Problems In Perforated Domains Research Articles

Uniform W1,p estimates and large-scale regularity for Dirichlet problems in perforated domains

In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, where ε represents the scale of the minimal distances between holes and η the ratio between the scale of sizes of holes and ε. We establish W1,p estimates for solutions with bounding constants depending explicitly on ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities

A boundary-value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain with periodic perforation by small cavities arranged along a fixed hypersurface at small distances one from another. The distances are proportional to a small parameter $\varepsilon$, and the linear sizes of the cavities are proportional to $\varepsilon\eta(\varepsilon)$, where $\eta(\varepsilon)$ is a function taking values in the interval $[0,1]$. The main result is a complete asymptotic expansion for the solution of the perturbed problem. The asymptotic expansion is a combination of an outer and an inner expansion; it is constructed using the method of matched asymptotic expansions. Both outer and inner expansions are power expansions in $\varepsilon$ with coefficients depending on $\eta$. These coefficients are shown to be infinitely differentiable with respect to $\eta\in(0,1]$ and uniformly bounded in $\eta\in[0,1]$. Bibliography: 38 titles.

Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities

В работе рассматривается краевая задача для эллиптического уравнения второго порядка с переменными коэффициентами в многомерной области, периодически перфорированной вдоль заданной гиперплоскости малыми полостями, расположенными на малых расстояниях друг от друга. Расстояния пропорциональны малому параметру $\varepsilon$, линейные размеры полостей - величине $\varepsilon\eta(\varepsilon)$, где $\eta(\varepsilon)$ - некоторая функция со значениями в отрезке $[0,1]$. Основной результат работы - полное асимптотическое разложение решения возмущенной задачи. Асимптотика строится на основе метода согласования асимптотических разложений в виде комбинации внешнего и внутреннего разложений. Оба этих разложения являются степенными по малому параметру $\varepsilon$ с коэффициентами, зависящими от $\eta$. Показано, что эти коэффициенты бесконечно дифференцируемы по $\eta\in(0,1]$ и равномерно ограничены по $\eta\in[0,1]$. Библиография: 38 названий.

DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

Wavelet-based Edge Multiscale Finite Element Method for Helmholtz problems in perforated domains

We introduce a new efficient algorithm for Helmholtz problems in perforated domains with the design of the scheme allowing for possibly large wavenumbers. Our method is based upon the wavelet-based...

Multiscale model reduction for the Allen–Cahn problem in perforated domains

In this paper, we consider a class of multiscale methods for the solution of nonlinear problem in perforated domains. These problems are of multiscale nature and their discretizations lead to large nonlinear systems. To discretize these problems, we construct a fine grid approximation using the finite element method with implicit time approximation and the Newton’s method. In order to solve these large systems efficiently, we will develop a model reduction procedure. To perform the model reduction, we construct a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM). The GMsFEM consists of an offline and online stages. In the offline stage, we construct multiscale basis functions based on the solution of some local spectral problems defined in the snapshot space. Then, we enrich the offline multiscale space by additional multiscale basis that handle non-homogeneous boundary condition. For the accurate solution of the nonlinear problem, we use two techniques on the online stage: (1) residual based multiscale basis functions and (2) residual based local correction. We will present numerical results for two-dimensional Allen–Cahn problems in perforated domains.

Multiscale hybridizable discontinuous Galerkin method for elliptic problems in perforated domains

We propose a multiscale hybridizable discontinuous Galerkin method in perforated domains. Perforations may have various shapes and sizes. Following Generalized Multiscale Finite Element method, local snapshot spaces are constructed by solving local spectral problems. Then multiscale finite element spaces for numerical traces are built on the coarse edges within the framework of hybridizable discontinuous Galerkin method. Some representative examples are presented to investigate numerical performance of the proposed method.

Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC)

In this paper, we present an upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations. Our methodology is based on the recently developed Non-local multicontinuum method (NLMC). The main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non-local effects. We will construct multiscale basis functions for the coarse regions and additional multiscale basis functions for perforations, with the aim of handling non-homogeneous boundary conditions on perforations. We start with describing our method for the Laplace equation, and then extending the framework for the elasticity problem and parabolic equations. The resulting upscaled model has minimal size and the solution has physical meaning on the coarse grid. We will present numerical results (1) for steady and unsteady problems, (2) for Laplace and Elastic operators, and (3) for Neumann and Robin non-homogeneous boundary conditions on perforations. Numerical results show that the proposed method can provide good accuracy and provide significant reduction on the degrees of freedoms.

Nonlocal problems in perforated domains

AbstractIn this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L∞(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

Homogenization of a class of singular elliptic problems in perforated domains

In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by ε-periodic holes of size ε. The quasilinear equations present a nonlinear singular lower order term fζ(uε), where uε is the solution of the problem at ε-level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new.The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of uε behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence {uε}, as ε→0. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem.The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as fζ(uε). This plays an essential role in order to get an almost everywhere convergence of the solution uε, needed in the study the asymptotic behavior of the problem.

Multiscale model reduction for transport and flow problems in perforated domains

Convection-dominated transport phenomenon is important for many applications. In these applications, the transport velocity is often a solution of heterogeneous flow problems, which results to a coupled flow and transport phenomena. In this paper, we consider a coupled flow (Stokes problem) and transport (unsteady convection–diffusion problem) in perforated domains. Perforated domains (see Fig. 1) represent void space outside hard inclusions as in porous media, filters, and so on. We construct a coarse-scale solver based on Generalized Multiscale Finite Element Method (GMsFEM) for a coupled flow and transport. The main idea of the GMsFEM is to develop a systematic approach for computing multiscale basis functions. We use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation. For the transport problem, we use Petrov–Galerkin mixed formulation, which provides a stability. As a first approach, we use the multiscale flow solution in constructing the basis functions for the transport equation. In the second approach, we construct multiscale basis functions for coupled flow and transport without solving global flow problem. The novelty of this approach is to construct a coupled multiscale basis function. Numerical results are presented for computations using offline basis. We also present an algorithm for adaptively adding online multiscale basis functions, which are computed using the residual information. Numerical examples using online GMsFEM show the speed up of convergence.

