collocation approximation to eigenvalues of an ordinary differential equation numerical illustrations we display the numerical results associated with the collocation of three eigenvalue problems using from one to four gauss points per partition interval in order to document the sharpness of the error bounds we have previously obtained the ordinary differential operators involved are real with constant coefficients two of the problems have an eigenvalue whose ascent exceeds one we propose an explanation for the observed manner in which a set of simple approximate eigenvalues can approach a single multiple eigenvalue

yang mills connections over manifolds with grassmann structure let m be a manifold with grassmann structure i e with an isomorphism of the cotangent bundle t m e h with the tensor product of two vector bundles e and h we define the notion of a half flat connection w in a vector bundle w m as a connection whose curvature f s2e 2h end w 2t m end w under appropriate assumptions for example when the grassmann structure is associated with a quaternionic kahler structure on m half flatness implies the yang mills equations inspired by the harmonic space approach we develop a local construction of holomorphic half flat connections w over a complex manifold with holomorphic grassmann structure equipped with a suitable linear connection any such connection w can be obtained from a prepotential by solving a system of linear first order odes the construction can be applied for instance to the complexification of hyper kahler manifolds or more generally to hyper kahler manifolds with admissible torsion and to their higher spin analogs it yields solutions

upper bounds on wavepacket spreading for random jacobi matrices a method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random possibly correlated jacobi matrix as an application one obtains sharp upper bounds on the diffusion exponents for random polymer models coinciding with the lower bounds obtained in a prior work the second application is an elementary argument not using multiscale analysis or the aizenman molchanov method showing that under the condition of uniformly positive lyapunov exponents the moments of the position operator grow at most logarithmically in time

optimal l error estimates for nonconforming and mixed finite element methods of lowest order for second order linear elliptic problems it is proved that thep1 nonconforming finite element method has the samel asymptotic accuracy as thep1 conforming one this result is applied to derive optimall error estimates for both the displacement and the stress fields of the lowest order raviart thomas mixed finite element method and a superconvergence result at the barycenter of each element

perturbation expansion for a one dimensional anderson model with off diagonal disorder the weak disorder expansion for a random schrodinger equation with off diagonal disorder in one dimension is studied the invariant measure the density of states and the lyapunov exponent are computed the most interesting feature in this model appears at the band center where the differentiated density of states diverges while the lyapunov exponent vanishes the invariant measure approaches an atomic measure concentrated on zero and infinity the results extend previous work of markos to all orders of perturbation theory

absence of localization in disordered systems with local correlation in an early work by dunlap et al 1990 it was conjectured using a matrix transfer approach that diagonal disordered systems with local correlation dimer models exhibit an absence of anderson localization in this letter we find explicitly these delocalized states and energies using a suitable unitary transformation and the symmetries related to the model moreover we expand these states around the critical energy and find a new 1 square root e ec divergence

existence uniqueness and continuation theorems for stochastic functional differential equations abstract the main aim of this paper is to develop some basic theories of stochastic functional differential equations sfdes firstly we establish stochastic versions of the well known picard local existence uniqueness theorem given by driver and continuation theorems given by hale and driver for functional differential equations fdes then we extend the global existence uniqueness theorems of wintner for ordinary differential equations odes driver for fdes and taniguchi for stochastic ordinary differential equations sodes to sfdes these show clearly the power of our new results

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

dynamical localization for the random dimer schrodinger operator we study the one dimensional random dimer model with hamiltonian h v where for all x mathbb z v 2x v 2x 1 and where the v 2x are i i d bernoulli random variables taking the values v v 0 we show that for all values of vand with probability one in the spectrum of his pure point if v 1 and v 1 sqrt 2 the lyapunov exponent vanishes only at the two critical energies given by e v for the particular value v 1 sqrt 2 respectively v sqrt 2 we show the existence of new additional critical energies at e 3 sqrt 2 respectively e 0 on any compact interval inot containing the critical energies the eigenfunctions are then shown to be semi uniformly exponentially localized and this implies dynamical localization for all q 0 and for all ell 2 mathbb z with sufficiently rapid decrease mathop sup limits t r psi i left q right kern 1pt left t right mathop sup limits t left langle p i left h omega right psi t left x right q p i left h omega right psi t right rangle infty here psi t e ih omega t psi and pi h is the spectral projector of h onto the interval i in particular if v 1 and v sqrt 2 these results hold on the entire spectrum so that one can take i h

