Higher-order Correctors Research Articles

High order correctors and two-scale expansions in stochastic homogenization

In this paper, we study high order correctors in stochastic homogenization. We consider elliptic equations in divergence form on $$\mathbb {Z}^d$$ , with the random coefficients constructed from i.i.d. random variables. We prove moment bounds on the high order correctors and their gradients under dimensional constraints. It implies the existence of stationary correctors and stationary gradients in high dimensions. As an application, we prove a two-scale expansion of the solutions to the random PDE, which identifies the first and higher order random fluctuations in a strong sense.

A note on iterations-based derivations of high-order homogenization correctors for multiscale semi-linear elliptic equations

This Note aims at presenting a simple and efficient procedure to derive the structure of high-order corrector estimates for the homogenization limit applied to a semi-linear elliptic equation posed in perforated domains. Our working technique relies on monotone iterations combined with formal two-scale homogenization asymptotics. It can be adapted to handle more complex scenarios including for instance nonlinearities posed at the boundary of perforations and the vectorial case, when the model equations are coupled only through the nonlinear production terms.

An efficient two-step algorithm for the incompressible flow problem

A new two-step stabilized finite element method for the 2D/3D stationary Navier---Stokes equations based on local Gauss integration is introduced and analyzed in this paper. The method consists of solving one Navier---Stokes problem based on the P1?P1 finite element pair and then solving a general Stokes problem based on the P2?P2 finite element pair, i.e., computes a lower order predictor and a higher order corrector. Moreover, the stability and convergence of the present method are deduced, which show that the new method provides an approximate solution with the convergence rate of the same order as the P2?P2 stabilized finite element solution solving the Navier---Stokes equations on the same mesh width. However, our method can save a large amount of computational time. Finally, numerical tests confirm the theoretical results of the method.

A Multiscale Approach and a Hybrid FE-FDTD Algorithm for 3D Time-Dependent Maxwell's Equations in Composite Materials

This paper discusses the multiscale analysis of the initial-boundary value problem for three-dimensional (3D) time-dependent Maxwell's equations in composite materials. The new contributions in this paper are the determination of higher-order correctors and an explicit convergence rate for the approximate solution. Consequently, a multiscale hybrid finite element finite-difference time-domain (FE-FDTD) method is presented. The numerical results demonstrate that this multiscale method has potential applications in engineering electromagnetics.

Multiscale analysis and numerical algorithm for the Schrödinger equations in heterogeneous media

In solid state physics, the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation (EMA) and its extension the multiband k · p method (see [9]). They have been particularly successful in the case of heterostructures (see, e.g. [4,9,11]). This paper discusses the multiscale analysis of the Schrödinger equation with rapidly oscillating coefficients. The new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions, the definition of boundary layer solutions, and higher-order correctors. Consequently, a multiscale finite element method and some numerical results are presented. As one of the main results of this paper, we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of Γ point, from the viewpoint of mathematics.

Multiscale Asymptotic Method for Maxwell's Equations in Composite Materials

In this paper we discuss the multiscale analysis of Maxwell's equations in composite materials with a periodic microstructure. The new contributions in this paper are the determination of higher-order correctors and the explicit convergence rate for the approximate solutions (see Theorem 2.3). Consequently, we present the multiscale finite element method and derive the convergence result (see Theorem 4.1). The numerical results demonstrate that higher-order correctors are essential for solving Maxwell's equations in composite materials.

Direct spherical and chromatic aberration correction for charged particle optical systems

A method to correct spherical and chromatic aberrations is described. The system provides complete physical and optical separation between first the first-order optics and higher order correctors. Calculations indicate subangstrom beams may be realized.

Improved complexity using higher-order correctors for primal-dual Dikin affine scaling

In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen. Roos and Terlaky enhances an asymptotical $$O(\sqrt n L)$$ complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems.