global attractor for the kirchhoff type equation with a strong dissipation abstract the paper studies the longtime behavior of the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the related continuous semigroup s t possesses in the phase space with low regularity a global attractor which is connected and an example is shown

a recursive approach to local mesh refinement in two and three dimensions abstract the newest vertex strategy for local refinement in two dimensions is reviewed and discussed a recursive algorithm for local refinement of tetrahedral meshes in three dimensions based on similar bisection strategy is described the recursive approach requires certain restrictions on the initial mesh on the other hand under this condition the refinement process can be kept as local as needed and it can be fully inverted simple data structures and derefinement algorithms are also outlined

occupation times of a ctrw on a lattice with anomalous sites abstract we develop formalism allowing us to derive results for the probability density of the occupation time of a set of sites on a lattice of a ctrw in which the pausing time density on the set can differ from those on the remaining lattice sites the problem is suggested by transillumination techniques used in optical imaging details are given for the case of a single site on the lattice and an anomalous pausing time density it is shown that if the random walk is transient i e in three or more dimensions when the variance of a single jump is finite the asymptotic density for the occupation time at the special point is a negative exponential in one dimension the random walk is recurrent here it is shown that the occupation time scales as t 1 2 these results extend to finite numbers of anomalous points

dynamical scaling in branching models for seismicity we propose a branching process based on a dynamical scaling hypothesis relating time and mass in the context of earthquake occurrence we show that experimental power laws in size and time distribution naturally originate solely from this scaling hypothesis we present a numerical protocol able to generate a synthetic catalog with an arbitrary large number of events the numerical data reproduce the hierarchical organization in time and magnitude of experimental interevent time distribution

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

a unified mixed finite element approximations of the stokes darcy coupled problem abstract in this paper we develop and analyze a unified approximation of the velocity pressure pair for the stokes darcy coupled problem in a plane domain it is well known that stable finite element approximations for the stokes problem may not be appropriate for darcy problem and for the coupling of fluid flow modeled by the stokes equations with porous media flow modeled by the darcy equation and therefore different spaces are commonly used for the discretizations of the darcy and the stokes problems in this work we proposed a modification of the darcy problem which allows us to apply the classical mini element to the whole coupled stokes darcy problem the proposed method is probably one of the cheapest method for continuous approximation of the coupled system has optimal accuracy with respect to solution regularity and has simple and straightforward implementations numerical experiments are also presented which confirm the excellent stability and accuracy of our method

hyperbolic parabolic singular perturbation for nondegenerate kirchhoff equations with critical weak dissipation we consider the hyperbolic parabolic singular perturbation problem for a nondegenerate quasilinear equation of kirchhoff type with weak dissipation this means that the dissipative term is multiplied by a coefficient b t which tends to 0 as t the case where b t 1 t p with p 1 but they remain true in the critical case p 1

linear response in complex systems ctrw and the fractional fokker planck equations abstract we consider the linear response of systems modelled by continuous time random walks ctrw and by fractional fokker planck equations under the influence of time dependent external fields we calculate the corresponding response functions explicitly the ctrw curve exhibits aging i e it is not translationally invariant in the time domain this is different from what happens under fractional fokker planck conditions

hyperbolic parabolic singular perturbation for quasilinear equations of kirchhoff type abstract we consider a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation for this purpose we show time decay estimates for hyperbolic parabolic singular perturbation problem for linear equations with a time dependent coefficient

a nystrom based finite element method on polygonal elements abstract we consider families of finite elements on polygonal meshes that are defined implicitly on each mesh cell as solutions of local poisson problems with polynomial data functions in the local space on each mesh cell are evaluated via nystrom discretizations of associated integral equations allowing for curvilinear polygons and non polynomial boundary data several experiments demonstrate the approximation quality of interpolated functions in these spaces

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

quadratic maximum entropy serendipity shape functions for arbitrary planar polygons in this paper we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons drawing on the work of bompadre et al 2012 1 and hormann and sukumar 2008 2 we adopt a relative entropy measure for signed positive or negative shape functions with nodal prior weight functions that have the appropriate zero set on the boundary of the polygon we maximize the objective functional subject to the constraints for quadratic completeness proposed by rand et al 2013 3 along an edge of a polygon the approximation is identical to univariate bernstein polynomials the choice of the nodal prior weight function ensures that the shape functions satisfy a weak kronecker delta property on each edge the shape functions are well defined for arbitrary planar polygons without self intersections on using a modified numerical integration scheme we show that the quadratic patch test is passed on polygonal meshes with convex and nonconvex elements numerical tests for the poisson equation on self similar trapezoidal meshes and quasiuniform polygonal meshes are presented which reveal the sound accuracy of the method and optimal rates of convergence in the l2l2 norm and the h1h1 seminorm are established

asymptotic distributions of continuous time random walks a probabilistic approach we provide a systematic analysis of the possible asymptotic distributions o one dimensional continuous time random walks ctrws by applying the limit theorems of probability theory biased and unbiased walks of coupled and decoupled memory are considered in contrast to previous work concerning decoupled memory and levy walks we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean obtaining asymmetric levy stable limits suprisingly it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution we discuss interesting properties of walks with an infinite mean waiting time between successive jumps

continuous time random walks and the fractional diffusion equation the average probability density p r t of random walks on fractals is revisited within the continuous time random walks formalism corrections to the accepted asymptotic stretched gaussian decay of p r t of the form ralpha are discussed it is shown that p r t obeys a diffusion equation with a fractional time derivative asymptotically and predictions about the value of alpha are presented

global solvability for the kirchhoff equations in exterior domains of dimension three abstract we consider the initial boundary value problem for the kirchhoff equations in exterior domains or in the whole space of dimension three and show that these problems admit time global solutions provided the norms of the initial data in the usual sobolev spaces of appropriate order are sufficiently small we obtain uniform estimates of the l 1 r norms with respect to time variable at each point in the domain of solutions of initial boundary value problem for the linear wave equations we then show that the estimates above yield the unique global solvability for the kirchhoff equations

finite dimensional attractors for the kirchhoff equation with a strong dissipation abstract the paper studies the existence of the finite dimensional global attractors and exponential attractors for the dynamical system associated with the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor for application the fact shows that for the concerned viscoelastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

residual based error estimate and quasi interpolation on polygonal meshes for high order bem based fem abstract only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes the bem based finite element method is one of the new discretization strategies which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally the article in hand addresses quasi interpolation operators for the approximation space over polygonal meshes to prove interpolation estimates the poincare constant is bounded uniformly for patches of star shaped elements these results give rise to the residual based error estimate for high order bem based fem and its reliability as well as its efficiency are proven such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive fem strategies numerical experiments show optimal rates of convergence for meshes with non convex elements on uniformly as well as on adaptively refined meshes

an attractor for the singularly perturbed kirchhoff equation with supercritical nonlinearity abstract we consider a quasilinear wave equation of kirchhoff type u t t 1 u 2 u u t f u g x where 0 is a small parameter without any growth restrictions on the nonlinearity f u we prove the existence of a finite dimensional global attractor on an appropriate bounded phase space the key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation

higher order bem based fem on polygonal meshes the bem based finite element method is reviewed and extended with higher order basis functions on general polygonal meshes these functions are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients they are treated by means of boundary integral formulations and are approximated using the boundary element method in the numerics to obtain higher order convergence a new approximation of the material coefficient is proposed since previous strategies are not sufficient following recent ideas error estimates are proved which guarantee quadratic convergence in the h 1 norm and cubic convergence in the l 2 norm the numerical realization is discussed and several experiments confirm the theoretical results

hyperbolic parabolic singular perturbation for mildly degenerate kirchhoff equations time decay estimates we consider degenerate kirchhoff equations with a small parameter e in front of the second order time derivative it is well known that these equations admit global solutions when e is small enough and that these solutions decay as t with the same rate of solutions of the limit problem of parabolic type in this paper we prove decay error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t concerning the decay rates it turns out that the difference decays faster than the two terms separately as t proofs involve a nonlinear step where we separate fourier components with respect to the lowest frequency followed by a linear step where we exploit weighted versions of classical energies

scaling range and cutoffs in empirical fractals fractal structures appear in a vast range of physical systems a literature survey including all experimental papers on fractals which appeared in the six physical review journals a char21 e and letters during the 1990s shows that experimental reports of fractal behavior are typically based on a scaling range ensuremath delta that spans only 0 5 char21 2 decades this range is limited by upper and lower cutoffs either because further data are not accessible or due to crossover bends focusing on spatial fractals a classification is proposed into a aggregation b porous media c surfaces and fronts d fracture and e critical phenomena most of these systems except for class e involve processes far from thermal equilibrium the fact that for self similar fractals in contrast to the self affine fractals of class c there are hardly any exceptions to the finding of ensuremath delta l 2 decades raises the possibility that the cutoffs are due to intrinsic properties of the measured systems rather than the specific experimental conditions and apparatus to examine the origin of the limited range we focus on a class of aggregation systems in these systems a molecular beam is deposited on a surface giving rise to nucleation and growth of diffusion limited aggregation like clusters scaling arguments are used to show that the required duration of the deposition experiment increases exponentially with ensuremath delta furthermore using realistic parameters for surfaces such as al 111 it is shown that these considerations limit the range of fractal behavior to less than two decades in agreement with the experimental findings it is conjectured that related kinetic mechanisms that limit the scaling range are common in other nonequilibrium processes that generate spatial fractals

