a fast collocation method for a static bond based linear peridynamic model abstract we develop a fast collocation method for a steady state bond based linear peridynamic model in two space dimensions the method reduces the computational work from o n 2 per krylov subspace iteration in a traditional collocation method to o n log n and the memory requirement from o n 2 in a collocation method to o n where n is the number of unknowns in the discrete system furthermore the method reduces the computational work of evaluating and assembling the stiffness matrix which often constitutes a major portion of the overall computational work from o n 2 in a collocation method to o n the significant reduction of the cpu times and storage in the fast collocation method is achieved by carefully exploring the structure of the stiffness matrix of the collocation method without any lossy compression involved in other words the fast method is evaluated in an equivalent but more efficient manner therefore the fast method generates identical numerical solutions as the traditional collocation method does and naturally inherits the stability and convergence properties that were already proved for a traditional collocation method numerical results are presented to show the utility of the fast method

the solvability of a strongly coupled nonlocal system of equations abstract we prove existence and uniqueness of strong pointwise solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of euclidean space the system of equations comes from a linearization of a nonlocal model of elasticity in solid mechanics it is a nonlocal analogue of the navier lame system of classical elasticity we use a well known semigroup technique that hinges on the strong solvability of the corresponding steady state elliptic system the leading operator is an integro differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field for an operator possessing an asymmetric kernel comparable to that of the fractional laplacian we prove the l 2 solvability of the elliptic system in a bessel potential space using the fourier transform and a priori estimates this l 2 solvability together with the hille yosida theorem is used to prove the well posedness of the wave type time dependent problem for the fractional laplacian kernel we extend the solvability to l p spaces using classical multiplier theorems

which functions may be the upper bohl function of the diagonal discrete linear time varying systems abstract in this paper we give the necessary and sufficient conditions for a function f to be an upper bohl function of the diagonal discrete linear time varying systems

traces for homogeneous sobolev spaces in infinite strip like domains abstract in this paper we construct a trace operator for homogeneous sobolev spaces defined on infinite strip like domains we identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse the intrinsic seminorm involves two features not encountered in the trace theory of bounded lipschitz domains or half spaces first due to the strip like structure of the domain the boundary splits into two infinite disconnected components the traces onto each component are not completely independent and the intrinsic seminorm contains a term that measures the difference between the two traces second as in the usual trace theory there is a term in the seminorm measuring the fractional sobolev regularity of the trace functions with a difference quotient integral however the finite width of the strip like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation the screened homogeneous fractional sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest and we study their basic properties we conclude the paper with applications of the trace theory to partial differential equations

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

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

nonlocal diffusion and peridynamic models with neumann type constraints and their numerical approximations abstract this paper studies nonlocal diffusion models associated with a finite nonlocal horizon parameter that characterizes the range of nonlocal interactions the focus is on the variational formulation associated with neumann type constraints and its numerical approximations we establish the well posedness for some variational problems associated and study their local limit as 0 a main contribution is to derive a second order convergence to the local limit we then discuss the numerical approximations including standard finite element methods and quadrature based finite difference methods we study their convergence in the nonlocal setting and in the local limit

the bond based peridynamic system with dirichlet type volume constraint in this paper the bond based peridynamic system is analysed as a non local boundary value problem with volume constraint the study extends earlier works in the literature on non local diffusion and non local peridynamic models to include non positive definite kernels we prove the well posedness of both linear and nonlinear variational problems with volume constraints the analysis is based on some non local poincare type inequalities and the compactness of the associated non local operators it also offers careful characterizations of the associated solution spaces such as compact embedding separability and completeness in the limit of vanishing non locality the convergence of the peridynamic system to the classical navier equations of elasticity with poisson ratio is demonstrated

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

nonlocal optimal design a new perspective about the approximation of solutions in optimal design abstract it is well known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations in the present work a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem the new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals the purpose of this paper is to prove existence of optimal design for the new model this work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional see 14

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

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 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

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

an asymptotically compatible formulation for local to nonlocal coupling problems without overlapping regions abstract in this paper we design and analyze an explicit partitioned procedure for a 2d dynamic local to nonlocal ltn coupling problem based on a new nonlocal robin type transmission condition the nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter characterizing the range of nonlocal interactions and the local subproblem is described by the classical heat equation we consider a heterogeneous system where the local and nonlocal subproblems present different physical properties and employ no overlapping region between the two subdomains we first propose a new generalization of classical local neumann type condition by converting the local flux to a correction term in the nonlocal model and show that the proposed neumann type boundary formulation recovers the local case as o 2 in the l norm we then extend the nonlocal neumann type boundary condition to a robin type boundary condition and develop a local to nonlocal coupling formulation with robin dirichlet transmission conditions to stabilize the explicit coupling procedure and to achieve asymptotic compatibility the choice of the coefficient in the robin condition is obtained via amplification factor analysis for the discretized system with coarse grids employing a high order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver the selection of optimal robin coefficients are verified with numerical experiments on heterogeneous and complicated domains with the developed optimal coupling strategy we numerically demonstrate the coupling framework s asymptotic convergence to the local limit with an o o h rate when there is a fixed ratio between the horizon size and the spatial discretization size h

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

a fast galerkin method with efficient matrix assembly and storage for a peridynamic model peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities which cannot be described properly by classical theory of solid mechanics however the operators in the peridynamic models are nonlocal so the resulting numerical methods generate dense or full stiffness matrices gaussian types of direct solvers were traditionally used to solve these problems which requires o n 3 of operations and o n 2 of memory where n is the number of spatial nodes this imposes significant computational and memory challenge for a peridynamic model especially for problems in multiple space dimensions a simplified model which assumes that the horizon of the material d o n 1 was proposed to reduce the computational cost and memory requirement to o n however the drawback is that the corresponding error estimate becomes one order suboptimal furthermore the assumption of d o n 1 does not seem to be physically reasonable since the horizon d represents a physical property of the material that should not depend on computational mesh size we develop a fast galerkin method for the non simplified peridynamic model by exploiting the structure of the stiffness matrix the new method reduces the computational work from o n 3 required by the traditional methods to o nlog 2n and the memory requirement from o n 2 to o n without using any lossy compression the significant computational and memory reduction of the fast method is better reflected in numerical experiments when solving a one dimensional peridynamic model with 2 1 4 16 384 unknowns the traditional method consumed cpu time of 6days and 11h while the fast method used only 3 3s in addition on the same computer with 128gb memory the traditional method with a gaussian elimination or conjugate gradient method ran out of memory when solving the problem with 2 1 6 131 072 unknowns in contrast the fast method was able to solve the problem with 2 2 8 268 435 456 unknowns using 3days and 11h of cpu time this shows the benefit of the significantly reduced memory requirement of the fast method

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

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

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

convergence on error correction methods for solving initial value problems higher order semi explicit one step error correction methods ecm for solving initial value problems are developed ecm provides the excellent convergence o h 2 p 2 one wants to get without any iteration processes required by most implicit type methods this is possible if one constructs a local approximation having a residual error o h p on each time step as a practical example we construct a local quadratic approximation further it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second order methods which can be improved into a semi explicit type ecm of the order of accuracy 6 the stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non stiff problems it should be remarked that the ecm approach developed here does not yield explicit methods but semi implicit methods of the rosenbrock type both ecm and rosenbrock s methods require to solve a few linear systems at each integration step but the ecm approach involves 2p 2 evaluations of the jacobian matrix per integration step whereas the rosenbrock method demands one evaluation only however it is much easier to get high order methods by using the ecm approach

