quadrilateral mesh revisited several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent the effect of the bi section condition on the degenerate mesh conditions is also checked

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit we present a mathematical analysis of the asymptotic preserving scheme proposed in m lemou and l mieussens siam j sci comput 31 2008 pp 334 368 for linear transport equations in kinetic and diffusive regimes we prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles this property is satisfied under an explicitly given cfl condition this condition tends to a parabolic cfl condition for small mean free paths and is close to a convection cfl condition for large mean free paths our analysis is based on very simple energy estimates

error reduction of the adaptive conforming and nonconforming finite element methods with red green refinement in this paper we analyze the convergence of the adaptive conforming and nonconforming p 1 finite element methods with red green refinement based on standard dorfler marking strategy since the mesh after refining is not nested into the one before the usual galerkin orthogonality or quasi orthogonality for newest vertex bisection does not hold for this case to overcome such a difficulty we develop some new quasi orthogonality instead under certain condition on the initial mesh condition a consequently we show convergence of the adaptive methods by establishing the reduction of some total errors to weaken the condition on the initial mesh we propose a modified red green refinement and prove the convergence of the associated adaptive methods under a much weaker condition on the initial mesh condition b furthermore we also develop an initial mesh generator which guarantee that all the interior triangles are equilateral triangles satisfy condition a and the other triangles containing at least one vertex on the boundary satisfy condition b

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

diffusion approximation and computation of the critical size this paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation to obtain this result we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size the paper also contains some simple numerical examples and some new results for the milne problem introduction the computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts first the computation of the critical size is much easier for the diffusion approximation than for the original transport equation second for the critical size one can consider a host media x nxq with x0 given and 17 a positive number the problem of the critical size is then reduced to the computation of the parameter zj it turns out that when the transport operator is almost conservative the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate on the other hand the physical boundary condition for the diffusion approxi mation is of the form

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

group theory and phase transitions abstract group theoretical implications in the landau theory of phase transitions are reviewed notwithstanding the large amount of work mainly concerned with construction of the landau thermodynamic potential this review emphasizes the group theoretical aspects of the actual minimization of the potential the progress of the last few decades is reviewed and some open questions stated

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

generalized su 2 spherical harmonics the generating functions for polynomial tensors based on each su 2 tensor of rank from 7 to 13 angular momentum 7 2 to 13 2 are given in a positive form suitable for interpretation in terms of an integrity basis an iterative procedure for extending the results to higher rank tensors is indicated

a posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models abstract in this paper we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond based peridynamic models in particular we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems a posteriori error estimators are consequently prompted the reliability and efficiency of the estimators are proved connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space numerical experiments 1d are also given to test the theoretical conclusions

symmetry and phase transitions abstract the problem of symmetry changes at phase transitions is discussed in the light of recent developments in the theory of phase transitions as well as in terms of classical landau theory various group theoretical criteria or selection rules are formulated and discussed from the point of view of how well they comply with the basic requirements of the theory of phase transitions i e the existence of the absolute minimum of the landau potential landau ginzburg wilson hamiltonian and the continuity of the transition the physical significance of these criteria is examined

bayesian estimation of change points using the general linear model bayesian analysis is applied to the general linear model to develop a framework for studying different types of change in the mean value of time series and linear regressions the output of the bayesian analysis is the posterior distribution of change point location and amplitude this information provides a rational and relatively objective basis for making decisions as to where to locate a change point several examples of hydrological applications are presented to demonstrate the utility of the methodology

a conservative nonlocal convection diffusion model and asymptotically compatible finite difference discretization in this paper we first propose a nonlocal convection diffusion model in which the convection term is constructed in a special upwind manner so that mass conservation and maximum principle are maintained in any space dimension the well posedness of the proposed nonlocal model and its convergence to the classical local convection diffusion model are established a quadrature based finite difference discretization is then developed to numerically solve the nonlocal problem and it is shown to be consistent and unconditionally stable we further demonstrate that the numerical scheme is asymptotically compatible that is the approximate solutions converge to the exact solution of the corresponding local problem when 0 0 and h 0h 0 numerical experiments are also performed to complement the theoretical analysis