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

a gronwall type lemma with parameter and its application to kirchhoff type nonlinear wave equation abstract the paper investigates a new gronwall type lemma with parameter 0 and establishes the uniform in time estimates as an application the paper studies the asymptotic behavior of the kirchhoff type equations with a strong dissipation and proves that the related solution semigroup possesses the global attractor

assessment of the relative ratio of correlated and uncorrelated waiting times in the southern california earthquakes catalogue abstract in the present study we investigated the interevent time interval distribution of earthquakes in southern california we analyzed and compared datasets of waiting times between consecutive earthquakes and time structure distorted datasets the aim of this study was to determine the proportion of waiting time values in the original catalogue that can be regarded as statistically distinguishable or indistinguishable from the baseline time intervals datasets where the original structure of the temporal distribution of earthquakes was disrupted we compiled two types of time structure distorted baseline sequences which comprised mean values of a shuffled original interevent data and b interevent times data from time randomized catalogues to compare the dynamical characteristics of the original and time structure distorted baseline sequences we used several data analysis methods such as power spectrum regression detrended fluctuation and multifractal detrended fluctuation analysis we also tested the nonlinear structure of the original and baseline sequences using the magnitude and sign scaling analysis method we calculated the z s c o r e in order to assess whether the interevent time values in the original dataset shared statistical similarity or dissimilarity with the time structure distorted baseline interevent data sequence we compared the interevent time values in the original dataset with the mean baseline interevent times computed for two types of time structure distorted sequences the results showed that about 30 of the original interevent times were statistically indistinguishable from the mean of the shuffled dataset and 10 from the mean of the time structure distorted baseline interevent dataset we performed similar analyses for other catalogues obtained from different parts of the world according to the results of this analysis the proportion of events in the original catalogues that were indistinguishable from sequences with disturbed time structure did not contradict the results obtained for the southern california catalogue

role of infinite invariant measure in deterministic subdiffusion statistical properties of the transport coefficient for deterministic subdiffusion are investigated from the viewpoint of infinite ergodic theory we find that the averaged diffusion coefficient is characterized by the infinite invariant measure of the reduced map we also show that when the time difference is much smaller than the total observation time the time averaged mean square displacement depends linearly on the time difference furthermore the diffusion coefficient becomes a random variable and its limit distribution is characterized by the universal law called the mittag leffler distribution

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

the ctrw in finance direct and inverse problems with some generalizations and extensions we study financial distributions within the framework of the continuous time random walk ctrw we review earlier approaches and present new results related to overnight effects as well as the generalization of the formalism which embodies a non markovian formulation of the ctrw aimed to account for correlated increments of the return

uncoupled continuous time random walk finite jump length probability density function in this work the uncoupled continuous time random walk model with the exponential jump length probability density function pdf is considered exact solutions for the pdf and second moment are obtained and investigated for different kinds of waiting time pdf

analysis of the discontinuous galerkin method for nonlinear convection diffusion problems abstract the subject matter is the analysis of the discontinuous galerkin finite element method applied to a nonlinear convection diffusion problem in the contrary to the standard fem the requirement of the conforming properties is omitted this allows us to consider general polyhedral elements with mutually disjoint interiors we do not require their convexity but assume only that they are star shaped we present an error analysis for the case of a nonsymmetric discretization of diffusion terms theoretical results are accompanied by numerical experiments

asymptotic solutions of decoupled continuous time random walks with superheavy tailed waiting time and heavy tailed jump length distributions we study the long time behavior of decoupled continuous time random walks characterized by superheavy tailed distributions of waiting times and symmetric heavy tailed distributions of jump lengths our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time we show that the probability density of the scaled walker position converges in the long time limit to a nondegenerate one only if the scaling function behaves in a certain way this function as well as the limiting probability density are determined in explicit form also we express the limiting probability density which has heavy tails in terms of the fox h function and find its behavior for small and large distances

an attractor for a nonlinear dissipative wave equation of kirchhoff type abstract in this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial boundary value problem of a quasilinear wave equation of kirchhoff type with a standard dissipation u t

