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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].
The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift
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.
Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems
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.
In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart--Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.
A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift
In this contribution we address a-posteriori error estimation in $L^\infty(L^2)$ for a heterogeneous multiscale finite element approximation of time-dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.
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