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

superconvergence of mixed covolume method for elliptic problems on triangular grids in this paper we consider the superconvergence of a mixed covolume method on the quasi uniform triangular grids for the variable coefficient matrix poisson equations the superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained with these superconvergence estimates we establish the superconvergence estimates and the l error estimates for the mixed covolume method for the elliptic problems based on the superconvergence of the mixed covolume method under the condition that the triangulation is uniform we construct a post processing method for the approximate velocity which improves the order of approximation of the approximate velocity

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

local correlations in one and two dimensional disordered systems in one dimensional disordered systems all states are localized but when you add local correlations such as a dimer impurity some states become delocalized in this paper a wide range of correlations are studied leading to a delocalization of states moreover second nearest neighbour models which also have extended states are investigated finally a new approach to the case of two dimensions is presented using a matrix transfer technique where explicit extended states are found

stability radii of linear systems with respect to stochastic perturbations abstract real and complex stability radii for stable linear systems with respect to structured stochastic multiperturbations are introduced and completely characterized

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

a discontinuous finite element method for solving a multiwell problem many physical materials of practical relevance can attain several variants of crystalline microstructure the appropriate energy functional is necessarily nonconvex and the minimization of the functional becomes a challenging problem a new numerical method based on discontinuous finite elements and a scaled energy functional is proposed it exhibits excellent convergence behavior for the energy second order as well as other crucial quantities of interest for general spatial meshes contrary to standard non conforming methods both theoretical analyses and numerical test calculations are presented and contrasted to other current finite element methods for this problem

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

perturbation theory for the random dimer model the authors study a one dimensional tight binding model with a random potential taking two values or nu with the restriction that on pairs of two neighbouring sites the potentials take the same value this random dimer model proposed by dunlap phillips and wu 1990 has a vanishing lyapunov exponent for the energy e or nu they compute the lyapunov exponent and the density of states perturbatively in the vicinity of this energy using the invariant measure formalism

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

nonconforming elements in least squares mixed finite element methods in this paper we analyze the finite element discretization for the first order system least squares mixed model for the second order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress respectively moreover on arbitrary regular quadrilaterals we propose new variants of both the rotated q1 nonconforming element and the lowest order raviart thomas element

critical energies in random palindrome models we investigate the occurrence of critical energies where the lyapunov exponent vanishes in random schrodinger operators when the potentials have some local order which we call random palindrome models we give necessary and sufficient conditions for the presence of such critical energies the commutativity of finite word elliptic transfer matrices finally we perform some numerical calculations of the lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet obtaining the exponent ruling the algebraic growth of the second momentum we also consider special random palindrome models with one letter bounded gap property the transport effects of such long range order are showed numerically

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

duality and yang mills fields on quaternionic kahler manifolds the concept of a self dual connection on a four dimensional riemannian manifold is generalized to the 4n dimensional case of any quaternionic kahler manifold the generalized self dual connections are minima of a modified yang mills functional it is shown that our definitions give a correct framework for a mapping theory of quaternionic kahler manifolds the mapping theory is closely related to the construction of yang mills fields on such manifolds some monopole like equations are discussed

quasi optimal pointwise error estimates for the reissner mindlin plate quasi optimal pointwise error estimates valid uniformly with respect to thickness are derived for the finite element approximation to the reissner mindlin plate recently proposed by arnold and falk the method approximates the rotation of fibers with conforming linear elements enriched with bubbles and the transverse displacement with nonconforming linear elements the present analysis is based on proving quite precise estimates in terms of thickness and meshsize for a regularized green s function of a singularly perturbed stokes like system

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

multi instantons in seven dimensions we consider the self dual yang mills equations in seven dimensions modifying the t hooft construction of instantons in d 4 we find n instanton 7d solutions which depend on 8n effective parameters and are e6 invariant

on sufficient conditions for nonexplosion of solutions to stochastic differential equations abstract in this paper we present sufficient conditions for nonexplosion of solutions to a stochastic differential equation theorems are proved by the comparison theorem 7 12 one of our theorems is a generalization of hasminskii s theorem 4 and narita s lemma 9 which is given by using a lyapunov function furthermore we consider nonexplosion theorems of solutions to a stochastic differential equation without using a lyapunov function

a new superconvergence property of nonconforming rotated q1 element in 3d abstract nonconforming rotated q 1 finite element method is used to approximate the general second order elliptic problem in 3d a new superconvergence property at eight vertices and six face centers of each element is proved several cheap numerical integration schemes are proposed for solving the discrete problem which include schemes with only two nodes all schemes yield optimal h 1 l 2 error bounds as well as the superconvergence property extensive numerical results are presented to confirm the theoretic prediction