infinite invariant density in a semi markov process with continuous state variables we report on a fundamental role of a non normalized formal steady state i e an infinite invariant density in a semi markov process where the state is determined by the interevent time of successive renewals the state describes certain observables found in models of anomalous diffusion e g the velocity in the generalized levy walk model and the energy of a particle in the trap model in our model the interevent time distribution follows a fat tailed distribution which makes the state value more likely to be zero because long interevent times imply small state values we find two scaling laws describing the density for the state value which accumulates in the vicinity of zero in the long time limit these laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density moreover we provide two distributional limit theorems for time averaged observables in these nonstationary processes we show that the infinite invariant density plays an important role in determining the distribution of time averages

arbitrary order trefftz like basis functions on polygonal meshes and realization in bem based fem polygonal meshes appear in more and more applications and the bem based finite element method fem turns out to be a forward looking approach the method uses trefftz like basis functions which are defined implicitly and are treated locally by means of boundary element methods bems the bem based finite element method is applicable on a variety of meshes including hanging nodes the aim of this presentation is to give a rigorous construction of h 1 conforming basis functions of a given arbitrary order yielding optimal rates of convergence in a finite element method for elliptic equations with the help of an interpolation operator approximation properties are proven which guarantee optimal rates of convergence in the h 1 as well as in the l 2 norm for finite element simulations these theoretical results are illustrated and verified by several numerical examples on polygonal meshes

limiting distributions of continuous time random walks with superheavy tailed waiting times we study the long time behavior of the scaled walker particle position associated with decoupled continuous time random walks which is characterized by superheavy tailed distribution of waiting times and asymmetric heavy tailed distribution of jump lengths both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution to analytically investigate the limiting density function we derive a number of different representations of this function and in this way establish its main properties we also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results

the partitioned element method in computational solid mechanics abstract a finite element like approximation method is proposed for solid mechanics applications in which the elements can take essentially arbitrary polygonal form a distinguishing feature of the method herein called the partitioned element method is a partitioning of the elements into quadrature cells over which the shape functions are taken to be piecewise linear the gradient and constant values for each cell are determined by minimizing a quadratic function which represents a combined smoothness and compatibility measure linear completeness of the shape function formulation is proved robustness in the presence of element non convexity and geometric degeneracy e g nearly coincident nodes are particular goals of the method convergence for various 2d linear elasticity problems is demonstrated and results for a finite deformation elastic plastic problem are compared to those of the standard fem

continuous time random walks under power law resetting we study continuous time random walks ctrw with power law distribution of waiting times under resetting which brings the walker back to the origin with a power law distribution of times between the resetting events two situations are considered under complete resetting the ctrw after the resetting event starts anew with a new waiting time independent of the prehistory under incomplete resetting the resetting of the coordinate does not influence the waiting time until the next jump we focus on the behavior of the mean squared displacement msd of the walker from its initial position on the conditions under which the probability density functions of the walker s displacement show universal behavior and on this universal behavior itself we show that the behavior of the msd is the same as in the scaled brownian motion sbm being the mean field model of the ctrw the intermediate asymptotics of the probability density functions pdf for ctrw under complete resetting provided they exist are also the same as in the corresponding case for sbm for incomplete resetting however the behavior of the pdf for ctrw and sbm is vastly different

self similar branching of aftershock sequences abstract in this paper we propose a branching aftershock sequence bass model for seismicity we suggest that the bass model is a preferred alternative to the widely studied epidemic type aftershock sequence etas model in the bass model an initial or seed earthquake is specified the subsequent earthquakes are obtained from the statistical distributions of magnitude time and location the magnitude scaling is based on a combination of the gutenberg richter scaling relation and the modified bath s law for the scaling relation of aftershocks relative to the magnitude of the seed earthquake omori s law specifies the distribution of earthquake times and a modified form of omori s law specifies the distribution of earthquake locations since the bass model is specified by the four scaling relations it is fully self similar this is not the case for etas we also give a deterministic version of bass and show that it satisfies tokunaga side branching statistics in a similar way to diffusion limited aggregation dla

generalized diffusion equation and analytical expressions to neutron scattering experiments abstract an integro differential diffusion equation with linear force based on the continuous time random walk model is considered the equation generalizes the ordinary and fractional diffusion equations analytical expressions related to neutron scattering experiments are presented and analyzed which can be used to describe for instance biological systems

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

recovery based error estimators for the vem and bem based fem abstract in this article we consider recovery based a posteriori error estimators for the virtual element method vem and the boundary element method based finite element method bem based fem both methods are galerkin methods on polygonal and polyhedral grids for the numerical solution of partial differential equations thus they are highly flexible and particularly efficient in combination with adaptive refinement our error estimator computes the distance between the finite element gradient and its post processed version to obtain an accurate approximation of the true error the post processing is realised by local averaging and is therefore easy to implement and fast furthermore we have found points of extraordinary accuracy so called stress points on specific regular elements for the bem based fem we demonstrate that when sampling these stress points the recovered gradient becomes superconvergent which means that it converges at a greater rate than the unprocessed gradient

