the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

improved convergence towards generalized euler mascheroni constant we propose new simple sequences approximating the euler mascheroni constant and its generalization which converge faster towards their limits than those considered by detemple d w detemple a quicker convergence to euler s constant am math monthly 100 5 1998 468 470 sintamarian a sintamarian a generalization of euler s constant numer algorithms 46 2 2007 141 151 and vernescu a vernescu a new accelerate convergence to the constant of euler gazeta matem ser a bucharest xvii xcvi 4 1999 273 278

inequities in the shanks renyi prime number race an asymptotic formula for the densities chebyshev was the first to observe a bias in the distribution of primes in residue classes the general phenomenon is that if a is a nonsquare mod q and b is a square mod q then there tend to be more primes congruent to a mod q than b mod q in initial intervals of the positive integers more succinctly there is a tendency for p x q a to exceed p x q b rubinstein and sarnak defined d q a b to be the logarithmic density of the set of positive real numbers x for which this inequality holds intuitively d q a b is the probability that p x q a p x q b when x is chosen randomly in this paper we establish an asymptotic series for d q a b that can be instantiated with an error term smaller than any negative power of q this asymptotic formula is written in terms of a variance v q a b that is originally defined as an infinite sum over all nontrivial zeros of dirichlet l functions corresponding to characters mod q we show how v q a b can be evaluated exactly as a finite expression in addition to providing the exact rate at which d q a b converges to 1 2a sq grows these evaluations allow us to compare the various density values d q a b as a and b vary modulo q by analyzing the resulting formulas we can explain and predict which of these densities will be larger or smaller based on arithme tic properties of the residue classes a and b mod q for example we show that if a is a prime power and a 0 is not then d q a 1 d q a 0 1 for all but finitely many moduli q for which both a and a 0 are nonsquares finally we establish rigorous numerical bounds for these densities d q a b and report on extensive calculations of them including for ex ample the determination of all 117 density values that exceed 9 10

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

on some euler mascheroni type sequences the aim of this paper is to introduce new sequences which extend the classical sequence convergent to the euler mascheroni constant some new techniques for obtaining faster converging sequences of euler mascheroni type are given we show how these sequences can be combined with the harmonic sequence to improve the rate of convergence

a class of integral approximations for the factorial function we introduce stirling s formula in a more general class of approximation formulas to extend the integral representation of liu z liu a new version of the stirling formula tamsui oxf j math sci 23 4 2007 389 392 finally an accurate approximation for the factorial function is established

monotone positive solutions of second order nonlinear differential equations we obtain an existence theorem for monotone positive solutions of nonlinear second order ordinary differential equations by using the schauder tikhonov fixed point theorem the result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in r n n greater than or equal to 3

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

computation of the gamma digamma and trigamma functions this paper gives an approximation for the gamma function that while different has the same form as one by lanczos siam j numer anal b1 1964 pp 86 96 both approximations correct stirlin

optimal distributed control of nonlocal steady diffusion problems a control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied the control is the right hand side forcing function and the objective of control is a standard matching functional a recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system when sufficient conditions on certain kernel functions and the volume constraints hold the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived we demonstrate the convergence as the nonlocal interactions vanish of the optimal nonlocal state to the optimal state of a local pde constrained control problem we also define continuous and discontinuous galerkin finite element discretizations of the optimality system for which we derive a priori error estimates numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained f

weighted averages of modular l values using an explicit relative trace formula on gl 2 we derive a formula for averages of modular l values in the critical strip weighting by fourier coefficients hecke eigenvalues and petersson norms as an application we show that a grh holds for these averages as the weight or the level goes to we also use the formula to give explicit zero free regions of the form im s 0 for some particular modular l functions

asymptotically compatible schemes and applications to robust discretization of nonlocal models many problems in nature being characterized by a parameter are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit numerical schemes that are convergent in both regimes offer robust discretizations which can be highly desirable in practice the asymptotically compatible schemes studied in this paper meet such objectives for a class of parametrized problems an abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits in particular the framework can be applied to nonlocal diffusion models and a general state based peridynamic system parametrized by the horizon radius recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter thus it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical dis

the dirichlet problem for nonlocal operators in this note we set up the elliptic and the parabolic dirichlet problem for linear nonlocal operators as opposed to the classical case of second order differential operators here the boundary data are prescribed on the complement of a given bounded set we formulate the problem in the classical framework of hilbert spaces and prove unique solvability using standard techniques like the fredholm alternative