a framework of change point detection for multivariate hydrological series under changing environments not only univariate but also multivariate hydrological series might become nonstationary nonstationarity in forms of change point or trend has been widely studied for univariate hydrological series while it attracts attention only recently for multivariate hydrological series for multivariate series two types of change point need to be distinguished i e change point in marginal distributions and change point in the dependence structure among individual variables in this paper a three step framework is proposed to separately detect two types of change point in multivariate hydrological series i e change point detection for individual univariate series estimation of marginal distributions and change point detection for dependence structure the last step is implemented using both the cramer von mises statistic cvm method and the copula based likelihood ratio test clr method for clr three kinds of copula model symmetric asymmetric and pair copula are employed to construct the dependence structure of multivariate series monte carlo experiments indicate that clr is far more powerful than cvm in detecting the change point of dependence structure this framework is applied to the trivariate flood series composed of annual maxima daily discharge amdd annual maxima 3 day flood volume and annual maxima 15 day flood volume of the upper hanjiang river china it is found that each individual univariate flood series has a significant change point and the trivariate series presents a significant change point in dependence structure due to the abrupt change in the dependence structure between amdd and annual maxima 3 day flood volume all these changes are caused by the construction of the ankang reservoir this article is protected by copyright all rights reserved

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

the green s functions for peridynamic non local diffusion in this work we develop the green s function method for the solution of the peridynamic non local diffusion model in which the spatial gradient of the generalized potential in the classical theory is replaced by an integral of a generalized response function in a horizon we first show that the general solutions of the peridynamic non local diffusion model can be expressed as functionals of the corresponding green s functions for point sources along with volume constraints for non local diffusion then we obtain the green s functions by the fourier transform method for unsteady and steady diffusions in infinite domains we also demonstrate that the peridynamic non local solutions converge to the classical differential solutions when the non local length approaches zero finally the peridynamic analytical solutions are applied to an infinite plate heated by a gauss source and the predicted variations of temperature are compared with the classical local solutions the peridynamic non local diffusion model predicts a lower rate of variation of the field quantities than that of the classical theory which is consistent with experimental observations the developed method is applicable to general diffusion type problems

little groups of irreps of o 3 so 3 and the infinite axial subgroups a new chain criterion for the determination of little groups is presented this result distinguishes massive rotationally inequivalent irrep basis functions and allows for multiple branching paths consequently it is suitable for application to lie groups applied to the groups o 3 so 3 d h d c v c h and c the result enables the enumeration of all possible little groups together with their associated basis functions these results are relevant to the determination of the symmetry of a material from its linear and nonlinear optical properties and to the choices of order parameters for symmetry breaking in liquid crystals

nonconforming discontinuous galerkin methods for nonlocal variational problems we present a nonconforming discontinuous galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints the nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions the key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin it works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions a convergence theory is established for two types of finite dimensional spaces with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

orientations of reflection rotation groups the authors extend the work of a previous paper 1980 in which they showed that some of the phase choices in the 3jm factors of certain point group embeddings affect the orientation of the symmetry axes the occurrence of such choices is unpredictable and their effect is sometimes subtle but once a set of 3jm factors has been calculated they can determine the orientations of the symmetry axes and mirror planes and investigate how these orientations vary with the phase choices the properties of the reflection rotation groups are obtained from the the isomorphic pure rotation groups but care is necessary in interpreting the effects of the corresponding operations of for example d3 and c3v

nonlocal convection diffusion problems and finite element approximations abstract nonlocal models are becoming increasingly popular in numerical simulations of important application problems ranging from anomalous diffusion in heterogeneous environment to crack and fracture in composites in this paper we propose and compare some nonlocal models for convection diffusion problems and study their finite element approximations by piecewise constant and piecewise bilinear basis functions our focus is on the convection dominated case through analysis and numerical experiments we demonstrate the advantage of the upwind models which are more consistent with the physical nature and allow more stable and robust numerical approximations on the other hand the central model may produce unphysical oscillations

