Analysis of the heterogeneous multiscale method for elliptic homogenization problems

Erratum to “A short and versatile finite element multiscale code for homogenization problems” [Comput. Methods Appl. Mech. Engrg. 198 (2009) 2839–2859

Elliptic homogenization problems in a domain $\Omega \subset \mathbb{R}^d$ with n + 1 separated scales are reduced to elliptic one-scale problems in dimension (n + 1)d. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work, and memory requirements comparable to those in a standard FEM for single-scale problems in $\Omega$, while it gives numerical approximations of the correct homogenized limit as well as of all first-order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results.

We analyze the random fluctuations of several multiscale algorithms, such as the multiscale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multiscale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one-dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multiscale algorithms in the presence of random environments with short-range and long-range correlations. For a given mesh size , we show that the fluctuations converge in distribution in the space of continuous paths to Gaussian processes as the correlation length . We next derive the limit of such Gaussian processes as and compare this limit with the distribution of the random fluctuations of the continuous model. When such limits agree, we conclude that the multiscale algorithm captures the random fluctuations accurately and passes the corrector test. This property serves as an interesting benchmark to assess the behavior of the multiscale algorithm in practical situations where the assumptions necessary for the theory of homogenization are not met. What we find is that the computationally more expensive methods MsFEM, and HMM with a choice of parameter , correctly capture the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM with correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.

Numerical solution of penalty formulations for constrained mechanical systems using heterogeneous multiscale methods

In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the $\mathbf{H}({curl})$- and the $H^{-1}$-norm, and we derive reliable and efficient localized residual-based a posteriori error estimates. Numerical experiments are presented to verify the a priori convergence results.

We propose an efficient heterogeneous multiscale finite element method based on a local least-squares reconstruction of the effective matrix using the data retrieved from the solution of cell problems posed on the vertices of the triangulation. The method achieves high order accuracy for high order macroscopic solver with essentially the same cost as the linear macroscopic solver. Optimal error bounds are proved for the elliptic problem. Numerical results demonstrate that the new method significantly reduces the cost without loss of accuracy.

A finite-element discretization, independent of the location of the interface, is proposed and analysed for linear elliptic and parabolic interface problems. We establish error estimates of optimal order in the H 1 -norm and almost optimal order in the L 2 -norm for elliptic interface problems. An extension to parabolic interface problems is also discussed and an optimal error estimate in the L 2 (0, T; H1 (Ω))-norm and an almost optimal order estimate in the L 2 (0, T; L 2 (Ω))-norrn are derived for the spatially discrete scheme. A fully discrete scheme based on the backward Euler method is analysed and an optimal order error estimate in the L 2 (0, T; H I (Ω))-norm is derived. The interfaces are assumed to be of arbitrary shape and smooth for our purpose.

Computation of effective resistivity in materials with microinclusions by a heterogeneous multiscale finite element method

Numerical methods for stochastic partial differential equations with multiple scales

Heterogeneous multiscale method for high energy-density matter: Connecting kinetic theory and molecular dynamics

In this paper, we suggest a new heterogeneous multiscale method (HMM) for the Helmholtz equation with high contrast. The method is constructed for a setting as in Bouchitte and Felbacq [C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 377-382], where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We revisit existing homogenization approaches for this special setting and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to the Helmholtz equation with discontinuous diffusion matrix. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and an a priori error estimate under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. Numerical experiments confirm our theoretical convergence results and examine the resolution condition. Moreover, the numerical simulation gives a good insight and explanation of the physical phenomenon of frequency band gaps.

We present improved convergence results for the boosting algorithm (BA) and demonstrate that an existing formulation of the heterogeneous multiscale methods (HMM) is accurate to first order only in the macro time step, regardless of the order of the numerical solvers employed. These results are obtained by considering the BA and two other formulations of HMM as special cases of a general formulation of HMM applied to dissipative stiff ordinary differential equations.

Spatial adaptive sampling in multiscale simulation

Heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients