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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.
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https://hal.archives-ouvertes.fr/hal-02922752
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Submitted on : Wednesday, August 26, 2020 - 2:56:25 PM
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Quentin Ayoul-Guilmard, Anthony Nouy, Christophe Binetruy. Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media★. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2018, 52 (3), pp.869-891. ⟨10.1051/m2an/2018022⟩. ⟨hal-02922752⟩

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