structural phase transitions in crystals broken symmetry isotropy groups a systematic procedure ensuring the determination of all isotropy groups of a given representation of a space group is presented for the first time isotropy groups play an important role in various areas of theoretical solid state physics for example only isotropy groups may occur in a structural phase transition driven by an order parameter belonging to a given representation the method uses the chain criterion directly on the image of the representation employing a labeling of the matrices by space group elements a notion of a substar of a wave vector associated with the representation is central to the method finally as a detailed illustration the method is applied to a structural phase transition in a15 systems driven by an x point order parameter the result agrees with previously reported ones

can the nonlocal characterization of sobolev spaces by bourgain et al be useful for solving variational problems we question whether the recent characterization of sobolev spaces by bourgain brezis and mironescu 2001 could be useful to solve variational problems on w 1 p omega to answer this we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier then for the approximated formulation we prove existence uniqueness and convergence of the solution to the unique solution of the initial formulation we show that these results can also be extended in the bv case interestingly this approximation leads to a unified implementation for sobolev spaces including with high p values and for the bv space finally we show how this theoretical study can indeed lead to a numerically tractable implementation and we give some image diffusion results as an illustration

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

a new approach for a nonlocal nonlinear conservation law we describe an approach to nonlocal nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux the spatially nonlocal operators we consider do not involve derivatives instead the spatial operator involves an integral that in a distributional sense reduces to a conventional nonlinear advective operator in particular we examine a nonlocal inviscid burgers equation which gives a basic form with which to characterize properties associated with well posedness and to examine numerical results for specific cases we describe the connection to a nonlocal viscous regularization which mimics the viscous burgers equation in an appropriate limit we present numerical results that compare the behavior of the nonlocal burgers formulation to the standard local case the developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theo

a nonlocal convection diffusion equation in this paper we study a nonlocal equation that takes into account convective and diffusive effects ut j u u g f u f u in rd with j radially symmetric and g not necessarily symmetric first we prove existence uniqueness and continuous dependence with respect to the initial condition of solutions this problem is the nonlocal analogous to the usual local convection diffusion equation ut u b f u in fact we prove that solutions of the nonlocal equation converge to the solution of the usual convection diffusion equation when we rescale the convolution kernels j and g appropriately finally we study the asymptotic behaviour of solutions as t when f u u q 1u with q 1 we find the decay rate and the first order term in the asymptotic regime

a convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models in this paper we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions the convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients such as the upper bound of the estimator the estimator reduction and the orthogonality property we also consider how the results are affected by the horizon parameter delta which characterizes the range of nonlocality numerical experiments are performed to verify our theoretical findings

conditioning analysis of nonlocal integral operators in fractional sobolev spaces we study the conditioning of nonlocal integral operators with singular and integrable kernels in fractional sobolev spaces these operators are used for instance in peridynamics formulation and nonlocal diffusion in one dimension 1d we present sharp quantification of the extremal eigenvalues in all three parameters size of nonlocality mesh size and regularity of the fractional sobolev space we accomplish sharpness both rigorously and numerically for the minimal eigenvalue we obtain sharpness analytically by using a nonlocal characterization of sobolev spaces we verify this estimate by exploiting the toeplitz property of the stiffness matrix however the analytical approach fails to give sharp quantification of the maximal eigenvalue hence in 1d we take an algebraic approach by directly working with the stiffness matrix entries which have complicated expressions due to all three parameters we systematically characterize the nonzero entries and dramatically simplify their expressions by u

