transient measures in the standard map abstract for an area preserving map each chaotic orbit appears numerically to densely cover a region an irregular component of nonzero area surprisingly the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component most prominently there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles we show that this phenomena is transient and therefore numerical ergodicity on the irregular component eventually obtains though the times involved are extremely long 1010 iterates a markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class for example a rotational circle however the density spikes do occur in a markov tree model that includes the effects of islands around islands

boundary circles for area preserving maps abstract we report on numerical observations of various properties of boundary circles for area preserving maps we find that the continued fraction expansion of their rotation number generally has a special form we describe how their rotation number changes as parameters are varied we observe a statistically self similar structure in the neighbourhood of boundary circles which is probably relevant to the long time tails of the decay of correlations in the irregular components

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

some remarks on the abundance of stable periodic orbits inside homoclinic lobes abstract we consider a family f of area preserving maps apms with a hyperbolic point h whose invariant manifolds form a figure eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes epl a domain typically dominated by chaotic behavior to this end we use the chirikov separatrix map sm as a model of the return to a fundamental domain containing lobes we obtain an explicit estimate valid for families f with central symmetry and close to an integrable limit of the relative measure of the set of parameters for which f has epl trajectories to get this estimate we look for epl of the sm with the lowest possible period the analytical results are complemented with quantitative numerical studies of the following families f of apms the sm family and we compare our analytical results with the numerical estimates the standard map stm family and we show how the results referring to the sm model apply to the epl visiting the lobes that the invariant manifolds of the stm hyperbolic fixed point form the conservative henon map family and we estimate the number of a particular type of symmetrical epl related to the separatrices of the 4 periodic resonant islands the results obtained can be seen as the quantitative analogs to those in simo and treschev 2008 9 although here we deal with the a priori stable situation instead

an element wise locally conservative galerkin lcg method for solving diffusion and convection diffusion problems an element wise locally conservative galerkin lcg method is employed to solve the conservation equations of diffusion and convection diffusion this approach allows the system of simultaneous equations to be solved over each element thus the traditional assembly of elemental contributions into a global matrix system is avoided this simplifies the calculation procedure over the standard global continuous galerkin method in addition to explicitly establishing element wise flux conservation in the lcg method elements are treated as sub domains with weakly imposed neumann boundary conditions the lcg method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging it is also shown in this paper that the proposed lcg method is identical to the standard global galerkin gg method at both steady and unsteady states for an inside node thus the method has all the advantages of the standard gg method while explicitly conserving fluxes over each element several problems of diffusion and convection diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the lcg method both linear and quadratic elements are used in the calculations for convection dominated problems petrov galerkin weighting and high order characteristic based temporal schemes have been implemented into the lcg formulation

conservation properties for the galerkin and stabilised forms of the advection diffusion and incompressible navier stokes equations abstract a common criticism of continuous galerkin finite element methods is their perceived lack of conservation this may in fact be true for incompressible flows when advective rather than conservative weak forms are employed however advective forms are often preferred on grounds of accuracy despite violation of conservation it is shown here that this deficiency can be easily remedied and conservative procedures for advective forms can be developed from multiscale concepts as a result conservative stabilised finite element procedures are presented for the advection diffusion and incompressible navier stokes equations

locally conservative galerkin and finite volume methods for two phase flow in porous media in this paper we propose a locally conservative galerkin lcg finite element method for two phase flow simulations in heterogeneous porous media the main idea of the method is to use the property of local conservation at steady state conditions in order to define a numerical flux at element boundaries it provides a way to apply standard galerkin finite element method in two phase flow simulations in porous media the lcg method has all the advantages of the standard finite element method while explicitly conserving fluxes over each element all the examples presented show that the formulation employed is accurate and robust while using less cpu time than finite volume method

the continuous galerkin method is locally conservative we examine the conservation law structure of the continuous galerkin method we employ the scalar advection diffusion equation as a model problem for this purpose but our results are quite general and apply to time dependent nonlinear systems as well in addition to global conservation laws we establish local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements these results are somewhat surprising and contradict the widely held opinion that the continuous galerkin method is not locally conservative

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

a locally conservative multiscale finite element method for multiphase flow simulation through heterogeneous and fractured porous media abstract a multiscale locally conservative galerkin mslcg method is proposed to accurately simulate multiphase flow in heterogeneous and fractured porous media mslcg employs a coarse partition of the fine grids and multiscale basis function for mapping the fine scale information to the coarse scale unknowns different from standard multiscale finite element method msfem the main improvement of our mslcg is to use the property of local conservation at steady state conditions to define a numerical flux at element boundaries mslcg provides a way to extend standard msfem to handle challenging multiphase flow problems in heterogeneous and fractured porous media mslcg preserves all the advantages of the standard msfem while it provides explicitly conservative fluxes through each element we present a number of representative numerical examples to demonstrate that our method is efficient and accurate for simulating multiphase flow through heterogeneous and fractured porous media

construction of locally conservative fluxes for high order continuous galerkin finite element methods abstract despite their robustness it is known that standard continuous galerkin finite element methods cgfems do not produce a locally conservative flux field as a result their application to solving model problems that are derived from conservation laws can be limited to remedy this issue some form of post processing must be performed on the cgfem solution in this work a simple post processing technique is proposed to obtain a locally conservative flux field from a cgfem solution one distinct advantage of the proposed method is that it produces continuous normal flux at the element s boundary the post processing is implemented on nodal centered control volumes that are constructed from the original finite element mesh the post processing method is performed by solving an independent set of low dimensional problems posed on each element the associated linear algebra systems are of dimension 1 2 k 1 k 2 where k is the polynomial degree of cgfem basis on a triangular mesh a theoretical investigation is conducted to confirm that the post processed solution converges in an optimal fashion to the true solution in the h 1 semi norm various numerical examples that demonstrate the performance of technique are given specifically a simulation of a model for single phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post processing technique

scaling invariance of the homoclinic tangle the structure of the homoclinic tangle of 11 2 degrees of freedom hamiltonian systems in the neighborhood of the saddle point is invariant under discrete rescaling of the system s parameters the rescaling constant is derived from the separatrix map and the melnikov formula invariant manifolds for the periodically modulated duffing oscillator are computed numerically to confirm this property the scaling is related to the recently found invariance of the separatrix map under a discrete renormalization group a possibility to extend the scaling invariance to different systems is demonstrated the equivalency conditions under which two systems have the similarity of their chaotic layer structure near the saddle are derived a numerical example shows a duffing oscillator and a pendulum acted on by different periodic perturbations with the same structure of the tangle

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

an obstruction method for the destruction of invariant curves abstract an important point in a conservative dynamical system which is a parameter depending perturbation of an integrable system is the destruction of the invariant curves for small perturbations many invariant curves subsist according to kam theorem an increase in the size of perturbations produces eventually the destruction of the curves the stochastic zone is then very large in this paper we present for the standard map a geometrical approach to obtain values of the parameter for which no invariant curves exist this is due to the existence of heteroclinic points as it has been observed using other methods there are several scaling properties in the phenomena which are also discussed here

locally conservative stabilized finite element methods for variably saturated flow standard galerkin finite element methods for variably saturated groundwater flow have several deficiencies for instance local oscillations can appear around sharp infiltration fronts without the use of mass lumping and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries here we consider conforming finite element discretizations based on a multiscale formulation along with recently developed local postprocessing schemes the resulting approach maintains the basic flexibility and appeal of traditional finite element methods while controlling nonphysical oscillations and producing element wise mass conservative velocity fields accuracy and efficiency of the proposed schemes are evaluated through a series of steady state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains the schemes are formulated for a generic nonlinear advection diffusion equation and are thus applicable to many other flow models

novel semi implicit locally conservative galerkin silcg methods application to blood flow in a systemic circulation abstract three novel locally conservative galerkin lcg methods in their semi implicit form are proposed for 1d blood flow modelling in arterial networks these semi implicit discretisations are the second order taylor expansion silcg te method the streamline upwind petrov galerkin silcg supg procedure and the forward in time and central in space silcg ftcs method in the lcg method enforcement of the flux continuity condition at the element interfaces allows to solve the discretised system of equations at element level for problems with a large number of degrees of freedoms this offers a significant advantage over the standard continuous galerkin cg procedure the well established fully explicit lcg method is used for assessing the accuracy of the proposed new methods results presented in this work demonstrate that the proposed silcg methods are stable and as accurate as the explicit lcg method among the three methods proposed the silcg ftcs method requires considerably lower number of iterations per element and thus requires lowest amount of cpu time on the other hand the silcg te and silcg supg methods are stable and accurate for larger time step sizes although the standard newton method requires evaluation of both the jacobian matrix and the residual for every single iteration which may be expensive for standard implicit solvers the computed results show that the maximum number of iterations per element for silcg te and silcg supg is less than unity less than 0 3 and 0 7 respectively also numerical experiments show that the jacobian matrix can be calculated only once per time step allowing to save a significant amount of computational time

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

control volume method for numerical simulation of two phase immiscible flow in two and three dimensional discrete fractured media 1 we provide a numerical procedure for the simulation of two phase immiscible and incompressible flow in two and three dimensional discrete fractured media the concept of cross flow equilibrium is used to reduce the fracture dimension from n to n 1 in the calculation of flow in the fractures this concept which is often referred to as the discrete fracture model has a significant effect on the reduction of computational time the spatial discretization is performed with the control volume method this method is locally conservative and allows the use of unstructured grids to represent complex geometries such as discrete fracture configurations the relative permeability is upwinded with a criterion based on the evaluation of the flux direction at the boundaries of the control volumes which is consistent with the physics of fluid flow the system of partial differential equations is decoupled and solved using the implicit pressure explicit saturation impes approach the algorithm has been successfully tested in two and three dimensional numerical simulations of wetting phase fluid injection such as water in discrete fractured media saturated by a nonwetting phase such as nonaqueous phase liquid or oil with mild to high nonlinearity in relative permeability and capillary pressure to the best of our knowledge results for simulations of two phase immiscible and incompressible flow in three dimensional discrete fractured media including capillary and gravity effects are the first to appear in the literature

renormalization for stochastic layers abstract chirikov s standard mapping for stochastic layers is derived in a simpler and more general way a one parameter renormalization transformation yields analytically many universal quantities defined in a previous renormalization theory expressions are given for the width of stochastic layers that correct several previous estimates

escape of particles in a time dependent potential well we investigate the escape of an ensemble of noninteracting particles inside an infinite potential box that contains a time dependent potential well the dynamics of each particle is described by a two dimensional nonlinear area preserving mapping for the variables energy and time leading to a mixed phase space the chaotic sea in the phase space surrounds periodic islands and is limited by a set of invariant spanning curves when a hole is introduced in the energy axis the histogram of frequency for the escape of particles which we observe to be scaling invariant grows rapidly until it reaches a maximum and then decreases toward zero at sufficiently long times a plot of the survival probability of a particle in the dynamics as function of time is observed to be exponential for short times reaching a crossover time and turning to a slower decay regime due to sticky regions observed in the phase space

enriched galerkin methods for two phase flow in porous media with capillary pressure abstract in this paper we propose an enriched galerkin eg approximation for a two phase pressure saturation system with capillary pressure in heterogeneous porous media the eg methods are locally conservative have fewer degrees of freedom compared to discontinuous galerkin dg and have an efficient pressure solver to avoid non physical oscillations an entropy viscosity stabilization method is employed for high order saturation approximations entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains the iterative and sequential implicit pressure and explicit saturation impes algorithms are treated in time numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of eg

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree

the principal eigenvalue for periodic nonlocal dispersal systems with time delay abstract the theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time periodic nonlocal dispersal cooperative system with time delay then we apply it to a nicholson s blowflies population model and obtain a threshold type result on its global dynamics