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

supg stabilization for the nonconforming virtual element method for advection diffusion reaction equations abstract we present the design convergence analysis and numerical investigations of the nonconforming virtual element method with streamline upwind petrov galerkin vem supg stabilization for the numerical resolution of convection diffusion reaction problems in the convective dominated regime according to the virtual discretization approach the bilinear form is split as the sum of a consistency and a stability term the consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form including the supg stabilization term polynomial projections can be computed exactly from the degrees of freedom the stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients the nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface optimal convergence properties of the method are proved in a suitable norm which includes contribution from the advective stabilization terms experimental results confirm the theoretical convergence rates

residual error estimate for bem based fem on polygonal meshes a conforming finite element method on polygonal meshes is reviewed which handles hanging nodes naturally trial functions are defined to fulfil the homogeneous pde locally and they are treated by means of local boundary integral equations using a quasi interpolation operator of clement type a residual based error estimate is obtained this a posteriori estimator can be used to rate the accuracy of the approximation over polygonal elements or it can be applied to an adaptive bem based fem the numerical experiments confirm our results and show optimal convergence for the adaptive strategy on general meshes

energy norm a posteriori error estimation for discontinuous galerkin methods in this paper we present a residual based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous galerkin approximations of elliptic problems the theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence 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

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

fem with trefftz trial functions on polyhedral elements the goal of this paper is to generalize a variant of the bem based fem for second order elliptic boundary value problems to three space dimensions here the emphasis lies on polyhedral meshes with polygonal faces where even non convex elements are allowed due to an implicit definition of the trial functions in the spirit of trefftz the strategy yields conforming approximations and is very flexible with respect to the meshes thus it gets into the line of recent developments in several areas the arising local problems are treated by two dimensional galerkin schemes coming from finite and boundary element formulations with the help of a new interpolation operator and its properties convergence estimates are proven in the h 1 as well as in the l 2 norm numerical experiments confirm the theoretical results

missing data in aftershock sequences explaining the deviations from scaling laws in this paper we extend the branching aftershock sequence model to study the role of missing data at short times and small amplitudes after a mainshock we apply this model which contains three parameters characterizing the missing data to the magnitude and temporal statistics of four aftershock sequences in california we find that the observed time dependent deviations of the frequency magnitude scaling from the gutenberg richter power law dependency can be described quantitatively by the model we also show that for the same set of parameters the model is able to explain quantitatively the observed magnitude dependent deviations of the temporal decay of aftershocks from omori s law in addition we show that the same sets of data can also reproduce quite well the various functional forms of the probability density functions of the return times between consecutive events with magnitudes above a prescribed threshold as well as the violation of scaling at short and intermediate time scales

generalized master equations for random walks with time dependent jump sizes in this work we develop a unified generalized master equation gme framework that extends the theory of continuous time random walks to include the cases when the jump sizes may have a delayed dependence on time and are not restricted to any particular class of distributions we compare and contrast analytical and numerical behavior of the corresponding master equations including the instantaneous vs delayed jump dependence on time and exponential vs mittag leffler interarrival times with the latter leading to fractional evolution equation we provide existence and uniqueness proofs for the resulting gmes

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

a high order method for evaluating derivatives of harmonic functions in planar domains we propose a high order integral equation based method for evaluating interior and boundary derivatives of harmonic functions that are specified by their dirichlet data in planar domains the tangential derivative of the given dirichlet data is used to form a complementary neumann problem whose solution is a harmonic conjugate of the function whose derivatives we seek we use a high order nystrom method to compute the dirichlet trace of the harmonic conjugate on the domain boundary the tangential derivative of this harmonic conjugate effected via an fft is the normal derivative of the original function because the original and conjugate harmonic functions are the real and imaginary parts of a complex analytic function we are able to use cauchy s integral formulas to compute function values and derivatives inside the domain several numerical experiments on smooth domains and domains with corners illustrate the rapid convergence and high accuracy of the proposed approach

comparison of the hurst and dea exponents between the catalogue and its clusters the california case abstract in this work we analyse the long run correlations of the seismic catalogue and of its clusters by means of the diffusion entropy analysis dea and the value of the hurst exponent first we calculate the values for the whole catalogue for the rate and the inter event times and distance distribution over time and subsequently we calculate the values for the declustered catalogue and the main clusters we find a wide variety of behaviours which depart from the poisson statistics

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization