Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|S761 - S784
|26 February 2021
Numerical upscaling for heterogeneous materials in fractured domains
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg 412 96, Sweden
2 Division of Applied Mathematics, UKK, Mälardalen University, Västerås 721 23, Sweden
* Corresponding author: firstname.lastname@example.org
Accepted: 11 August 2020
We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.
Mathematics Subject Classification: 35J15 / 65N12 / 65N15 / 65N30
Key words: Generalized finite element method / localized orthogonal decomposition / porous media / fracture / Darcy flow
© EDP Sciences, SMAI 2021
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