Related papers: Computational homogenization of time-harmonic Maxw…
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…
This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis --- which has been made possible by recent contributions…
In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed…
We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on…
We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + \lambda E = \partial_E F(x,E) &&\quad \text{in}\Omega\\ &\nu\times E = 0 &&\quad \text{on}\partial\Omega \end{aligned}…
This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional…
We present a numerical method for solving the free-space Maxwell's equations in three dimensions using compact convolution kernels on a rectangular grid. We first rewrite Maxwell's Equations as a system of wave equations with auxiliary…
We introduce a novel multi-resolution Localized Orthogonal Decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets…
We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite…
In this paper, we study the regularity of the solutions of Maxwell's equations in a bounded domain. We consider several different types of low regularity assumptions to the coefficients which are all less than Lipschitz. We first develop a…
We consider H(curl)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by…
In this work, we develop a numerical homogenization approach for the fully nonlinear Landau-Lifshitz equation with rough coefficients, including non-periodicity and nonseparable scales. Direct numerical resolution of such multiscale…
We derive $H_{\text{curl}}$-error estimates and improved $L^2$-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale…
We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or…
The paper deals with periodic homogenization problem for a para\-bo\-lic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic…
This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with…
This paper provides an a~priori error analysis of a localized orthogonal decomposition method (LOD) for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient…
We develop a numerical homogenization method for fourth-order singular perturbation problems within the framework of heterogeneous multiscale method. These problems arise from heterogeneous strain gradient elasticity and elasticity models…
We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete {\em a priori} error analysis for the case of…
When homogenizing elliptic partial differential equations, the so-called corrector problem is pivotal to compute the macroscale effective coefficients from the microscale information. To solve this corrector problem in the periodic setting,…