Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media

Abstract

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain D such that D̅ is the union of cells {D̅i}i∈I and we introduce a two-scale representation by identifying any function v(x) defined on D with a bi-variate function v(i,y), where i ∈ I relates to the index of the cell containing the point x and y ∈ Y relates to a local coordinate in a reference cell Y. We introduce a weak formulation of the problem in a broken Sobolev space V(D) using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying V(D) with a tensor product space ℝI⊗ V(Y) of functions defined over the product set I × Y. Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.