Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems
Patrick Henning
Networks & Heterogeneous Media 2012, 7(3): 503-524 doi: 10.3934/nhm.2012.7.503
In this work, we are concerned with the convergence of the multiscale finite element method (MsFEM) for elliptic homogenization problems, where we do not assume a certain periodic or stochastic structure, but an averaging assumption which in particular covers periodic and ergodic stochastic coefficients. We also give a result on the convergence in the case of an arbitrary coupling between grid size $H$ and a parameter $\epsilon$. $\epsilon$ is an indicator for the size of the fine scale which converges to zero. The findings of this work are based on the homogenization results obtained in [B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (4), 2011].
keywords: convergence MsFEM Elliptic homogenization multiscale methods.
The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift
Patrick Henning Mario Ohlberger
Networks & Heterogeneous Media 2010, 5(4): 711-744 doi: 10.3934/nhm.2010.5.711
This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.
keywords: Finite Element scheme multiscale methods HMM Advection-diffusion equation error estimate.
Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems
Patrick Henning Mario Ohlberger
Discrete & Continuous Dynamical Systems - S 2015, 8(1): 119-150 doi: 10.3934/dcdss.2015.8.119
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.
keywords: A posteriori estimate multiscale methods. HMM monotone operator

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