local and pollution error estimation for finite element approximations of elliptic boundary value problems this paper addresses the issue of local elementwise error estimation of finite element approximations of elliptic boundary value problems the characterization of element error as local and pollution error component is presented and the relationship between energy norms of local errors and that predicted by means of a posteriori error estimators is investigated in addition techniques for calculating element indicators of local error pollution error and other error indicators are presented

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

poincare husimi representation of eigenstates in quantum billiards for the representation of eigenstates on a poincare section at the boundary of a billiard different variants have been proposed we compare these poincare husimi functions discuss their properties and based on this select one particularly suited definition for the mean behavior of these poincare husimi functions an asymptotic expression is derived including a uniform approximation we establish the relation between the poincare husimi functions and the husimi function in phase space from which a direct physical interpretation follows using this a quantum ergodicity theorem for the poincare husimi functions in the case of ergodic systems is shown

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

on level spacing distributions for 2d non normal gaussian random matrices the level spacing distribution of general non normal gaussian random 2d matrices is derived in particular tridiagonal matrices have no level repulsion and show a halfsided gaussian distribution general non normal matrices show strong level repulsion the repulsion exponent is 2 0log

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

distribution and fluctuation properties of transition probabilities in a system between integrability and chaos the authors study the statistical properties of transition probabilities or generalized intensities equal to the squared matrix elements of operators having a classical limit they first generalize the shnirelman theorem to include semiclassically integrable states and use this to generalize the feingold peres formula for the average value of generalized intensities they perform an unfolding procedure to separate the smooth mean part of the intensities as a function of frequency from its fluctuating part by applying the generalized feingold peres formula this formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos the average values of the intensities as a function of frequency are excellently described by the generalized feingold peres formula

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

inverse anisotropic mean curvature flow and a minkowski type inequality abstract in this paper we show that the inverse anisotropic mean curvature flow in r n 1 initiating from a star shaped strictly f mean convex hypersurface exists for all time and after rescaling the flow converges exponentially fast to a rescaled wulff shape in the c topology as an application we prove a minkowski type inequality for star shaped f mean convex hypersurfaces

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

a new approach to the classical stokes flow problem part ii series solutions and higher order applications this is the second in a series of papers on the stokes flow past an arbitrary axisymmetrical body the truncated series solutions of the two infinite systems of simultaneous ordinary differential equations with variable coefficients are obtained for an arbitrary truncation order n each series solution together with logarithmic terms is shown to be convergent in the entire physical interval of interest by the construction of the complete solutions of the systems the corresponding hydrodynamical problem formulated in terms of the stream function has been solved as a specimen numerical application the drag on a prolate spheroid is computed and compared with the exact one highly accurate numerical results have been achieved depending on the b a ratio of the spheroid

a comparative study on low memory iterative solvers for fft based homogenization of periodic media in this paper we assess the performance of four iterative algorithms for solving non symmetric rank deficient linear systems arising in the fft based homogenization of heterogeneous materials defined by digital images our framework is based on the fourier galerkin method with exact and approximate integrations that has recently been shown to generalize the lippmann schwinger setting of the original work by moulinec and suquet from 1994 it follows from this variational format that the ensuing system of linear equations can be solved by general purpose iterative algorithms for symmetric positive definite systems such as the richardson the conjugate gradient and the chebyshev algorithms that are compared here to the eyre milton scheme the most efficient specialized method currently available our numerical experiments carried out for two dimensional elliptic problems reveal that the conjugate gradient algorithm is the most efficient option while the eyre milton method performs comparably to the chebyshev semi iteration the richardson algorithm equivalent to the still widely used original moulinec suquet solver exhibits the slowest convergence besides this we hope that our study highlights the potential of the well established techniques of numerical linear algebra to further increase the efficiency of fft based homogenization methods