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

mortar method for nonconforming finite elements a mortar method for second order elliptic problems is considered using the nonconforming finite element proposed by douglas et al j douglas j e santos d sheen x ye nonconforming galerkin methods based on quadrilateral elements for second order elliptic problems m2an math model numer anal 33 1999 747 770 the error estimate of optimal order is proved in the broken energy norm numerical experiments which confirm the convergence analysis are provided

some comments on the adhm construction in 4k dimensions a class of completely solvable gauge field equations is investigated these equations are shown to be closely analogous to the four dimensional self duality equations a similar geometrical interpretation is exhibited and a generalisation of the adhm construction is shown to provide solutions

oxidation of self duality to 12 dimensions and beyond using partial curvature flows and the transitive action of subgroups of rm o d mathbb z on the indices 1 ldots d of the components of the yang mills curvature in an orthonormal basis we obtain a nested system of equations in successively higher dimensions d each implying the yang mills equations on d dimensional riemannian manifolds possessing special geometric structures this matryoshka of self duality equations contains the familiar self duality equations on riemannian fourfold as well as their generalisations on complex kahler threefold and on seven and eight dimensional manifolds with g2 and spin 7 holonomy the matryoshka allows enlargement oxidation to a remarkable system in 12 dimensions invariant under sp 3 there are hints that the underlying geometry is related to the sextonions a six dimensional algebra between the quaternions and octonions

numerical approach to double well problems this paper deals with two sided numerical energy estimates of a solution to two dimensional double well problems these estimates are obtained by computing the energy appropriate for the polyconvex envelope and an upper estimate of the energy appropriate for the rank one convex envelope of the energy density examples of computed laminated microstructures are shown

superconvergence analysis and error expansion for the wilson nonconforming finite element in this paper the wilson nonconforming finite element is considered for solving a class of two dimensional second order elliptic boundary value problems superconvergence estimates and error expansions are obtained for both uniform and non uniform rectangular meshes a new lower bound of the error shows that the usual error estimates are optimal finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

nonconforming immersed finite element spaces for elliptic interface problems abstract in this paper we use a unified framework introduced in chen and zou 1998 to study two nonconforming immersed finite element ife spaces with integral value degrees of freedom the shape functions on interface elements are piecewise polynomials defined on sub elements separated either by the actual interface or its line approximation in this unified framework we use the invertibility of the well known sherman morison systems to prove the existence and uniqueness of ife shape functions on each interface element in either a rectangular or triangular mesh furthermore we develop a multi edge expansion for piecewise functions and a group of identities for nonconforming ife functions which enable us to show the optimal approximation capability of these ife spaces

liouville theorem for the yang mills self duality equations it is shown that under certain conditions one may associatively matrix multiply lie algebra valued matrices with a componentwise lie bracket using this a simple algebraic constraint on a lie algebra valued antisymmetric n n matrix f which in n 4 is essentially self duality or anti self duality is described somewhat in analogy with liouville s theorem for the cauchy riemann equations in n 2 it is shown that for n 4 the constraint implies that the lie subalgebra generated by the matrix elements f decomposes into copies of s o n plus a few degenerate cases the result may be relevant to the structure of the quantum chromodynamic vacuum

analysis of the finite element approximation of microstructure in micromagnetics the solution to nonconvex variational problems is often characterized by microstructure the variational problem for the magnetization field in micromagnetics has a nonconvex constraint and energy minimizing sequences of magnetization fields can have oscillations whose scale converges to zero but whose amplitude remains finite it is shown that the finite element approximation of the magnetization field for this variational problem does not converge pointwise as the mesh is refined but that nonlinear functions of the approximate magnetization fields converge weakly or equivalently all local spatial averages of nonlinear functions of the approximate magnetization field converge as the mesh is refined which implies the convergence as the mesh is refined of the probability distribution of the approximate magnetization fields in local spatial domains a norm is given to measure the convergence of this microstructure and we prove a rate of convergence in this norm

some solutions of the yang mills equations in dimension greater than four the yang mills equations for gauge fields of an arbitrary semisimple lie group g in euclidean spaces rn with dimension n 4 is considered for gauge fields aa we introduce the ansatze which reduce the yang mills equations in rn to a set of nonlinear differential matrix equations rouhani ward equations this permits us to write out the explicit form of cosmological plane wave and other types of solutions

delocalization phenomena in one dimensional models with long range correlated disorder a perturbative approach we study the nature of electronic states in one dimensional continuous models with weak correlated disorder using a perturbative approach we compute the inverse localization length lyapunov exponent up to terms proportional to the fourth power of the potential this makes it possible to analyse the delocalization transition which takes place when the disorder exhibits specific long range correlations we find that the transition consists of a change of the lyapunov exponent which switches from a quadratic to a quartic depending on the strength of the disorder within the framework of the fourth order approximation we also discuss the different localization properties which distinguish gaussian from non gaussian random potentials

correlation induced metal insulator transition in the one dimensional anderson model abstract we study the nature of the electronic states in tight binding one dimensional models with long range correlated disorder in particular we study both diagonal and off diagonal chains the energies are considered to be in such a sequency to describe the trace of a fractional brownian motion with a specified spectral density s k 1 k using a renormalization group technique we show that for random on site energy sequences with anti persistent increments 2 the lyapunov coefficient inverse localization length vanishes within a finite range of energy values revealing the presence of an anderson like metal insulator transition in the case of off diagonal disorder a phase of delocalized states becomes stable for any 1

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

a local post processing technique for improving the accuracy in mixed finite element approximations a general simple post processing technique is given in this paper under certain conditions if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection pu then by this method a better approximation for u can be easily obtained essentially the leading term of the error between u and its original approximation can be computed it is shown how the method may be applied to various mixed type finite element methods

on the estimate of solutions of perturbed linear differential equations abstract b aulbach j math anal appl 112 1985 317 327 discussed the estimate of solutions of linearly perturbed linear differential equations dx dt a t b t x where a t b t are continuous matrices the purpose of this paper is to generalize his results that is we study the estimate of solutions of a perturbed linear differential equation dx dt a t x f t x where f t x is a continuous function which has suitable conditions and using our theorems g r sell s theorems funkcial ekvac 10 1967 1 13 are induced simply as corollaries

mixed methods using standard conforming finite elements abstract we investigate the mixed finite element method mfem for solving a second order elliptic problem with a lowest order term as might arise in the simulation of single phase flow in porous media we find that traditional mixed finite element spaces are not necessary when a positive lowest order i e reaction term is present hence we propose to use standard conforming finite elements q k q k d on rectangles or p k p k d on simplices to solve for both the pressure and velocity field in d dimensions the price we pay is that we have only sub optimal order error estimates with a delicate superconvergence analysis we find some improvement for the simplest pair q k q k d with any k 1 or for p 1 p 1 d when the mesh is uniform and the solution has one extra order of regularity we also prove similar results for both parabolic and second order hyperbolic problems numerical results using q 1 q 1 2 and p 1 p 1 2 are presented in support of our analysis these observations allow us to simplify the implementation of the mfem especially for higher order approximations as might arise in an hp adaptive procedure

analysis of a class of nonconforming finite elements for crystalline microstructures an analysis is given for a class of nonconforming lagrange type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure these elements were first proposed and analyzed in 1992 by rannacher and turek for the stokes equation our analysis highlights the features of these elements which make them effective for the computation of microstructure new results for superconvergence and numerical quadrature are also given

discontinuous galerkin methods for non linear elasticity this paper presents the formulation and a partial analysis of a class of discontinuous galerkin methods for quasistatic non linear elasticity problems these methods are endowed with several salient features the equations that define the numerical scheme are the euler lagrange equations of a one field variational principle a trait that provides an elegant and simple derivation of the method in consonance with general discontinuous galerkin formulations it is possible within this framework to choose different numerical fluxes numerical evidence suggests the absence of locking at near incompressible conditions in the finite deformations regime when piecewise linear elements are adopted finally a conceivable surprising characteristic is that as demonstrated with numerical examples these methods provide a given accuracy level for a comparable and often lower computational cost than conforming formulations stabilization is occasionally needed for discontinuous galerkin methods in linear elliptic problems in this paper we propose a sufficient condition for the stability of each linearized non linear elastic problem that naturally includes material and geometric parameters the latter needed to account for buckling we then prove that when a similar condition is satisfied by the discrete problem the method provides stable linearized deformed configurations upon the addition of a standard stabilization term we conclude by discussing the complexity of the implementation and propose a computationally efficient approach that avoids looping over both elements and element faces several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous galerkin method within the class copyright 2006 john wiley sons ltd

on the accuracy of nonconforming quadrilateral q 1 element approximation for the navier stokes problem this paper deals with the nonconforming quadrilateral q1 element approximation of the navier stokes equation by using the so called velocity pressure mixed formulation a convergence condition is presented for the nonconforming stokes element it is shown that the condition is not only sufficient for the convergence but also in some sense is necessary the quasi optimal maximum norm error estimates of the velocity and its first derivatives and of the pressure are derived for the nonconforming quadrilateral q1 element for the stationary navier stokes problem the analysis is based on the weighted inf sup condition and the technique of weighted sobolev norm

lifschitz tails for the anderson model we present in an expository way an elementary rigorous proof patterned after an argument of kirsch martinelli that the anderson model has lifschitz tails in very great generality

