Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

Patrick Henning; Mario Ohlberger

doi:10.3934/dcdss.2015.8.119

Article Contents

2015, Volume 8, Issue 1: 119-150. Doi: 10.3934/dcdss.2015.8.119

This issue Previous Article Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems Next Article Two-Scale numerical simulation of sand transport problems

Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

Patrick Henning^1, and
Mario Ohlberger^2,

1.
Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala
2.
Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster

Received: February 2012

Revised: April 2013

Published: February 2015

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Abstract

In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the $L^2$-error between the HMM approximation and the solution of the macroscopic limit equation. The a posteriori error estimate is obtained in a general heterogeneous setting with scale separation without assuming periodicity or stochastic ergodicity. The applicability of the method and the usage of the a posteriori error estimate for adaptive local mesh refinement is demonstrated in numerical experiments. The experimental results underline the applicability of the a posteriori error estimate in non-periodic homogenization settings.

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Mathematics Subject Classification: Primary: 35J60, 65G99, 74Q15; Secondary: 65N30.

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