nodal type collocation methods for hypersingular integral equations and nonlocal diffusion problems abstract in this paper we study nodal type quadrature rules for approximating hypersingular integrals and their applications to numerical solution of finite part integral equations and nonlocal diffusion problems we first derive explicit expressions for the quadrature coefficients and establish corresponding error estimates some collocation schemes are then constructed based on these rules to numerically solve certain type of finite part integral equations and nonlocal diffusion problems in one dimension and condition number and optimal error estimates for the proposed schemes are also rigorously obtained on uniform grids these schemes are of toeplitz structure which results in many advantages in developing fast linear solvers various numerical experiments are also performed to illustrate the theoretical results

on mesh geometry and stiffness matrix conditioning for general finite element spaces the performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation pde models it is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix in this paper for typical second order elliptic problems some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes the derivation of such relations for general high order elements is based on a new trace formula for the element stiffness matrix it is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces these results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing

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

robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients abstract as alternatives to partial differential equations pdes nonlocal continuum models given in integral forms avoid the explicit use of conventional spatial derivatives and allow solutions to exhibit desired singular behavior it is of practical interest to develop robust numerical schemes not only for the numerical solution of nonlocal models but also for the evaluation of suitably defined derivatives of solutions the latter motivates the development of a nonlocal analog of gradient recovery for numerical solution of pdes for structure mechanical models this leads to a posteriori nonlocal stress analysis we illustrate that when smooth solutions are found in nonlocal models one may compute local gradients of nonlocal solutions using conventional techniques like that for pdes more generically however we present a framework for stress analysis of nonlocal solutions based on nonlocal gradient operators and their asymptotically compatible discretization we demonstrate that the nonlocal gradient recovery is consistent in the local limit and is more advantageous than using local gradients of nonlocal solutions superconvergence properties of some special nonlocal gradient operators are identified for nonlocal continuum models moreover methods are presented to preserve such features in the numerical discretization both computational observations and theoretical insights are provided to substantiate our findings

linearized state based peridynamics for 2 d problems summary in this work the authors formulate a 2 d linearized ordinary state based peridynamic model of elastic deformations and compute the stiffness matrix for 2 d plane stress strain conditions this model is then verified by testing the recovery of elastic properties for given poisson s ratios in the range 0 1 0 45 the convergence behavior of peridynamic solutions in terms of the size of the nonlocal region by comparison with the classical local mechanics model is also discussed the degree to which the peridynamic surface effect influences the recovery of elastic properties is examined and stress strain recovery values are found to have a definite influence on the results the technique used here can provide the basis for applying 2 d peridynamic models to the study of fatigue failure and quasi static fracture problems copyright 2016 john wiley sons ltd

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

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

numerical approximations for a nonlocal evolution equation in this paper we study numerical approximations of continuous solutions to the nonlocal p laplacian type diffusion equation u t t x int omega j x y u t y u t x p 2 u t y u t x dy first we find that a semidiscretization in space of this problem gives rise to an ode system whose solutions converge uniformly to the continuous one as the mesh size goes to zero moreover the semidiscrete approximation shares some properties of the continuous problem it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity next we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties in addition we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile finally we present some numerical experiments that illustrate our results

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

an hp galerkin method with fast solution for linear peridynamic models in one dimension the computational work and memory requirement are bottlenecks for galerkin finite element methods for peridynamic models because of their non locality in this paper fast galerkin and hp galerkin finite element methods are introduced and analyzed to solve a steady state peridynamic model we present a fast solution technique to accelerate non square toeplitz matrixvector multiplications arising from piecewise linear piecewise quadratic and piecewise cubic galerkin methods this fast solution technique is based on a fast fourier transform and depends on the special structure of coefficient matrices and it helps to reduce the computational work from o n3 required by traditional methods to o nlog2n and the memory requirement from o n2 to o n without using any lossy compression where n is the number of unknowns the peridynamic model admits solutions having jump discontinuities for problems with discontinuous solutions we therefore introduce a piecewise constant galerkin method and give an h and p refinement algorithm then we develop fast hp galerkin methods based on hybrid piecewise constant piecewise linear and piecewise constant piecewise quadratic finite element approximations the new method reduces the computational work from o n3 required by the traditional methods to o nlog2n and the memory requirement from o n2 to o n

