Elliptic Homogenization Problems Research Articles

Finite element approximation of elliptic homogenization problems in nondivergence-form

We use uniform W2,p estimates to obtain corrector results for periodic homogenization problems of the form A(x/ε):D2uε = f subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.

Homogenization and exact controllability for problems with imperfect interface

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual \begin{document}$ L^2 $\end{document} , in an \begin{document}$ \varepsilon $\end{document} -periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.

Asymptotically Compatible Schemes for Stochastic Homogenization

In this paper, we study the numerical solution of elliptic homogenization problems with stationary and ergodic coefficients characterized by a small length scale $\varepsilon$. The issue of asympto...

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {Le} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in ℝn. We are able to show that the uniform W1,p estimate of second order elliptic systems holds for \(\frac{{2n}}{{n + 1}} - \delta 0 is independent of e and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W1,p estimate is true for \(\frac{3}{2} - \delta < p < 3 + \delta \) if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞.

Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains

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.

A time dependent approach for removing the cell boundary error in elliptic homogenization problems

This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O ( e / ? ) error in the computation, where e is the size of the microscopic variations in the media and ? is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of e / ? in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O ( e / ? ) error to O ( e ) in general non-periodic media.

Convergence Rates for General Elliptic Homogenization Problems in Lipschitz Domains

The paper extends the results obtained by Kenig, Lin, and Shen [Arch. Ration. Mech. Anal., 203 (2012), pp. 1009--1036] to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no smoothness on the coefficients. We do not repeat their arguments. Instead we find the new weighted-type estimates for the smoothing operator at scale $\varepsilon$, and combining some techniques developed by Shen in [preprint, arXiv:1505.00694v1, 2015] leads to our main results. In addition, we also obtain sharp $O(\varepsilon)$ convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Shen for elasticity systems in [preprint, arXiv:1505.00694v1, 2015]. Also, this work may be regarded as the extension of [T. Suslina, Mathematika, 59 (2013), pp. 463--476; T. Suslina SIAM J. Math. Anal., 45 (2013), pp. 3453--3493] developed by Suslina concerned with the bounded Lipschitz domain.

Uniform bound and convergence for elliptic homogenization problems

Uniform bound and convergence for the solutions of elliptic homogenization problems are concerned. The problem domain has a periodic microstructure; it consists of a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let \(\epsilon \in (0,1)\) denote the size ratio of the period to the whole domain, and let \(\omega ^2\in (0,1)\) denote the permeability ratio of the disconnected matrix block subset to the connected subregion. For elliptic equations with diffusion depending on the permeability, the elliptic solutions are smooth in the connected subregion but change rapidly in the disconnected matrix block subset. More precisely, the solutions in the connected subregion can be bounded uniformly in \(\omega ,\epsilon \) in Holder norm, but not in the matrix block subset. It is known that the elliptic solutions converge to a solution of some homogenized elliptic equation as \(\omega ,\epsilon \) converge to 0. In this work, the \(L^p\) convergence rate for \( p\in (2,\infty ]\) is derived. Depending on strongly coupled or weakly coupled case, the convergence rate is related to the factors \(\omega ,\epsilon ,\frac{\omega }{\epsilon }\) for the former and related to the factors \(\omega ,\epsilon \) for the latter.

An isogeometric analysis for elliptic homogenization problems

A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rationalB-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Besides the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Furthermore, the nearly optimal quadrature rule for IGA (Auricchio et al., 2012) introduced recently is utilized to reduce significantly the number of micro problems, which is the main factor contributing into the computational cost in heterogeneous multiscale method. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. An efficient coupling between degrees of macro and micro basis functions is introduced. Numerical results show the excellent performance of the proposed method.

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

A fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed. In contrast to previous results obtained for such problems in dimension $d\leq2$ for the $H^1$ norm and for a semi-discrete formulation [W.E, P. Ming and P. Zhang, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156], we obtain optimal convergence results for dimension $d\leq3$ and for a fully discrete method, which takes into account the microscale discretization. In addition, our results are also valid for quadrilateral finite elements, optimal a-priori error estimates are obtained for the $H^1$ and $L^2$ norms, improved estimates are obtained for the resonance error and the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems.

Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

A reduced basis nite element heterogeneous multiscale method (RB-FE-HMM) for a class of nonlinear homogenization elliptic problems of nonmonotone type is introduced. In this approach, the solutions of the micro problems needed to estimate the macroscopic data of the homogenized problem are selected by a Greedy algorithm and computed in an o ine stage. It is shown that the use of reduced basis (RB) for nonlinear numerical homogenization reduces considerably the computational cost of the nite element heterogeneous multiscale method (FE-HMM). As the precomputed microscopic functions depend nonlinearly on the macroscopic solution, we introduce a new a posteriori error estimator for the Greedy algorithm that guarantees the convergence of the online Newton method. A priori error estimates and uniqueness of the numerical solution are also established. Numerical experiments illustrate the e ciency of the proposed method.

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

AbstractWe present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Adaptive reduced basis finite element heterogeneous multiscale method

An adaptive reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is proposed for elliptic problems with multiple scales. The multiscale method is based on the RB-FE-HMM introduced in [A. Abdulle, Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys. 231 (21) (2012) 7014–7036]. It couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains. Unlike classical numerical homogenization methods, the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions (the reduced basis) obtained by a greedy algorithm in an offline stage. In this paper we present a residual-based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest. For both type of adaptive strategies, rigorous a posteriori error estimates are derived and corresponding error estimators are proposed. In contrast to the adaptive finite element heterogeneous multiscale method (FE-HMM), there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement. Up to an offline preliminary stage, the RB-FE-HMM has the same computational complexity as a standard adaptive FEM for the effective problem. Two and three dimensional numerical experiments confirm the efficiency of the RB-FE-HMM and illustrate the improvements compared to the adaptive FE-HMM.

Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs

In this paper Monte Carlo finite element approximations for elliptic homogenization problems with random coefficients, which oscillate on $n\in\mathbb{N}$ a priori known, separated microscopic length scales, are considered. The convergence of multilevel Monte Carlo finite element discretizations is analyzed. In particular, it is considered that the multilevel finite element discretization resolves the finest physical length scale, but the coarsest finite element mesh does not, so that the so-called resonance case occurs at intermediate multilevel Monte Carlo sampling levels. It is shown that for first order finite elements in two space dimensions, the multilevel Monte Carlo finite element method converges at the same rate as the corresponding single-level Monte Carlo finite element method, despite the majority of samples being underresolved in the multilevel Monte Carlo finite element estimator. It is proved that switching to a hierarchic multiscale finite element method such as the finite element heterogeneous multiscale method to compute the multilevel Monte Carlo finite element estimator, when only meshes are used which underresolve all physical length scales, implies optimal convergence. Specifically, both methods proposed here allow one to obtain estimates of the expectation of the random solution, with accuracy versus work that is identical to the solution of a single deterministic problem. In the case of the finite element heterogeneous multiscale method the estimate is, moreover, robust with respect to the physical length scales. Numerical experiments corroborate our analytical findings.

Error Control Based Model Reduction for Parameter Optimization of Elliptic Homogenization Problems

In this work we are considered with parameter optimization of elliptic multiscale problems with macroscopic optimization functionals and microscopic material design parameters. An efficient approximation is obtained by the reduced basis approach. A posteriori error estimates for the reduced forward problem are obtained in the periodic homogenization setting, using the so called two scale weak formulation of the multiscale problem. The resulting error indicators allow for an efficient offline/online decomposition and are used for an efficient reduced basis construction, both for the homogenization limit, as well as for the approximation of the corresponding cell problems.

Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems

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].

Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

A new finite element method for the efficient discretization of elliptic homogenization problems is proposed. These problems, characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed. Recently, the finite element heterogeneous multiscale method (FE-HMM) has been proposed for such problems, based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh. Departing from the approach used in the FE-HMM, we show that interpolation techniques based on the reduced basis methodology (an offline-online strategy) allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions. This new method, called the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is significantly more efficient than the FE-HMM for high order macroscopic discretizations and for three-dimensional problems, when the repeated computation of micro problems over the whole computational domain is expensive. A priori error estimates of the RB-FE-HMM are derived. Numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method.

Convergence Rates in L 2 for Elliptic Homogenization Problems

We study rates of convergence of solutions in L 2 and H 1/2 for a family of elliptic systems $${\{\mathcal{L}_\varepsilon\}}$$ with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of $${\{\mathcal{L}_\varepsilon\}}$$ . Most of our results, which rely on the recently established uniform estimates for the L 2 Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains.

Layer potential methods for elliptic homogenization problems

AbstractIn this paper we use the method of layer potentials to study L2 boundary value problems in a bounded Lipschitz domain Ω for a family of second‐order elliptic systems with rapidly oscillating periodic coefficients. Defining ${\cal L}_\varepsilon = - {\rm div}(A(\varepsilon ^{ - 1} X)\nabla )$, under the assumption that A(X) is elliptic, symmetric, periodic, and Hölder‐continuous, we establish the solvability of the L2 Dirichlet, regularity, and Neumann problems for ${\cal L}_\varepsilon (u_\varepsilon ) = 0$ in Ω with optimal estimates uniform in ε &gt; 0. © 2010 Wiley Periodicals, Inc.

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.