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### Open Access Journals

NHM

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].

NHM

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.

DCDS-S

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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