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

a convergent convex splitting scheme for the periodic nonlocal cahn hilliard equation in this paper we devise a first order in time second order in space convex splitting scheme for the periodic nonlocal cahn hilliard equation the unconditional unique solvability energy stability and ell infty 0 t ell 4 l 0 t l 4 stability of the scheme are established using the a priori stabilities we prove error estimates for our scheme in both the ell infty 0 t ell 2 l 0 t l 2 and ell infty 0 t ell infty l 0 t l norms the proofs of these estimates are notable for the fact that they do not require point wise boundedness of the numerical solution nor a global lipschitz assumption or cut off for the nonlinear term the ell 2 l 2 convergence proof requires no refinement path constraint while the one involving the ell infty l norm requires only a mild linear refinement constraint s le c h s c h the key estimates for the error analyses take full advantage of the unconditional ell infty 0 t ell 4 l 0 t l 4 stability of the numerical solution and an interpolation estimate of the form left phi right 4 le c left phi right 2 alpha left nabla h phi right 2 1 alpha alpha frac 4 d 4 d 1 2 3 4 c 2 h 2 1 4 d 4 d 1 2 3 which we establish for finite difference functions we conclude the paper with some numerical tests that confirm our theoretical predictions

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

fractional order orlicz sobolev spaces abstract in this paper we define the fractional order orlicz sobolev spaces and prove its convergence to the classical orlicz sobolev spaces when the fractional parameter s 1 in the spirit of the celebrated result of bourgain brezis mironescu we then deduce some consequences such as convergence of the modulars and convergence of solutions for some fractional versions of the g operator as the fractional parameter s 1

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

a radial basis function galerkin method for inhomogeneous nonlocal diffusion abstract we introduce a meshfree discretization for a nonlocal diffusion problem using a localized basis of radial basis functions our method consists of a conforming radial basis of local lagrange functions for a variational formulation of a volume constrained nonlocal diffusion equation we also establish an l 2 error estimate on the local lagrange interpolant the stiffness matrix is assembled by a special quadrature routine unique to the localized basis combining the quadrature method with the localized basis produces a well conditioned sparse symmetric positive definite stiffness matrix we demonstrate that both the continuum and discrete problems are well posed and present numerical results for the convergence behavior of the radial basis function method we explore approximating the solution to inhomogeneous differential equations by solving inhomogeneous nonlocal integral equations using the proposed radial basis function method

stabilized linear semi implicit schemes for the nonlocal cahn hilliard equation abstract comparing with the well known classic cahn hilliard equation the nonlocal cahn hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials on the other hand it evidently brings more computational costs in numerical simulations thus efficient and accurate time integration schemes are highly desired in this paper we propose two energy stable linear semi implicit methods with first and second order temporal accuracies respectively for solving the nonlocal cahn hilliard equation the temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly while the spatial discretization is carried out by the fourier collocation method with fft based fast implementations the energy stabilities are rigorously established for both methods in the fully discrete sense numerical experiments are conducted for a typical case involving gaussian kernels we test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one in addition long time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay

a policy iteration algorithm for the american put option and free boundary control problems abstract we develop a solution method for american put options that directly employs the policy iteration principle of dynamic programming the method iteratively improves exercise policies obtains monotonically increasing value functions and converges quadratically under reasonable assumptions we present a numerical implementation that exhibits these features the same principle is also applied to obtain a monotonically improving policy iteration scheme for general free boundary optimal control problems

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

a coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions we develop and analyze an optimization based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions the approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations the objective is to minimize their mismatch on the overlap of the nonlocal and local domains and the controls are virtual volume constraints and boundary conditions when some assumptions on the kernel functions hold we prove that the resulting optimization problem is well posed and discuss its implementation using sandia s agile software components toolkit the latter provides the groundwork for the development of engineering analysis tools while numerical results for nonlocal diffusion in three dimensions illustrate key properties of the optimization based coupling method