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 $\overline{D}$ is the union of cells $\{\overline{D_i}{\}}_{i\in I}$ 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 $i\in I$ relates to the index of the cell containing the point $x$ and $y\in Y$ 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 ${\mathbb{R}}^I\otimes V(Y)$ of functions defined over the product set $I\times Y$. Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.

Changed the choice of test spaces V(D) and X (with regard to regularity) and the argumentation thereof. Corrected proof of proposition 3. Corrected wrong multiplicative factor in proposition 4 and its proof (was 2 instead of 1). Added remark 6 at the end of section 2. Extended remark 7. Added references. Some minor improvements (typos, typesetting)