trivariate statistical analysis of extreme rainfall events via the plackett family of copulas 1 recent hydrologic studies on multivariate stochastic analysis have indicated that copulas perform well for bivariate problems in particular the frank family of archimedean copulas has been a popular choice for a dependence model however there are limitations to extending such archimedean copulas to trivariate and higher dimensions with very specific restrictions on the kinds of dependencies that can be modeled in this study we examine a non archimedean copula from the plackett family that is founded on the theory of constant cross product ratio it is shown that the plackett family not only performs well at the bivariate level but also allows a trivariate stochastic analysis where the lower level dependencies between variables can be fully preserved while allowing for specificity at the trivariate level as well the feasible range of plackett parameters that would result in valid 3 copulas is determined numerically the trivariate plackett family of copulas is then applied to the study of temporal distribution of extreme rainfall events for several stations in indiana where the estimated parameters lie in the feasible region on the basis of a given rainfall depth and duration conditional expectations of rainfall features such as expected peak intensity time to peak and percentage cumulative rainfall at 10 cumulative time increments are evaluated the results of this study suggest that while the constant cross product ratio theory was conventionally applied to discrete type random variables it is also applicable to continuous random variables and that it provides further flexibility for multivariate stochastic analyses of rainfall

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

efficiency of a posteriori bem error estimates for first kind integral equations on quasi uniform meshes in the numerical treatment of integral equations of the first kind using boundary element methods bem the author and e p stephan have derived a posteriori error estimates as tools for both reliable computation and self adaptive mesh refinement so far efficiency of those a posteriori error estimates has been indicated by numerical examples in model situations only this work affirms efficiency by proving the reverse inequality based on best approximation on inverse inequalities and on stability of the discretization and complementary to our previous work an abstract approach yields a converse estimate this estimate proves efficiency of an a posteriori error estimate in the bem on quasi uniform meshes for symm s integral equation for a hypersingular equation and for a transmission problem

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

continuous time random walks on bounded domains a useful perspective to take when studying anomalous diffusion processes is that of a continuous time random walk and its associated generalized master equation we derive the generalized master equations for continuous time random walks that are restricted to a bounded domain and compare numerical solutions with kernel density estimates of the probability density function computed from simulations the numerical solution of the generalized master equation represents a powerful tool in the study of continuous time random walks on bounded domains

multivariate hydrological frequency analysis using copulas 1 this article presents the modeling of multivariate extreme values using copulas our approach allows us to model the dependence structure independently of the marginal distributions which is not possible with standard classical methods the methodology has been applied on two different problems in hydrology the first application is concerned with the combined risk in the framework of frequency analysis four copulas have been tested on peak flows from the watershed of peribonka in quebec canada the second application relates to the joint modeling of peak flows and volumes three copulas have been applied to the watershed of the rimouski river in quebec canada this approach using copulas is promising since it allows us to take into account a wide range of correlation which can happen in hydrology

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

properties of poincare generating functions for polynomial covariants tensors we show that irreducible polynomial covariants tensors based on a given irreducible representation of a finite group must have a definite p phase i e degree modulo p where p is the order of the center of the image of the group we also show that the numerator of the poincare function is often a polynomial symmetric around a given degree and we derive several interesting properties of the poincare function

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

green s functions in non local three dimensional linear elasticity in this paper we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical local formulation to the corresponding deformations in the peridynamic non local formulation owing to the linearity of the problem the response to a point load can be used to obtain the response to general body force loading functions by superposition using laplace and fourier transforms we thus obtain an integral representation for the three dimensional peridynamic solution with the help of green s functions we illustrate this new theoretical result by dynamic and static examples in one and three dimensions in addition to this main result we also derive the non local three dimensional jump conditions as well as the weak formulation of peridynamics together with the associated finite element discretization

space groups and their isotropy subgroups the theory of space groups and their subgroups is related to that of their irreducible representations several important theorems concerning the determination of isotropy subgroups are proved in particular we show that the translation subgroups of isotropy subgroups are in one to one correspondence with certain subsets called substars of the star characterizing the irreducible representation of the space group

variational theory and domain decomposition for nonlocal problems abstract in this article we present the first results on domain decomposition methods for nonlocal operators we present a nonlocal variational formulation for these operators and establish the well posedness of associated boundary value problems proving a nonlocal poincare inequality to determine the conditioning of the discretized operator we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix we then introduce a nonlocal two domain variational formulation utilizing nonlocal transmission conditions and prove equivalence with the single domain formulation a nonlocal schur complement is introduced we establish condition number bounds for the nonlocal stiffness and schur complement matrices supporting numerical experiments demonstrating the conditioning of the nonlocal one and two domain problems are presented