implicit restart lanczos as an eigensolver this paper investigates the efficiency of the implicit restart lanczos and simple without reorthogonalization lanczos algorithms as eigensolvers for large scale computations in molecular and chemical physics using the cardioid billiard and the hydrogen cyanide hydrogen isocyanide hcn hnc molecule as model systems we demonstrate superior efficiency of implicit restart lanczos compared to the simple lanczos algorithm a modified implementation of implicit restart lanczos is also presented which works with a smaller krylov space with associated savings in memory and can handle larger basis sets than the usual implicit restart lanczos it also enables getting all eigenpairs of a matrix or all eigenvalues below a threshold where the number of such is not known before hand which is more difficult with the usual implicit restart algorithm

symmetrization with respect to the anisotropic perimeter and applications in this paper we introduce a new type of symmetrization which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function in order to prove sharp comparison results for solutions of a class of homogeneous dirichlet fully nonlinear elliptic problems of second order and for suitable anisotropic hessian integrals

estimating and localizing the algebraic and total numerical errors using flux reconstructions this paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization and an arbitrary iterative algebraic solver the derived bounds are based on the flux reconstruction techniques do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain we also discuss bounds on the discretization error their application for constructing mathematically justified stopping criteria for iterative algebraic solvers global and local efficiency of the total error upper bound and the relationship to the previously published estimates on the algebraic error theoretical results are illustrated on numerical experiments for higher order finite element approximations and the preconditioned conjugate gradient method they in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over thecomputational domain and illustrate the associate cost

numerical study of a three dimensional generalized stadium billiard we study a generalized three dimensional stadium billiard and present strong numerical evidence that this system is completely chaotic in this convex billiard chaos is generated by the defocusing mechanism the construction of this billiard uses cylindrical components as the focusing elements and thereby differs from the recent approach pioneered by bunimovich and rehacek commun math phys 189 729 1997 we investigate the stability of lower dimensional invariant manifolds and discuss bouncing ball modes c 2000 the american physical society

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

failure of semiclassical methods to predict individual energy levels the authors argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite and when the number of energy levels goes to infinity by this they mean that the average relative error of the semiclassical eigenvalues in units of the mean level spacing typically increases indefinitely as the energy goes to infinity or is at least bounded from below this they show for the case of the integrable circular billiard and the one dimensional potential uo cos2 alpha x by comparing the torus quantized semiclassical eigenenergies with the exact results since all the various semiclassical methods such as gutzwiller s and bogomolny s are reduced to the torus quantization in integrable cases they believe that their conclusion is generally valid they have theoretical arguments and strong numerical evidence for the case of the circular billiard that nevertheless the statistical properties of the exact energy spectra are correctly reproduced by the semiclassical approximations it is numerically found that the energy level spacing distribution and the spectral rigidity for the exact spectrum and for the semiclassical spectrum are in excellent agreement even for finite spectra where they both deviate from the limiting poissonian behaviour so they suggest that the nonuniversal approach to the limiting energy level statistics is also correctly described by the semiclassical theory they discuss the validity of semiclassical methods in the light of their negative and positive findings in addition they find the surprising result for the previously mentioned special cases that the error distribution of the semiclassical approximation is stationary i e it is independent of the energy

maximum norms of chaotic quantum eigenstates and random waves abstract the growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated the maximum norms provide a measure for localization effects in eigenfunctions an upper bound for the maxima of random superpositions of random functions is derived for the random wave model this gives the bound c ln e in the semiclassical limit e the growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic the maximum norms of these systems are numerically shown to be conform with the random wave model furthermore the distribution of the locations of the maximum norms is discussed

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

amplitude distribution of eigenfunctions in mixed systems we study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space for an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and a good agreement with numerical computations for the family of limacon billiards is found the natural extension of our result to more general systems e g with a potential is also discussed

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

distribution of the discretization and algebraic error in numerical solution of partial differential equations abstract in the adaptive numerical solution of partial differential equations local mesh refinement is used together with a posteriori error analysis in order to equilibrate the discretization error distribution over the domain since the discretized algebraic problems are not solved exactly a natural question is whether the spatial distribution of the algebraic error is analogous to the spatial distribution of the discretization error the main goal of this paper is to illustrate using standard boundary value model problems that this may not hold on the contrary the algebraic error can have large local components which can significantly dominate the total error in some parts of the domain the illustrated phenomenon is of general significance and it is not restricted to some particular problems or dimensions to our knowledge the discrepancy between the spatial distribution of the discretization and algebraic errors has not been studied in detail elsewhere

anisotropic isoperimetric inequalities involving boundary momentum perimeter and volume abstract we consider a scale invariant functional involving the anisotropic p momentum the anisotropic perimeter and the volume we show that the wulff shape associated with the finsler norm f considered and centered at the origin is the unique minimizer of the anisotropic functional taken into consideration among all bounded convex sets

dynamical localization in chaotic systems spectral statistics and localization measure in the kicked rotator as a paradigm for time dependent and time independent systems we study the kicked rotator in the classically fully chaotic regime using izrailev s n dimensional model for various n 4000 which in the limit n tends to the quantized kicked rotator we do treat not only the case k 5 as studied previously but also many different values of the classical kick parameter 5 k 35 and many different values of the quantum parameter ke 5 60 we describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time periodic and time independent autonomous fully chaotic or and mixed type hamilton systems we generalize the scaling variable l n to the case of anomalous diffusion in the classical phase space by deriving the localization length l for the case of generalized classical diffusion we greatly improve the accuracy and statistical significance of the numerical calculations giving rise to the following conclusions 1 the level spacing distribution of the eigenphases or quasienergies is very well described by the brody distribution systematically better than by other proposed models for various brody exponents br 2 we study the eigenfunctions of the floquet operator and characterize their localization properties using the information entropy measure which after normalization is given by loc in the interval 0 1 the level repulsion parameters br and loc are almost linearly related close to the identity line 3 we show the existence of a scaling law between loc and the relative localization length now including the regimes of anomalous diffusion the above findings are important also for chaotic eigenstates in time independent systems batistic and robnik j phys a math gen 43 215101 2010 arxiv 1302 7174 2013 where the brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates even in the mixed type systems after separation of regular and chaotic eigenstates

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

eigenvalue dependence on small pseudointegrable perturbation of a two dimensional box the eigenvalues of a hard walled parallelogram shaped enclosure were investigated using quantum perturbation theory the perturbation being a small angle distortion of a pair of parallel sides of the rectangle a comparison of perturbation theory results and exact computations for low lying states showed good agreement the distribution of eigenvalues grossly disturbed from their value in the nearby regular system was studied and found to be systematic

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach recovering mass balance in any situation we present novel h div and h1 liftings of given piecewise polynomials over a hierarchy of simplicial meshes based on a global solve on the coarsest mesh and on local solves on patches of mesh elements around vertices on subsequent mesh levels this in particular allows to lift a given algebraic residual in connection with approaches lifting the total residual we show how to obtain guaranteed fully computable and constant free upper and lower a posteriori bounds on the algebraic total and discretization errors here we consider the model poisson equation discretized by the conforming finite element method of arbitrary order and including an arbitrary iterative solver we next formulate safe stopping criteria ensuring that the algebraic error does not dominate the total error we also prove efficiency i e equivalence of our upper total and algebraic estimates with the total and algebraic errors respectively up to a generic constant this constant is polynomial degree independent for the total error numerical experiments illustrate sharp control of all error components and accurate prediction of their spatial distribution in several test problems including cases where some classical estimators fail the h div liftings at the same time allow to recover mass balance for any problem any numerical discretization and any situation such as inexact solution of nonlinear algebraic systems or algorithm failure which we believe is of independent interest we demonstrate this mass balance recovery in a simulation of immiscible incompressible two phase flow in porous media

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

stopping criteria for adaptive finite element solvers we consider a family of practical stopping criteria for linear solvers for adaptive finite element methods for symmetric elliptic problems a contraction property between two consecutive levels of refinement of the adaptive algorithm is shown when a family of smallness criteria for the corresponding linear solver residuals are assumed on each level or refinement more importantly based on known and new results for the estimation of the residuals of the conjugate gradient method we show that the smallness criteria give rise to practical stopping criteria for the iterations of the linear solver which guarantees that the inexact adaptive algorithm converges a series of numerical experiments highlights the practicality of the theoretical developments

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

a new approach to the classical stokes flow problem part i methodology and first order analytical results the problem of determining the axisymmetric stokes flow past an arbitrary body the boundary shape of which can be represented by an analytic function is examined by developing an exact method an appropriate nonorthogonal coordinate system is introduced and it is shown that the hilbert space to which the stream function belongs is spanned by the set of gegenbauer polynomials based on the physical argument that the drag on a body should be finite the partial differential equation of the original problem is then reduced to two simultaneous vector differential equations by the truncation of this infinite dimensional system to the one dimensional subspace an explicit analytic solution to the stokes equation valid for all bodies in question is obtained as a first approximation

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

a model for the computation of quantum billiards with arbitrary shapes an expansion method for the stationary schrodinger equation of a three dimensional quantum billiard system whose boundary is defined by an arbitrary analytic function is introduced the method is based on a coordinate transformation and an expansion in spherical harmonics the effectiveness is verified and confirmed by a numerical example which is a billiard system depending on a parameter

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

the algebraic quantisation of the birkhoff gustavson normal form the author develops an algebraic quantisation method for the birkhoff gustavson normal form for this purpose the weyl quantisation rule is used the method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities the author gives a brief review of the normal form derives some of its general properties and finds a general analytic solution for the fourth degree normal form for hamiltonians of two degrees of freedom in particular this includes the henon heiles system the author compares the results of specific examples with other works the question of canonically invariant quantisation the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed the author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals if the normal form accidentally terminates so that the classical system is integrable then this implies quantum integrability of the normal form hamiltonian

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

some estimates of the rate of convergence for the cascadic conjugate gradient method abstract the paper deals with a cascadic conjugate gradient method shortly called the ccg algorithm which was proposed by p deuflhard and can be considered as a simpler version of a multigrid multilevel method we define it recurrently for discrete self adjoint positive definite problems on a sequence of grids on the coarsest grid the linear discrete algebraic system is solved directly on the finer grids the system is iteratively solved by the conjugate gradient method where the starting guess is an interpolation of the approximated solution on the previous grid any preconditioning or restriction to coarser grids is not implemented nevertheless the ccg algorithm has the same optimal property compared to multigrid methods namely the algorithm converges with a rate which is independent of the number of unknowns and the number of grids as an example this property is proved for elliptic second order dirichlet problems in two dimensional convex polygonal bounded domains for ensuring convergence the number of iterations on each grid level has to increase from finer to coarser grids the optimal dependence of these numbers is established with respect to the mesh size and the number of unknowns the theory has been presented in an abstract setting which allows the application to both finite element and finite difference methods

intermediate statistics in the regime of mixed classical dynamics we explain the arguments in support of the berry robnik 1984 17 2413 picture of the energy level statistics in the asymptotic strict far semiclassical limit of sufficiently small where the entire energy spectrum can be represented as a statistically independent superposition of regular and irregular level sequences the regular ones obeying the poissonian statistics and the irregular ones the rmt statistics goe or gue we generalize the results to describe not only the level spacing distribution the number variance and the delta statistics but also arbitrary statistics the probability to find k levels inside an interval of length l after unfolding very useful and effective approximations for are described we demonstrate very clearly that this regime is excellently described by this picture for k as high as the outer energy scale even when taking into account the regular component and only one the dominant chaotic component this we show numerically for the compactified standard map and for the quartic 2d billiard

a note on conjugate gradient convergence the one dimensional discrete poisson equation on a uniform grid with n points produces a linear system of equations with a symmetric positive definite coefficient matrix hence the conjugate gradient method can be used and standard analysis gives an upper bound of o n on the number of iterations required for convergence this paper introduces a systematically defined set of solutions dependent on a parameter beta and for several values of beta presents exact analytic expressions for the number of steps k beta tau n needed to achieve accuracy tau the asymptotic behavior of these expressions has the form o n alpha as n rightarrow infty and o tau gamma as tau rightarrow 0 in particular two choices of beta corresponding to nonsmooth solutions give alpha 0 i e iteration counts independent of n this is in contrast to the standard bounds the standard asymptotic convergence behavior alpha 1 is seen for a relatively smooth solution numerical examples illustrate and supplement the analysis