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.

Existence and uniqueness results for a class of singular elliptic problems in perforated domains

In this paper we prove some existence, uniqueness and regularity results for a weak solution of a quasilinear elliptic nonlinear problem posed in a domain perforated by one or more holes. The nonlinear term is of the form \(f\zeta (u)\), where \(\zeta (s)\) is singular at \(s=0\). On the boundary of the holes we impose a nonlinear Robin condition, while on the exterior boundary we prescribe a homogeneous Dirichlet condition. The difficulty here arises in dealing simultaneously with the quasilinear matrix field, the singular datum and the nonlinear Robin condition. To show the existence of a solution we approximate the problem with a sequence of nonsingular problems for which the existence of a solution is proved via the Schauder fixed-point theorem. The main tool when passing to the limit in the approximate problem is to split the integral of the singular term into the sum of two integrals, one on the set where the solution is very close to the singularity and one where it is far from it. To obtain the uniqueness of the solution we need to require additional assumptions on the quasilinear term and a monotonicity property for the singular one. Under suitable stronger hypotheses on the data, we also prove the boundedness of the solution.

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.

Mixed GMsFEM for second order elliptic problem in perforated domains

We consider a class of second order elliptic problems in perforated domains with homogeneous Neumann boundary condition. It is well-known that numerically solving these problems require a very fine computational mesh, and some model reduction techniques are therefore necessary. We will develop a new model reduction technique based on the generalized multiscale finite element method (GMsFEM). The GMsFEM has been applied successfully to second order elliptic problems in perforated domains with Dirichlet boundary conditions Chung et al. (2015). However, due to the use of multiscale partition of unity functions, the same method cannot be applied to the case with Neumann boundary conditions. The aim of this paper is to develop a new mixed GMsFEM, based on a piecewise constant approximation for pressure and a multiscale approximation for velocity, giving a mass conservative method. The method can handle the Neumann boundary condition naturally. The multiscale basis functions for velocity are constructed by some carefully chosen local snapshot spaces and local spectral decompositions. The spectral convergence of the method is analyzed. Moreover, by using some local error indicators, the basis functions can be added locally and adaptively. We also consider an online procedure for the construction of new basis functions in the online stage in order to capture the distant effects. We will present some numerical examples to show the performance of the method.

Mesoscale Models and Approximate Solutions for Solids Containing Clouds of Voids

For highly perforated domains the paper addresses a novel approach to study mixed boundary value problems for the equations of linear elasticity in the framework of mesoscale approximations. There are no assumptions of periodicity involved in the description of the geometry of the domain. The size of the perforations is small compared to the minimal separation between neighboring defects and here we discuss a class of problems in perforated domains, which are not covered by the homogenization approximations. The mesoscale approximations presented here are uniform. Explicit asymptotic formulas are supplied with the remainder estimates. Numerical illustrations, demonstrating the efficiency of the asymptotic approach developed here, are also given.

Homogenization of optimal control problems in perforated domains via periodic unfolding method

This article aims to study the limiting behaviors of optimal control problems based on an elliptic boundary value problem with highly oscillating coefficients in a periodically perforated domain. We consider two different types of cost functionals, the -cost functional and the Dirichlet cost functional. We use the periodic unfolding method for perforated domains to homogenize the problems. Moreover, we prove that under this method, only the energy corresponding to the former cost functional converges.

Homogenization and correctors of a class of elliptic problems in perforated domains

Homogenization and correctors of a class of elliptic problems in perforated domains

Homogenization of a class of elliptic problems with nonlinear boundary conditions in domains with small holes

We consider a class of second order elliptic problems in a domain of RN , N > 2, ε-periodically perforated by holes of size r(ε) , with r(ε)/ε → 0 as ε → 0. A nonlinear Robin-type condition is prescribed on the boundary of some holes while on the boundary of the others as well as on the external boundary of the domain, a Dirichlet condition is imposed. We are interested in the asymptotic behavior of the solutions as ε → 0. We use the periodic unfolding method introduced in [Cioranescu, D., Damlamian, A. and Griso, G., Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 99–104]. The method allows us to consider second order operators with highly oscillating coefficients and so, to generalize the results of [Cioranescu, D., Donato, P. and Zaki, R., Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal., Vol. 53 (2007), No. 4, 209–235].

Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions

This article is devoted to the homogenization of a quasilinear elliptic equation with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, depending on a real parameter γ ≥ 1. We suppose that the data satisfy some suitable hypotheses which ensure, as proved by the authors in Cabarrubias and Donato [B. Cabarrubias and P. Donato, Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math. (2) (2011) (to appear)], the existence and the uniqueness of a solution of the problem. In particular, suitable growth conditions are assumed on the nonlinear boundary term, as done in Cioranescu–Donato–Zaki [D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), pp. 209–235]. On the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [M. Chipot, Elliptic Equations: An Introductory Course, Birkhauser Verlag AG, Germany, 2009] and weaker than a Lipschitz condition are prescribed. We study the convergence to a limit problem, which is identified by using the periodic unfolding method. We also prove the well-posedness of the limit system. To do that, we show that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows.