell's equations on rough domains. The material law for Maxwell's equations may also rapidly oscillate between eddy current type approximations and their hyperbolic non-approximated counter parts. Applications to models in nonlocal response theory used in quantum theory and the description of meta-materials, to fourth order elliptic problems as well as to homogenisation problems on Riemannian manifolds are provided.