Γ-convergence of multiscale periodic functionals depending on the time-derivative and the curl operator

We study the Γ-convergence of a family of multiscale periodic quadratic integral functionals defined in a product space, whose densities depend on the time-derivative and on the curl of solenoidal fields, through the multiscale convergence in time–space and the multiscale Young measures in time–space associated with relevant sequences of pairs. An explicit representation of the Γ-limit density is given by means of an infinite dimensional minimization problem.     

Numerical analysis of nonlinear multiharmonic eddy current problems

Summary This work is devoted to non-linear eddy current problems and their numerical treatment by the so-called multiharmonic approach. Since the sources are usually alternating currents, we propose a truncated Fourier series expansion instead of a costly time-stepping scheme. Moreover, we suggest to introduce some regularization parameter that ensures unique solvability not only in the factor space of divergence-free functions, but also in the whole space H(curl). Finally, we provide a rigorous estimate for the total error that is due to the use of truncated Fourier series, the regularization technique and the spatial finite element discretization.This work has been supported by the Austrian Science Fund Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under the grants SFB F013, P 14953 and START Y192.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.     

Embedding polynomial matrices of one variable

A non square matrix with coefficients in K[z] can (if a condition on its minors is satisfied) be embedded into a square matrix with determinant 1. Finding theoretically and in an algorithmic way an embedding of small degree is solved by a construction with vector bundles on the projective line over K.     

Sharp Hardy–Leray inequality for axisymmetric divergence-free fields

We show that the sharp constant in the classical n-dimensional Hardy–Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for n = 2 without the axisymmetry assumption.     

On coefficients of polynomials over finite fields

In this paper we study the relation between coefficients of a polynomial over finite field Fq and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m|q−1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m?q−1 and . As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q3/2) operations.     

Asymptotic solutions of the Navier-Stokes equations and systems of stretched vortices filling a three-dimensional volume

We construct asymptotic solutions of the Navier-Stokes equations describing periodic systems of vortex filaments entirely filling a three-dimensional volume. Such solutions are related to certain topological invariants of divergence-free vector fields on the two-dimensional torus. The equations describing the evolution of of such a structure are defined on a graph which is the set of trajectories of a divergence-free field.     

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant rth mean curvature Sr. We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with Sr+1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.     

Asymptotic solutions of the Navier-Stokes equations describing periodic systems of localized vortices

We construct asymptotic solutions of the Navier-Stokes equations describing the two-phase Taylor-scale structures consisting of periodic systems of localized vortex filaments. Such solutions are related to certain topological invariants of divergence-free vector fields on the two-dimensional cylinder or the torus. The equations describing the evolution of the vortex system are defined on a graph which is the set of trajectories of a divergence-free field.     

Asymptotic solutions of Navier-Stokes equations and topological invariants of vector fields and Liouville foliations

We construct asymptotic solutions of the Navier-Stokes equations. Such solutions describe periodic systems of localized vortices and are related to topological invariants of divergence-free vector fields on two-dimensional cylinders or tori and to the Fomenko invariants of Liouville foliations. The equations describing the evolution of a vortex system are given on a graph that is a set of trajectories of the divergence-free field or a set of Liouville tori.     

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.     

Linear complexity profile of m-ary pseudorandom sequences with small correlation measure

We estimate the linear complexity profile of m-ary sequences in terms of their correlation measure, which was introduced by Mauduit and Sárközy. For prime m this is a direct extension of a result of Brandstätter and the second author. For composite m, we define a new correlation measure for m-ary sequences, relate it to the linear complexity profile and estimate it in terms of the original correlation measure. We apply our results to sequences of discrete logarithms modulo m and to quaternary sequences derived from two Legendre sequences.     

Error estimate for finite volume scheme

We study the convergence of a finite volume scheme for the linear advection equation with a Lipschitz divergence-free speed in R ^d . We prove a h ^1/2-error estimate in the L ^∞(0,t;L ¹)-norm for BV data. This result was expected from numerical experiments and is optimal.     

Linear recurrences with polynomial coefficients

We relate sequences generated by recurrences with polynomial coefficients to interleaving and multiplexing of sequences generated by recurrences with constant coefficients. In the special case of finite fields, we show that such sequences are periodic and provide linear complexity estimates for all three constructions.     

On Γ-convergence in divergence-free fields through Young measures

The explicit characterization of the limit energy density of sequences of general non-periodic functionals defined on divergence-free fields is achieved by means of the div-Young measure associated with relevant sequences of functions through a minimization problem.     

Modified edge finite elements for photonic crystals

We consider Maxwell’s equations with periodic coefficients as it is usually done for the modeling of photonic crystals. Using Bloch/Floquet theory, the problem reduces in a standard way to a modification of the Maxwell cavity eigenproblem with periodic boundary conditions. Following [8], a modification of edge finite elements is considered for the approximation of the band gap. The method can be used with meshes of tetrahedrons or parallelepipeds. A rigorous analysis of convergence is presented, together with some preliminary numerical results in 2D, which fully confirm the robustness of the method. The analysis uses well established results on the discrete compactness for edge elements, together with new sharper interpolation estimates.     

Restriction varieties and geometric branching rules

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.