Numerical upscaling for heterogeneous materials in fractured domains
Journal article, 2021

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.


Darcy flow

Porous media

Generalized finite element method

Localized orthogonal decomposition


Fredrik Hellman

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Axel Målqvist

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Siyang Wang

Mälardalens högskola

Mathematical Modelling and Numerical Analysis

0764-583X (ISSN) 1290-3841 (eISSN)

Vol. 55 S761-S784

Subject Categories

Computational Mathematics

Control Engineering

Mathematical Analysis



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