Related papers: Finite Element Approximation of Elliptic Homogeniz…
We study the homogenization of the equation $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain $\Omega\subset \mathbb{R}^n$ subject to a Dirichlet boundary condition and the numerical approximation of…
We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for…
We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in…
For a family of second-order elliptic systems in divergence form with rapidly oscillating almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the…
We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative…
In this paper we investigate the homogenization problem with a non-homogeneous Dirichlet condition. Our aim is to give error estimates with boundary data in $H^{1/2}(\partial\Omega)$. The tools used are those of the unfolding method in…
We develop a quantitative theory of stochastic homogenization in the more general framework of differential forms. Inspired by recent progress in the uniformly elliptic setting, the analysis relies on the study of certain subadditive…
This paper is concerned with the homogenization of Dirichlet problem of elliptic systems in a bounded, smooth domain of finite type. Both the coefficients of the elliptic operator and the Dirichlet boundary data are assumed to be periodic…
In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial…
A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such…
In this paper, we consider stochastic homogenization of elliptic equations with unbounded and non-uniformly elliptic coefficients. Extending subadditive arguments, we get an estimate for the rate of the convergence of the solution of the…
We consider an homogenization problem for the second order elliptic equation $- \Delta u^{\varepsilon} + \dfrac{1}{\varepsilon} V(./\varepsilon) u^{\varepsilon} + \nu u^{\varepsilon} =f$ when the highly oscillatory potential $V$ belongs to…
We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$…
We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media…
This paper is devoted to the study of uniform $W^{1,\frac{np}{n-p}}$- and $W^{2,p}$-estimates for viscosity solutions to fully nonlinear, uniformly elliptic, periodic homogenization problems, up to boundaries, subject to Dirichlet boundary…
This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order…
In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex…
In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic operator $A_\varepsilon$ of order $2m\ge 4$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the…
For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the…
Homogenization is studied for a nonlinear elliptic boundary-value problem with a large nonlinear potential. More specifically we are interested in the asymptotic behavior of a sequence of p-Laplacians of the form $$…