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

on a class of nonlocal wave equations from applications we study equations from the area of peridynamics which is a nonlocal extension of elasticity the governing equations form a system of nonlocal wave equations we take a novel approach by applying operator theory methods in a systematic way on the unbounded domain n we present three main results as main result 1 we find that the governing operator is a bounded function of the governing operator of classical elasticity as main result 2 a consequence of main result 1 we prove that the peridynamic solutions strongly converge to the classical solutions by utilizing for the first time strong resolvent convergence in addition main result 1 allows us to incorporate local boundary conditions in particular into peridynamics this avenue of research is developed in companion papers providing a remedy for boundary effects as main result 3 employing spherical bessel functions we give a new practical series representation of the solution which allows straightforward numerical treatment with symbolic computation

selection rules for symmetry breaking and symmetry restoration in continuous transitions abstract using landau s phenomenological theory as a framework subduction and chain subduction criteria are discussed as methods for obtaining directly selection rules for symmetry breaking for symmetry restoration induction and chain induction criteria give selection rules minimal symmetry breaking and minimal symmetry restoration are related to satisfaction of chain subduction and induction criteria

analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations we consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models two classes of discretization methods are presen

a spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions abstract the nonlocal diffusion model describes the diffusion process of solutes in complex media properly while the classical theory of partial differential equations can not provide an appropriate description the purpose of this paper is to provide illustrations from both theoretical and numerical perspectives of the computation of nonlocal diffusion models by legendre collocation methods compared to local numerical methods legendre collocation methods can achieve a fixed accuracy with much fewer unknowns whenever the computational domain is regular and the solutions are sufficiently smooth this paper is a groundwork towards efficient high order methods including spectral and spectral element methods for nonlocal diffusion equations with volume constrained boundary conditions

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

voronoi based peridynamics and cracking analysis with adaptive refinement summary the most common technique for the numerical implementation of peridynamic theory is the uniform discretization together with constant horizon however unlike the nonuniform discretization and varying horizons it is not a natural and intrinsic component of the adaptive refinement analysis and multiscale modeling besides it encounters discretization difficulty in analyzing irregular structures therefore to analyze problems with nonuniform discretization and varying horizons and solve the resulting problems of ghost forces and spurious wave reflection the dual horizon peridynamics based on uniform discretization is extended to the nonuniform discretization based on voronoi diagrams for which we call it voronoi based peridynamics we redefine the damage definition as well next an adaptive refinement analysis method based on the proposed voronoi based peridynamics and its numerical implementation is introduced finally the traditional bond based peridynamics and the voronoi based peridynamics with or without refinement are used to simulate 4 benchmark problems the examples of 2 d quasi static elastic deformation 2 d wave propagation 2 d dynamic crack growth and 3 d simulation of the kalthoff winkler experiment demonstrate the efficiency and effectivity of the proposed voronoi based peridynamics further the adaptive refinement analysis can be used to obtain reasonable crack path and crack propagation speed with reduced computational burden

pattern selection with o 3 symmetry abstract for each irreducible representation of so 3 and o 3 we determine up to conjugacy all isotropy subgroups and identify in particular the maximal isotropy subgroups each isotropy subgroup corresponds to a possible planform for the spherical benard problem using an equivariant branching lemma of cicogna 1 we prove for each of these representations the existence of solutions corresponding to a number of different planforms thus extending substantially the work of busse 2 3 and sattinger 4 we also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable