Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Numerical stochastic homogenization by quasilocal effective diffusion tensors

Pages: 637 – 651



Dietmar Gallistl (Department of Applied Mathematics, University of Twente, Enschede, The Netherlands)

Daniel Peterseim (Institut für Mathematik, Universität Ausgburg, Germany)


This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.


numerical homogenization, multiscale method, upscaling, a priori error estimates, a posteriori error estimates, uncertainty, modeling error estimate, model reduction

2010 Mathematics Subject Classification

35R60, 65N12, 65N15, 65N30, 74Q05

Copyright © 2019 Dietmar Gallistl and Daniel Peterseim

D. Gallistl acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

Received 13 April 2017

Accepted 22 January 2019

Published 30 August 2019