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

reflectionless modes in chains with large size homogeneous impurities the authors study the spectra and localization properties of tight binding chains with randomly distributed binary blocks of arbitrary size m one having site energies epsilon a and the other epsilon b respectively they demonstrate that for block size m greater than one and delta mod epsilon a epsilon b less than a critical value delta c m perfect transmission resonance modes exist in the band their number is proportional to m and they occur via dominant 1 e2 divergencies of the localization length no transmission is found for delta delta c m the results are understood by solving exactly the scattering problem from a single homogeneous impurity block of arbitrary size m in the limit of hard impurities m to infinity transmission stops only when delta delta c infinity 4v v being the intersite matrix element that is when the pure a and b bands become detached

singular continuous electron spectrum for a class of circle sequences we derive substitution rules for a class of binary quasiperiodic sequences generated by circle maps whose rotation numbers are obtained from the precious means the nature of the electron spectra for the corresponding diatomic chains is studied in the nearest neighbour on site tight binding approximation using the transfer matrix technique by studying properties of the trace maps we find that the spectrum is purely singular continuous in most of the studied cases

the nature of electronic states in a disordered chain with long ranged hopping amplitudes abstract in this work we investigate the nature of the electronic states in a one dimensional tight binding model with on site energetic disorder and long ranged hopping amplitudes these are assumed to follow a power law decay behavior with distance in the form h r 1 r with 1 direct numerical diagonalization finite size scaling and a spectral statistics analysis show that as 1 the localization length exhibits an exponential growth as the upper band edge is approached strong finite size corrections are present in this regime near the upper band edge with the one electron states displaying a strong level repulsion even at large although finite chains

analysis of a finite element method for the drift diffusion semiconductor device equations the multidimensional case an explicit finite element method for numerically solving the drift diffusion semiconductor device equations in two space dimensions is analyzed the method is based on the use of a mixed finite element method for the approximation of the electric field and a discontinuous upwinding finite element method for the approximation of the electron and hole concentrations the mixed method gives an approximate electric field in the precise form needed by the discontinuous method which is trivially conservative and fully parallelizable it is proven that the method produces uniformly bounded concentrations and electric fields and that it converges to the exact solution provided there is a convergent subsequence of the electron concentrations numerical simulations are presented that display the performance of the method and indicate the behavior of the solution

multigrid and multilevel methods for nonconforming q 1 elements in this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second order elliptic problems using nonconforming rotated q 1 finite elements in two space dimensions in particular for the case of square partitions and the laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the w cycle with any number of smoothing steps and close to optimal condition number estimates for v cycle preconditioners this is in contrast to most of the other nonconforming finite element discretizations where only results for w cycles with a sufficiently large number of smoothing steps and variable v cycle multigrid preconditioners are available some numerical tests including also a comparison with a preconditioner obtained by switching from the nonconforming rotated q 1 discretization to a discretization by conforming bilinear elements on the same partition illustrate the theory

a study of one dimensional correlated disordered systems using the invariant measure method the behaviour of electronic states of one dimensional correlated disordered systems which are modelled by a tight binding hamiltonian is studied analytically using the invariant measure method the approach of bovier is generalized to include the possibility of different site energies and nearest neighbour hopping integrals inside the correlated sites or the cluster the process is further elaborated by applying to the symmetric random trimer model which contains in it many hitherto known models of this category an alternative mathematical definition of the exceptional energy from the invariant measure density along with physical arguments substantiating it is presented furthermore the procedure for obtaining exceptional energies is outlined and applied to the symmetric random trimer model to derive conditions for obtaining doubly degenerate exceptional energies the lyapunov exponent or the inverse localization length of states around the exceptional energy is found to vary as in the leading order n denotes the degeneracy of the exceptional energy the density of states at the exceptional energies are calculated we further propose that one dimensional correlated disordered systems can be mapped to a lloyd model in which the width of the distribution of site energies is determined by the reflection coefficient of the cluster embedded in the lattice of the other constituent the importance of our results is discussed