Related papers: Two-scale homogenization of abstract linear time-d…
The present paper concerns a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of corrector results (i.e., strong convergences of…
We develop an efficient $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and…
This paper is devoted to the homogenization of Shr\"odinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and…
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…
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…
Partial differential equations (PDEs) are central to computational electromagnetics (CEM) and photonic design, but classical solvers face high costs for large or complex structures. Quantum Hamiltonian simulation provides a framework to…
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,…
In an abstract Hilbert space setting, we discuss many linear phenomena of mathematical physics. The functional analytic framework presented is used to address continuous dependence of the solution operators $\mathcal{S}(\mathcal{M})$ of…
We develop the stochastic two-scale convergence method in the framework of Orlicz-Sobolev spaces, in order to deal with the homogenization of coupled stochastic-periodic problems in such spaces. One fundamental in this topic is the…
It is well-known under the name of `periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R^d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of…
We consider the problem of homogenizing the Maxwell equations for periodic composites. The analysis is based on Bloch-Floquet theory. We calculate explicitly the reflection coefficient for a half-space, and derive and implement a…
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…
We analyze the concepts of analytically weak solutions of stochastic differential equations (SDEs) in Hilbert spaces with time-dependent unbounded operators and give conditions for existence and uniqueness of such solutions. Our studies are…
We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic…
We consider an elliptic differential operator $A_\varepsilon = - \frac{d}{dx} g(x/\varepsilon) \frac{d}{dx} + \varepsilon^{-2} V(x/\varepsilon)$, $\varepsilon > 0$, with periodic coefficients acting in $L_2(\mathbb{R})$. For the…
We present a notion of almost periodicity wich can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept…
This work is devoted to the homogenization of elliptic equations in high-contrast media in the so-called 'double-porosity' resonant regime, for which we solve two open problems of the literature. First, we prove qualitative stochastic…
The paper deals with homogenization of divergence form second order parabolic operators whose coefficients are periodic in spatial variables and random stationary in time. Under proper mixing assumptions, we study the limit behaviour of the…
A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with…
This paper addresses the issue of homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic…