sparse tensor finite elements for elliptic multiple scale problems locally periodic elliptic multiscale problems in a bounded lipschitz domain d subset mathbb r n with k geq 2 separated scales are reduced to an elliptic system of k coupled anisotropic elliptic one scale problems in a cartesian product domain of total dimension kn e g cite n89 a ab cdg2008 in cite hs sst08 it has been shown how thesecoupled elliptic problems could be solved by sparse tensor wavelet finite element methods in log linear complexity with respect to the number n of degrees of freedom required by multilevel solvers for elliptic one scale problems in d with the same convergence rate in the present paper the high dimensional one scale limiting problems are discretized by a sparse tensor product finite element method fem with standard one scale fe basis functions as used in engineering fe codes sparse tensorizationand multilevel preconditioning is achieved by a bpx multilevel iteration we show that the resulting sparse tensor multilevel fem resolves all physical length scales throughout the domain with efficiency i e accuracy versus work and memory comparable to that of multigrid solvers for elliptic one scale problems in the physical domain d in particular our sparse tensor fem gives numerical approximations of the correct homogenized limit as well as compressed numerical representationsof all first order correctors throughout the physical domain with performance independent of the physical problem s scale parameters numerical examples with standard fe shape functions and bpx multilevelpreconditioners for elliptic problems with k 2 separated physical scales in spatial dimension n 2 confirm the theoretical results in particular the present approach allows to avoid the construction of wavelet fe bases necessary in previous work cite hs sst08 while achieving resolution of all scales throughout the physical domain in log linear complexity with the logarithmic exponent behaving linearly in the number k of scales

numerical homogenization of the acoustic wave equations with a continuum of scales in this paper we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients namely when the bulk modulus and the density of the medium are only bounded we show that under a cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are l 2 instead h 1 with respect to euclidean coordinates and the solution itself is in l 0 t h 2 instead of l 0 t h 1 with respect to euclidean coordinates then we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales and present error estimates of the method it follows that by pre computing the associated harmonic coordinates it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity

sparse tensor finite elements for elastic wave equation with multiple scales we consider an elastic wave equation whose moduli depend on n separable microscopic scales and are periodic with respect to each scale in a domain d in r d by using multiscale convergence we deduce an n 1 d dimensional problem that contains all the macroscopic and microscopic information on the multiscale equation the full tensor product finite element approach is natural to solve this high dimensional multiscale homogenized equation but is too expensive practically we develop a sparse tensor finite element approach that achieves essentially equal accuracy as the full tensor product finite elements but requires an essentially equal number of degrees of freedom as for solving a problem in r d restricting our consideration to the case of zero initial wave displacement we deduce a corrector with a homogenization error estimate in the case of two scales from this numerical correctors are constructed from the finite element solutions of the high dimensional multiscale homogenized problem with an error of the order of the sum of the homogenization and the finite element errors for the general multiscale problems we construct numerical correctors from the finite element solutions but without an explicit error numerical examples on two and three scale elastic wave equations in two dimensions confirm our analysis

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

homogenization of the maxwell equations at fixed frequency the homogenization of the maxwell equations at fixed frequency is addressed in this paper the bulk homogenized electric and magnetic properties of a material with a periodic microstructure are found from the solution of a local problem on the unit cell by suitable averages the material can be anisotropic and satisfies a coercivity condition the exciting field is generated by an incident field from sources outside the material under investigation a suitable sesquilinear form is defined for the interior problem and the exterior calderon operator is used to solve the exterior radiating fields the concept of two scale convergence is employed to solve the homogenization problem a new a priori estimate is proved as well as a new result on the correctors

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the 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 operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

on the approximation of electromagnetic fields by edge finite elements part 2 in the second part of this series of papers we consider highly oscillatory media in this situation the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load on the other hand undersampling by using a coarse mesh might lead to inaccurate results to overcome these difficulties and to improve the ratio between accuracy and computational costs homogenization techniques can be used in this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for maxwells equations in frequency domain this scheme follows the design principles of heterogeneous multiscale methods we prove convergence to the effective solution of the multiscale maxwells equations in a periodic setting and give numerical experiments in accordance to the stated results

complex anti self dual connections on a product of calabi yau surfaces and triholomorphic curves we study the adiabatic limit of a sequence of anti self dual connections on unitary bundles over a product of two compact calabi yau surfaces m n by scaling metrics to shrink n to a point we show that after fixing gauge transformations a subsequence of the n components of these connections converges to a triholomorphic curve from m away from a cayley cycle in m n to the moduli space cal m n of instantons on m n modulo gauge equivalence in the hausdorff topology and converges on the blow up locus to a family which is parameterized by the cayley cycle of triholomorphic curves from c 2 to cal m n

quaternionic maps between quaternionic kahler manifolds in this paper we consider quaternionic maps between quaternionic kahler manifolds we will derive a monotonicity inequality a bochner type formula and small energy regularity for quaternionic maps

quaternion structure on the moduli space of yang mills connections soit m h un 2 tore complexe avec une metrique plate ou une surface k3 a metrique de ricciplate soit p un fibre g principal sur m g est un groupe de lie compact simple et l espace des modules des connexions irreductibles anti auto duales sur p alors la variete de rieman admet une structure de quaternion constante de facon covariante son groupe d holonomie est contenu dans sp n 4n dim r

a generalized lower bound theorem for balanced manifolds a simplicial complex of dimension d 1 is said to be balanced if its graph is d colorable juhnke kubitzke and murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes we establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak lefschetz property as a corollary we show that if delta is an arbitrary balanced triangulation of any closed homology manifold of dimension d 1 ge 3 then 2h 2 delta d 1 h 1 delta ge 4 d atopwithdelims 2 tilde beta 1 delta tilde beta 0 delta thus verifying a conjecture by klee and novik to prove these results we develop the theory of flag h vectors

on the generalized lower bound conjecture for polytopes and spheres in 1971 mcmullen and walkup posed the following conjecture which is called the generalized lower bound conjecture if p is a simplicial d polytope then its h vector h0 h1 hd satisfies h 0 leq h 1 leq ldots leq h left lfloor d left 2 right right rfloor moreover if hr 1 hr for some r leq frac 1 2 d then p can be triangulated without introducing simplices of dimension d r

a multiscale approach and a hybrid fe be algorithm for heterogeneous scattering of maxwell s equations abstract in this paper we discuss the multiscale approach for the scattering problem of maxwell s equations in a heterogeneous material with a periodic microstructure the new contributions in this paper are the determination of the multiscale correctors and the strong convergence in the norm of the space h curl with an explicit rate for the approximate solutions see theorem 2 4 consequently we present a multiscale hybrid finite element method boundary element method fe be the numerical examples are carried out to validate the theoretical results of this paper

high dimensional finite element method for multiscale nonlinear monotone parabolic equations abstract we develop in this paper an essentially optimal finite element fe method for solving locally periodic nonlinear monotone parabolic equations in a domain d r d that depend on n separable microscopic scales for nonlinear multiscale equations it is not possible to form the homogenized equation explicitly numerically the method solves the multiscale homogenized equation which is obtained from multiscale convergence this equation contains all the necessary information the solution to the homogenized equation which approximates the solution to the multiscale equation macroscopically and the scale interacting terms which provide the microscopic information however it is posed in a high dimensional tensorized domain we develop the sparse tensor product fe method for this equation that uses an essentially optimal number of degrees of freedom to obtain an approximation for the solution within a prescribed accuracy we then construct numerical correctors from the fe solution in the two scale case we derive a new homogenization error from which an explicit error for the numerical corrector is established it is the sum of the fe error and the homogenization error numerical examples illustrate the theoretical results