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

new fourier transform containing a pair of complex euler γ functions with a monomial mathematical and physical applications one of the goals of this paper is to propose an elementary method to find a general formula for a new fourier transform containing a pair of complex euler functions with a monomial sm in terms of gauss s hypergeometric functions 2f1 some mathematical results have been obtained and discussed using this transform furthermore physical applications in the phase space quantum mechanics requiring the expectation values of position and momentum operators for a quantum system endowed with position dependent effective mass are presented

refinements of gurland s formula for pi the aim of this paper is to refine gurland s formula for approximating pi we prove the complete monotonicity of a class of functions related to gurland s formula and then we state some sharp inequalities

asymptotic integration of second order nonlinear differential equations via principal and nonprincipal solutions let u and v denote respectively the principal and nonprincipal solutions of the second order linear equation p t x q t x 0 defined on some half line of the form t it is shown that for any given real numbers a and b the nonlinear equation p t x q t x f t x has a solution x t asymptotic to av t bu t as t under some reasonable conditions on the function f examples are given to illustrate the results

asymptotically linear solutions of differential equations via lyapunov functions we discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of lyapunov functions the approach is new in this field and allows for simpler proofs of general results regarding emden fowler like equations

on the treatment of boundary conditions for bond based peridynamic models abstract in this paper we propose two approaches to apply boundary conditions for bond based peridynamic models there has been in recent years a renewed interest in the class of so called non local models which include peridynamic models for the simulation of structural mechanics problems as an alternative approach to classical local continuum models however a major issue which is often disregarded when dealing with this class of models is concerned with the manner by which boundary conditions should be prescribed our point of view here is that classical boundary conditions since applied on surfaces of solid bodies are naturally associated with local models the paper describes two methods to incorporate classical dirichlet and neumann boundary conditions into bond based peridynamics the first method consists in artificially extending the domain with a thin boundary layer over which the displacement field is required to behave as an odd function with respect to the boundary points the second method resorts to the idea that peridynamic models and local models should be compatible in the limit that the so called horizon vanishes the approach consists then in decreasing the horizon from a constant value in the interior of the domain to zero at the boundary so that one can directly apply the classical boundary conditions we present the continuous and discrete formulations of the two methods and assess their performance on several numerical experiments dealing with the simulation of a one dimensional bar

chebyshev series expansion of inverse polynomials the chebyshev series expansion n 0 antn x of the inverse of a polynomial j 0kbjtj x is well defined if the polynomial has no roots in 1 1 if the inverse polynomial is decomposed into partial fractions the an are linear combinations of simple functions of the polynomial roots also if the first k of the coefficients an are known the others become linear combinations of these derived recursively from the bj s on a closely related theme finding a polynomial with minimum relative error towards a given f x is approximately equivalent to finding the bj in f x 0k bj tj x 1 k 1 an tn x a newton algorithm produces these if the chebyshev expansion of f x is known

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

asymptotic to polynomials solutions for nonlinear differential equations abstract this article is concerned with solutions that behave asymptotically like polynomials for nth order n 1 nonlinear ordinary differential equations for each given integer m with 1 m n 1 sufficient conditions are presented in order that for any real polynomial of degree at most m there exists a solution which is asymptotic at to this polynomial conditions are also given which are sufficient for every solution to be asymptotic at to a real polynomial of degree at most n 1 the application of the results obtained to the special case of second order nonlinear differential equations leads to improved versions of the ones contained in the recent paper by lipovan glasg math j 45 2003 179 and of other related results existing in the literature

enabling fast stable and accurate peridynamic computations using multi time step integration abstract peridynamics is a nonlocal extension of classical continuum mechanics that is well suited for solving problems with discontinuities such as cracks this paper extends the peridynamic formulation to decompose a problem domain into a number of smaller overlapping subdomains and to enable the use of different time steps in different subdomains this approach allows regions of interest to be isolated and solved at a small time step for increased accuracy while the rest of the problem domain can be solved at a larger time step for greater computational efficiency performance of the proposed method in terms of stability accuracy and computational cost is examined and several numerical examples are presented to corroborate the findings

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

convergence studies in meshfree peridynamic simulations meshfree methods are commonly applied to discretize peridynamic models particularly in numerical simulations of engineering problems such methods discretize peridynamic bodies using a set of nodes with characteristic volume leading to particle based descriptions of systems in this paper we perform convergence studies of static peridynamic problems we show that commonly used meshfree methods in peridynamics suffer from accuracy and convergence issues due to a rough approximation of the contribution of nodes near the boundary of the neighborhood of a given node to numerical integrations we propose two methods to improve meshfree peridynamic simulations the first method uses accurate computations of volumes of intersections between neighbor cells and the neighborhood of a given node referred to as partial volumes the second method employs smooth influence functions with a finite support within peridynamic kernels numerical results demonstrate great improvements in accuracy and convergence of peridynamic numerical solutions when using the proposed methods

a fast and faithful collocation method with efficient matrix assembly for a two dimensional nonlocal diffusion model abstract the peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities which cannot be described properly by the classical theory of solid mechanics however the operator in the peridynamic theory is nonlocal so the resulting numerical methods generate dense or full coefficient matrices which require o n 2 of memory where n is the number of unknowns in the discretized system gaussian types of direct solvers which were traditionally used to solve these problems require o n 3 of operations furthermore due to the singularity of the kernel in the peridynamic model the evaluation and assembly of the coefficient matrix can be very expensive numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main computational cost the significantly increased computational work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications especially in multiple space dimensions we develop a fast and faithful collocation method for a two dimensional nonlocal diffusion model which can be viewed as a scalar valued version of a peridynamic model without using any lossy compression but rather by exploiting the structure of the coefficient matrix the new method reduces the evaluation and assembly of the coefficient matrix by o n reduces the computational work from o n 3 required by the traditional methods to o n log 2 n and the memory requirement from o n 2 to o n numerical results are presented to show the utility of the fast method

on the solution set for a class of sequential fractional differential equations we establish here that under some simple restrictions on the functional coefficient a t the solution set of the fractional differential equation 0d tx a t x 0 splits between eventually small and eventually large solutions as t where 0d t designates the riemann liouville derivative of the order 0 1

fast and accurate implementation of fourier spectral approximations of nonlocal diffusion operators and its applications abstract this work is concerned with the fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions for radially symmetric kernels the nonlocal operators under consideration are diagonalizable in the fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or fourier symbols consisting of possibly singular and highly oscillatory integrals for a large class of fractional power like kernels we propose a new approach based on reformulating the fourier symbols both as coefficients of a series expansion and solutions of some simple ode models we then propose a hybrid algorithm that utilizes both truncated series expansions and high order runge kutta ode solvers to provide fast evaluation of fourier symbols in both one and higher dimensional spaces it is shown that this hybrid algorithm is robust efficient and accurate as applications we combine this hybrid spectral discretization in the spatial variables and the fourth order exponential time differencing runge kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal allen cahn equations nonlocal cahn hilliard equations and nonlocal phase field crystal models numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

two series expansions for the logarithm of the gamma function involving stirling numbers and containing only rational coefficients for certain arguments related to π 1 abstract in this paper two new series for the logarithm of the function are presented and studied their polygamma analogs are also obtained and discussed these series involve the stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to 1 in particular for any value of the form ln 1 2 n 1 and k 1 2 n 1 where k stands for the kth polygamma function is positive rational greater than 1 6 n is integer and k is non negative integer these series have rational terms only in the specified zones of convergence derived series converge uniformly at the same rate as n ln m n 2 where m 1 2 3 depending on the order of the polygamma function explicit expansions into the series with rational coefficients are given for the most attracting values such as ln 1 ln 2 1 ln 1 2 1 1 1 2 1 and k 1 besides in this article the reader will also find a number of other series involving stirling numbers gregory s coefficients logarithmic numbers also known as bernoulli numbers of the second kind cauchy numbers and generalized bernoulli numbers finally several estimations and full asymptotics for gregory s coefficients for cauchy numbers for certain generalized bernoulli numbers and for certain sums with the stirling numbers are obtained in particular these include sharp bounds for gregory s coefficients and for the cauchy numbers of the second kind

an application of integral inequality to second order nonlinear oscillation abstract a simple result concerning integral inequalities enables us to give an alternative proof of waltman s theorem lim t t 0 a s ds implies oscillation of the second order nonlinear equation y t a t f y t 0 to prove an analog of wintner s theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations assuming that lim t t 0 a s ds exists and to give an alternative proof and to extend a result of butler an often used condition on the coefficient a t is given a more familiar equivalent form and an oscillation criterion involving this condition is established

monotonicity properties inequalities and asymptotic expansions associated with the gamma function we define x by the equality x 1 x e x 8 x 3 4 x 2 x 1 30 1 6 e x we call x the remainder of ramanujan s formula in this paper we present some properties for x including monotonicity properties inequalities and asymptotic expansions furthermore we present some full asymptotic expansions for the gamma function related to the nemes ramanujan and burnside formulas

on the asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas in this paper we present new asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas

high order parameter approximation for von mises fisher distributions abstract this paper concerns the issue of the maximum likelihood estimation mle for the concentration parameters of the von mises fisher vmf distributions which are crucial to directional data analysis in particular we study the numerical approximation approach for solving the implicit nonlinear equation arising from building the mle of the concentration parameter of vmf distributions in addition we address the implementation of i s x the modified bessel function of the first kind which is the most time consuming and fundamental ingredient in the proposed approximation scheme of the mle for the main contribution of this paper is two fold the first is to present a two steps halley based method for exploring a high order approximation of the mle for which can significantly contribute to the improvement of estimation accuracy the second is to develop a novel approach for the implementation of i s x which can make the substantial improvement of computation efficiency for computing the mle approximation for the numerical experiments were conducted to compare the proposed schemes with those in the existing works by tanabe et al 1 and sra 2 the experimental results show that given the same amount of computation as their methods the proposed high order scheme can achieve much more accurate approximations while our implementation of i s x is preferable yet desirable for high dimensional applications

dirichlet s principle and wellposedness of solutions for a nonlocal p laplacian system abstract we prove dirichlet s principle for a nonlocal p laplacian system which arises in the nonlocal setting of peridynamics when p 2 this nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain our analysis uses nonlocal versions of integration by parts techniques that resemble the classical green and gauss identities the nonlocal energy functional associated with this elliptic type system exhibits a general kernel which could be weakly singular the coercivity of the system is shown by employing a nonlocal poincare s inequality we use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy from which we obtain the wellposedness of this steady state diffusion system

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 simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

monotone positive solutions for a class of second order nonlinear differential equations it is shown that the second order nonlinear differential equation p t x q t x f t x x t 0 has monotone positive solutions asymptotic to a prescribed nonprincipal solution v t at of the corresponding linear homogeneous equation p t x q t x 0 t 0 where email protected c 0 0 email protected c 0 r and email protected c 0 xrxr r an example is given to illustrate the obtained result

positive solutions of second order differential equations we prove a nonoscillation result for second order nonlinear ordinary differential equations an application to the schrodinger equation is also given c 2004 elsevier ltd all rights reserved

the asymptotic series of the generalized stirling formula the aim of this paper is to construct some asymptotic expansions which produce increasingly accurate approximations of the gamma function

best estimates of the generalized stirling formula the aim of this paper is to introduce an approximations family of the factorial function that contains stirling s formula burnside s formula and gosper s formula the parameters which provide the best approximations are indicated finally numerical computations are made to show the superiority of our formulas over other known formulas

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

integral inequalities of gronwall bellman bihari type and asymptotic behavior of certain second order nonlinear differential equations abstract several integral inequalities similar to gronwall bellmann bihari inequalities are obtained these inequalities are used to discuss the asymptotic behavior of certain second order nonlinear differential equations

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

on the asymptotic integration of nonlinear differential equations the aim of the present paper is twofold firstly the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations the class of second order equations with bihari like nonlinearity secondly some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations

conditioning and error analysis of nonlocal operators with local boundary conditions tubitak mfag 115f473 european commission marie curie career integration 293978 united states department of energy doe 1120 1120 99

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

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

asymptotic integration of nonlinear ϕ laplacian differential equations abstract the aim of the present paper is to study the existence of solutions to initial values problems for a laplace like operator we generalize the results of agarwal 2007 1 section 4 theorem 3 and the result of philos 2004 20 theorem 5 to the laplacian like problems

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

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

global existence of solutions with prescribed asymptotic behavior for second order nonlinear differential equations where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel we always assume without mentioning that the function f t u v is continuous in the domain d t u v t t0 u v r where t0 1 it is supposed also that all equations and inequalities involving t hold for t t0 unless otherwise speci9ed the symbols o and o have their standard meanings as t and we use the notation r def 0 eq 1 is often used for mathematical modelling of various physical chemical and biological systems and attracts constant interest of researchers a great deal of papers published during the last three decades are concerned with local and global existence of solutions of eq 1 and its numerous particular cases uniqueness of solutions

existence theory for single and multiple solutions to singular positone boundary value problems abstract in this paper we present some old and new existence results for singular boundary value problems our nonlinearity may be singular in its dependent variable

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

computing the gamma function using contour integrals and rational approximations some of the best methods for computing the gamma function are based on numerical evaluation of hankel s contour integral for example temme evaluates this integral based on steepest descent contours by the trapezoid rule here we investigate a different approach to the integral the application of the trapezoid rule on talbot type contours using optimal parameters recently derived by weideman for computing inverse laplace transforms relatedly we also investigate quadrature formulas derived from best approximations to exp z on the negative real axis following cody meinardus and varga the two methods are closely related and both converge geometrically we find that the new methods are competitive with existing ones even though they are based on generic tools rather than on specific analysis of the gamma function

on the asymptotic integration of a class of sublinear fractional differential equations we estimate the growth in time of the solutions to a class of nonlinear fractional differential equations d0 x x0 f t x which includes d0 x x0 h t x with 0 1 for the case of slowly decaying coefficients h the proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x t o ta when t for 1 1 a 0 our result can be thought of as a noninteger counterpart of the classical bihari asymptotic integration result for nonlinear ordinary differential equations by a carefully designed example we show that in some circumstances such an estimate is optimal