Nonlocal H-convergence

Waurick, Marcus (2018) Nonlocal H-convergence. Calculus of Variations and Partial Differential Equations, 57. 159. ISSN 1432-0835

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    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.