The construction of periodic unfolding operators on some compact Riemannian manifolds
S. Dobberschütz, Michael Böhm · Advances in Pure and Applied Mathematics · 2014
The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the “flat” Euclidean space ℝn${\mathbb {R}^n}$. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of ℝn${\mathbb {R}^n}$ can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.