minimal surfaces in kahler surfaces and ricci curvature soit m une surface immergee dans une surface de kahler n soient p le nombre de points tangents complexes et q le nombre de points tangents anticomplexes denombres selon leur multiple cite on deduit une version locale de p q x m x ou x m est la caracteristique d euler de m et x la caracteristique d euler du fibre normal de m dans n

quaternionic maps between hyperkahler manifolds quaternionic maps q maps between hyperkahler manifolds are quaternionic analogue of cauchy riemann equations between kahler manifolds we provide a necessary and sufficient condition on when a quaternionic map becomes holomorphic with respect to some complex structures in the hyperkahler 2 spheres and give examples of q maps which cannot be holomorphic when the domain is real 4 dimensional we analyze the structure of the blow up set of a sequence of q maps and show that the singular set of a stationary q map is h1 rectifiable

a new heterogeneous multiscale method for time harmonic maxwell s equations 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 us to introduce fine scale correctors that are not subject to a cumbersome divergence free constraint and which can hence easily be implemented to analyze the method we first revisit classical homogenization theory for time harmonic maxwell equations and derive a new homogenization result that makes use of the divergence regularization in the two scale homogenized equation we then show that the hmm is equivalent to a discretization of this equation in particular writing both problems in a fully coupled two scale formulation is the crucial starting point for a corresponding numerical analysis of the method with this approach we are able to prove rigorous a priori error estimates in the mathbf h curl and the h 1 norm and we derive reliable and efficie

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

highly neighborly centrally symmetric spheres abstract in 1995 jockusch constructed an infinite family of centrally symmetric 3 dimensional simplicial spheres that are cs 2 neighborly here we generalize his construction and show that for all d 3 and n d 1 there exists a centrally symmetric d dimensional simplicial sphere with 2n vertices that is cs d 2 neighborly this result combined with work of adin and stanley completely resolves the upper bound problem for centrally symmetric simplicial spheres

rigidity and the lower bound theorem 1 for an arbitrary triangulated d 1 manifold without boundaryc withf 0 vertices andf 1 edges define gamma c f 1 df 0 left begin array 20 c d 1 2 end array right barnette proved that c 0 we use the rigidity theory of frameworks and in particular results related to cauchy s rigidity theorem for polytopes to give another proof for this result we prove that ford 4 if c 0 thenc is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope other results a we prove a lower bound conjectured by bjorner for the number ofk faces of a triangulated d 1 manifold with specified numbers of interior vertices and boundary vertices b ifc is a simply connected triangulatedd manifold d 4 and lk v c 0 for every vertexv ofc then c 0 lk v c is the link ofv inc c letc be a triangulatedd manifold d 3 then ske11 d 2 can be embedded in skel1 c iff c 0 d is thed dimensional simplex d ifp is a 2 simpliciald polytope then f 1 p geqq df 0 p left begin array 20 c d 1 2 end array right related problems concerning pseudomanifolds manifolds with boundary and polyhedral manifolds are discussed

high dimensional finite elements for two scale maxwell wave equations abstract we develop an essentially optimal numerical method for solving two scale maxwell wave equations in a domain d r d the problems depend on two scales one macroscopic scale and one microscopic scale solving the macroscopic two scale homogenized problem we obtain the desired macroscopic and microscopic information this problem depends on two variables in r d one for each scale that the original two scale equation depends on and is thus posed in a high dimensional tensorized domain the straightforward full tensor product finite element fe method is exceedingly expensive we develop the sparse tensor product fes that solve this two scale homogenized problem with essentially optimal number of degrees of freedom i e the number of degrees of freedom differs by only a logarithmic multiplying factor from that required for solving a macroscopic problem in a domain in r d only for obtaining a required level of accuracy numerical correctors are constructed from the fe solution we derive a rate of convergence for the numerical corrector in terms of the microscopic scale and the fe mesh width numerical examples confirm our analysis