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

Ayoul-Guilmard, Quentin; Nouy, Anthony; Binetruy, Christophe

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 869-891.

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 $\left\{{\overline{D}}_i\right\}i\in I$ and we introduce a two-scale representation by identifying any function $v\left(x\right)$ 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 $V\left(D\right)$ using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying $V\left(D\right)$ with a tensor product space ${ℝ}^1\otimes V\left(Y\right)$ 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.

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