on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

finite volume element methods for non definite problems the error estimates for finite volume element method applied to 2 and 3 d non definite problems are derived a simple upwind scheme is proven to be unconditionally stable and first order accurate

wave chaos in acoustics and elasticity interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area especially in elasticity and acoustics the theory has indeed been developed primarily in a quantum context it is increasingly becoming clear however that important applications lie in the field of mechanical vibrations and acoustics we provide an overview over basic concepts in this emerging field of wave chaos this ranges from ray approximations of the green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics both experimentally and from a theoretical numerical point of view

subdeterminant approach for pseudo orbit expansions of spectral determinants in quantum maps and quantum graphs we study the implications of unitarity for pseudo orbit expansions of the spectral determinants of quantum maps and quantum graphs in particular we advocate to group pseudo orbits into subdeterminants we show explicitly that the cancellation of long orbits is elegantly described on this level and that unitarity can be built in using a simple subdeterminant identity which has a nontrivial interpretation in terms of pseudo orbits this identity yields much more detailed relations between pseudo orbits of different lengths than was known previously we reformulate newton identities and the spectral density in terms of subdeterminant expansions and point out the implications of the subdeterminant identity for these expressions we analyze furthermore the effect of the identity on spectral correlation functions such as the autocorrelation and parametric cross correlation functions of the spectral determinant and the spectral form factor

quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 as observables we take the l 2 functions on the interval the proof is based on the periodic orbit expansion of a majorant of the quantum variance specifically given a one dimensional lebesgue measure preserving map of an interval we consider an increasingly refined sequence of partitions of the interval to this sequence we associate a sequence of graphs whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

eigenstate structure in graphs and disordered lattices we study wave function structure for quantum graphs in the chaotic and disordered regime using measures such as the wave function intensity distribution and the inverse participation ratio the result is much less ergodicity than expected from random matrix theory even though the spectral statistics are in agreement with random matrix predictions instead analytical calculations based on short time semiclassical behavior correctly describe the eigenstate structure

the local time distribution of a particle diffusing on a graph we study the local time distribution of a brownian particle diffusing along the links on a graph in particular we derive an analytic expression of its laplace transform in terms of the green function on the graph we show that the asymptotic behaviour of this distribution has non gaussian tails characterized by a nontrivial large deviation function

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

multigrid for the mortar finite element method a multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed these problems are posed on domains partitioned into subdomains each of which is independently triangulated in a multilevel fashion the multilevel mortar finite element spaces based on such triangulations which need not align across subdomain interfaces are in general not nested suitable grid transfer operators and smoothers are developed which lead to a variable v cycle preconditioner resulting in a uniformly preconditioned algebraic system computational results illustrating the theory are also presented

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

scattering theory on graphs we consider the scattering theory for the schrodinger operator dx2 v x on the graphs made of one dimensional wires connected to external leads we derive two expressions for the scattering matrix on arbitrary graphs one involves matrices that couple arcs oriented bonds the other involves matrices that couple vertices we discuss a simple way to tune the coupling between the graph and the leads the efficiency of the formalism is demonstrated with a few known examples

a system of mixed equilibrium problems fixed point problems of strictly pseudo contractive mappings and nonexpansive semi groups the purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions for a system of mixed equilibrium problems the set of common fixed points for an infinite family of strictly pseudo contractive mappings and the set of common fixed points for nonexpansive semi groups in hilbert space under suitable conditions some strong convergence theorem are proved the results presented in the paper extend and improve some recent results

general iterative methods for a one parameter nonexpansive semigroup in hilbert space abstract let h be a hilbert space and f a fixed contractive mapping with coefficient 0 1 a a strongly positive linear bounded operator with coefficient 0 consider two iterative methods that generate the sequences x n y n by the algorithm respectively i x n i n a 1 t n 0 t n t s x n d s n f x n ii y n 1 i n a 1 t n 0 t n t s y n d s n f y n where n and t n are two sequences satisfying certain conditions and t s s 0 is a one parameter nonexpansive semigroup on h it is proved that the sequences x n y n generated by the iterative method i and ii respectively converge strongly to a common fixed point x f which solves the variational inequality a f x x z 0 z f

the mortar finite volume method with the crouzeix raviart element for elliptic problems abstract we construct and analyse a mortar finite volume method for the discretization for selfadjoint elliptic boundary value problems in r 2 this method is based on the mortar crouzeix raviart non conforming finite element spaces we prove the optimal order h 1 norm and l 2 norm error estimates between the exact solution and the mortar finite volume approximation of the elliptic problems

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

occupation times distribution for brownian motion on graphs considering a brownian motion on a general graph we study the joint law for the occupation times on all the bonds in particular we show that the laplace transform of this distribution can be expressed as the ratio of two determinants we give two formulations with arc or vertex matrices for this result and discuss a simple example

three point correlations for quantum star graphs we compute the three point correlation function for the eigenvalues of the laplacian on quantum star graphs in the limit where the number of edges tends to infinity this extends a work by berkolaiko and keating where they get the two point correlation function and show that it follows neither poisson nor random matrix statistics it makes use of the trace formula and combinatorial analysis

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

a finite volume method based on the crouzeix raviart element for elliptic pde s in two dimensions we introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in mathbb r 2 the method is based on nonuniform triangulations with piecewise linear nonconforming spaces we prove optimal order error estimates in the l 2 norm and a mesh dependent h 1 norm

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

value distribution of the eigenfunctions and spectral determinants of quantum star graphs we compute the value distributions of the eigenfunctions and spectral determinant of the schrodinger operator on families of star graphs the values of the spectral determinant are shown to have a cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related this is in contrast to the spectral determinants of random matrices for which the logarithm is known to satisfy a gaussian limit distribution the value distribution of the eigenfunctions also differs from the corresponding random matrix result we argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of seba type billiards

scattering and transport properties of tight binding random networks we study numerically scattering and transport statistical properties of tight binding random networks characterized by the number of nodes n and the average connectivity we use a scattering approach to electronic transport and concentrate on the case of a small number of single channel attached leads we observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements the conductance probability distribution w t the average conductance the shot noise power p and the elastic enhancement factor f by varying from small 0 to large 1 values we also show that all these quantities are invariant for fixed n moreover we proposes a heuristic and universal relation between and p and the disorder parameter

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

periodic orbit theory and the statistical analysis of scaling quantum graph spectra the explicit solution to the spectral problem of quantum graphs found recently by dabaghian and blumel phys rev e 68 055201 r 2003 70 046206 2004 jetp lett 77 530 2003 is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics including the distribution for the nearest neighbor separations sn kn kn 1 and the distribution of the spectral oscillations around the average deltakn kn kn

multigrid for the mortar element method for p1 nonconforming element in this paper a multigrid algorithm is presented for the mortar element method for p1 nonconforming element based on the theory developed by bramble pasciak xu in 5 we prove that the w cycle multigrid is optimal i e the convergence rate is independent of the mesh size and mesh level meanwhile a variable v cycle multigrid preconditioner is constructed which results in a preconditioned system with uniformly bounded condition number

a mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense abstract let x be a uniformly convex banach space such that its dual x has the kadec klee property c a bounded closed convex subset of x and t t t t 0 a one parameter nonlinear semigroup which is asymptotically nonexpansive in the intermediate sense we show that every continuous almost orbit of t is weakly almost convergent to a common fixed point of t a discrete version of our main result is also included

universal spectral statistics in quantum graphs we prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations as predicted by random matrix theory the stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph

uniform convergence and preconditioning method for mortar mixed element method for nonselfadjoint and indefinite problems abstract in this paper a mortar version of the modified mixed element method is proposed for nonselfadjoint and indefinite second order elliptic problems it is proven that the modified mortar mixed element method is equivalent to a modified mortar nonconforming element method based on this equivalence the existence uniqueness and uniform convergence of the solution for the modified mortar mixed element method are shown under minimal regularity assumption meanwhile an optimal error estimate for the modified mortar mixed element method is obtained under h2 smoothness assumption furthermore an additive schwarz preconditioning method for solving the discrete problem is provided and the nearly optimal convergence rate for the preconditioned gmres method is proven under the minimal regularity assumption

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

orbit bifurcations and the scarring of wave functions we extend the semiclassical theory of scarring of quantum eigenfunctions n q by classical periodic orbits to include situations where these orbits undergo generic bifurcations it is shown that n q 2 averaged locally with respect to position q and the energy spectrum en has structure around bifurcating periodic orbits with an amplitude and length scale whose dependence is determined by the bifurcation in question specifically the amplitude scales as and the length scale as and values of the scar exponents and are computed for a variety of generic bifurcations in each case the scars are semiclassically wider than those associated with isolated and unstable periodic orbits moreover their amplitude is at least as large and in most cases larger in this sense bifurcations may be said to give rise to superscars the competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed we argue that there is a resulting universal scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase space dynamics finally a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps

families of line graphs and their quantization any directed graph g with n vertices and j edges has an associated line graph l g where the j edges form the vertices of l g we show that the non zero eigenvalues of the adjacency matrices are the same for all graphs of such a family ln g we give necessary and sufficient conditions for a line graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions line graphs may therefore serve as models to study the semiclassical limit of large matrix size of a quantum dynamics on graphs with fixed classical behaviour

mortar upwind finite volume element method for convection diffusion problems abstract in this paper we consider the mortar upwind finite volume element method with local linear conforming finite element space for the convection diffusion problems and show that the mortar upwind finite volume element approximations derived are convergent with optimal order in the h 1 norm

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries