ANALYSIS AND COMPARISON OF DIFFERENT APPROXIMATIONS TO NONLOCAL DIFFUSION AND LINEAR PERIDYNAMIC EQUATIONS

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1 SIAM J. NUMER. ANAL. Vol. xx, No. x, pp. to appear c 214 Society for Industrial and Applied Mathematics ANALYSIS AND COMPARISON OF DIFFERENT APPROXIMATIONS TO NONLOCAL DIFFUSION AND LINEAR PERIDYNAMIC EQUATIONS XIAOCHUAN TIAN AND QIAND DU Abstract. 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 presented, including standard finite element methods and quadrature based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight to applications and simulations of nonlocal models. Key words. nonlocal diffusion, peridynamic models. numerical approximation, finite element, finite difference, discrete maximum principle, convergence analysis, local limit AMS subject classifications. 82C21, 65R2, 65M6, 46N2, 45A5 1. Introduction. The peridynamic (PD) model proposed by Silling [29] is an integral-type nonlocal continuum model of materials which provides an alternative set-up to classical continuum mechanics based on partial differential equations. As PD avoids the explicit use of spatial derivatives, its effectiveness in modeling materials singularities has been shown in numerical simulations of crack nucleation and growth, fracture and failure of composites, polycrystals and nanofiber networks [4, 2, 27, 31, 32]. Linear scalar PD operators also share similarities with nonlocal diffusion operators, as pointed out in [1], thus making the study of PD relevant to the study of general nonlocal diffusion (ND) models in various applications [2, 3, 5, 7, 8, 12, 16, 17, 18, 19, 21, 23, 26]. One of the common features of the PD and ND models considered here is the introduction of a horizon parameter δ, which characterizes the range (maximum radius) of nonlocal interactions [12, 29]. As δ, nonlocal effect diminishes and the limits provide natural links between nonlocal models and classical (local) differential equation models when the latter are well-defined. Meanwhile, to simulate PD and ND models numerically, various discretization methods have been studied and implemented, including finite difference, finite element, quadrature and particle based methods [6, 9, 14, 2, 22, 28, 3, 34, 35]. With more and more numerical simulations being carried out, it is becoming increasingly important to validate and verify the computational methods and simulation findings. These development motivates our current work. The main contributions of the study presented here are the analysis and comparison of several different discretization schemes of linear PD and ND equations. While some of these schemes are standard like conforming finite element approximations, some other discretizations are new such as the This work is supported in part by the U.S. Department of Energy grant DE-SC5346, U.S. NSF grant DMS and AFOSR MURI center for Material Failure Prediction through peridynamics. Department of Mathematics, Pennsylvania State University, University Park, PA 1682, tian x@math.psu.edu Department of Mathematics, Pennsylvania State University, University Park, PA 1682, qdu@math.psu.edu 1

2 2 XIAOCHUAN TIAN AND QIANG DU maximum-principle preserving quadrature based difference schemes. Our aims are to present methods which not only can be derived in an intuitive and straightforward fashion, but also are broadly applicable so that they exemplify representative methods developed to solve general nonlocal models with both integrable and non-integrable kernels. Our discussions serve as a systematic examination of several basic numerical analysis issues concerning nonlocal problems whose counterparts in the local partial differential equation (PDE) setting have become standard textbook materials. We demonstrate how the schemes remain valid for more general nonlocal interaction kernels and show that many discrete schemes preserve the discrete versions of properties associated with continuum models. Comparisons and connections between different schemes are also made. Through careful analysis of the approximations to a simple one-dimensional model problem, we are able to elucidate some expected and some unexpected convergence properties both in the nonlocal setting and in suitable local limits. In particular, when the horizon parameter δ is coupled with the mesh size, we analyze how potentially unreliable numerical results may be produced by many discrete schemes which are known to converge in the nonlocal setting with a fixed positive δ. Such findings offer important guidance to future theoretical studies and algorithmic development as well as practical applications of PD and ND models. The paper is organized as follows. In section 2, we introduce the nonlocal Dirichlet type constrained value problems of a nonlocal model. In section 3, we present both quadrature based finite difference and finite element discretization schemes. In section 4, we make some comparison of these discretizations by studying the properties of nonlocal stiffness matrices they generate, discussing the discrete maximum principle and examining their local limits. Sections 5 and 6 are devoted to convergence analysis. We focus on two types of convergence, one being the convergence to solution of the nonlocal problem for a fixed horizon δ as the grid size h goes to zero; another being convergence to the local limiting solution, when δ and h go to zero simultaneously. In section 7, we give related numerical results to further substantiate our theoretical studies. Finally, we give some conclusions in section The nonlocal operator and the model equation. We consider here the homogeneous Dirichlet volume constrained value problem associated with a one dimensional nonlocal linear PD/ND model. We refer to [1, 24] for more discussions on related mathematical theory. Let Ω be a finite bar in R. Without loss of generality, we take Ω = (, 1). A nonlocal operator L δ is defined as, for any function u = u(x) : Ω R, (2.1) L δ u(x) = (u(y) u(x))γ δ (x, y)dy, x Ω B δ (x) with B δ (x) = {y R : y x < δ} denoting a neighborhood centered at x of radius δ which is the horizon parameter, and γ δ (x, y) : Ω Ω R being a symmetric nonlocal kernel (influence function), i.e., γ δ (x, y) = γ δ (y, x), and γ δ (x, y) = if y / B δ (x). The following nonlocal constrained value problem on a nonzero measure volume is our main subject of interests here: (2.2) { Lδ u = f δ on Ω, u = on Ω I. The notation Ω I = ( δ, ) (1, 1 + δ) is used above to denote the interaction domain (between points inside and outside Ω) where u is constrained to have value zero. The

3 Different approximations to nonlocal diffusion and linear peridynamic equations 3 constraint (2.2) is a natural extension of Dirichelet boundary condition for classical PDEs [1]. We note that the operator L can be written as L = Dγ δ D where D and D are the basic nonlocal divergence operator and its adjoint defined in a nonlocal vector calculus given in [12]. Such a formulation draws the natural analogy between nonlocal operators and local 2nd order elliptic differential operators (or more specifically, second order derivative operators in our current setting). For simplicity, we consider only the case that γ δ is non-negative and radial, i.e. γ δ (x, y) = γ δ ( y x ) with the support of γ δ (z) contained in [, δ]. We in addition assume that its second order moment is positive and finite, i.e., (2.3) < C δ = s 2 γ δ (s)ds <. The assumption (2.3) is the most general condition for a well-defined elastic modulus (in the context of PD) or a well-defined diffusion constant (in the context of ND). Such a condition allows for more general kernels which may or may not be locally integrable. To connect with the local limit, we assume that (2.4) C δ C >, f δ f, as δ. Under assumptions (2.3) and (2.4), as δ, the solutions of nonlocal problems (2.2) converge [1, 25, 35] to the solution of two-point boundary value problem: (2.5) { Cu = f on Ω, u = on Ω = {} {1}. One of the main contributions of our work here is to elucidate how such limiting behaviors can or cannot be preserved in various discrete approximations. We note that there have been previous attempts to study and compare numerical methods for nonlocal PD models for a variety choices of parameters and nonlocal interaction kernels, for instance, [6, 9]. One important factor that influences the convergence of numerical methods is the choice of the horizon δ. In [6], three types of numerical convergence were discussed for quadrature based approximations of the PD integral equation. Uniform convergence to local classical solution was shown numerically if the number of grid points increases faster than the horizon decreases. Both continuous and discontinuous finite element approximations of PD models were studied in [9] where carefully designed numerical experiments indicate that, when the ratio of δ and mesh size h is fixed as a constant, piecewise constant approximations fail to converge in general, while piecewise linear approximations still give the desired limit. Recently, there have been more studies on numerical approximations of nonlocal models both analytically and computationally [9, 14, 15, 2, 22, 28, 3, 34, 35] including results on a posteriori error estimations and convergent adaptive approximations. Yet, a number of important issues remain open. For example, most existing quadrature based difference approximations are only properly defined for special nonlocal interaction kernels such as ones that are locally integrable even though there is no such limitation from the modeling perspective. Also, little attention has been paid to explore connections between different approaches and to investigate similarities and differences between corresponding nonlocal stiffness matrices. While properties of stiffness matrices associated with PDE models have been studies in textbooks, they have not been fully analyzed for the nonlocal counterpart except for condition number estimates [2, 1, 35]. Another important issue for the simulation of nonlocal models

4 4 XIAOCHUAN TIAN AND QIANG DU is related to the convergence of numerical experiments with a variety of choices of δ and h. For instance, while the horizon δ was taken as a material parameter in many computational studies, it is also a popular engineering practice to suitably couple δ with h in order to control the possible growth in the number of neighboring interacting computation cells (or grid points) within the range of nonlocal interactions or to limit the bandwidth of nonlocal stiffness matrices for efficient numerical evaluations. These issues are discussed in later sections. 3. Numerical Schemes. In this section two classes of discrete schemes for (2.2) are introduced, including a new class of difference/quadrature schemes which preserves the discrete maximum principle (DMP), and conforming finite element Galerkin approximations with piecewise constant and piecewise linear finite element spaces. We also make a comparison of resulting linear systems (nonlocal stiffness matrices) generated by those methods. It is standard textbook material that for the local problem (2.5) on a uniform mesh, the simple second order centre finite difference, the standard finite volume and the continuous piecewise linear finite element discretization generate same linear systems of equations for (2.5), it is thus interesting to check if there are such similarities in the nonlocal case. As our study shows, while the linear systems are more complicated in a nonlocal world, some similarities can still be drawn. In terms of convergence studies, we are concerned with several different limiting processes in this work, namely: 1) limit of nonlocal continuum models as δ ; 2) limit of discrete schemes for nonlocal models as δ for a fixed h; 3) limit of discrete schemes for nonlocal models with a fixed δ as h ; and 4) limit of the discrete schemes for the nonlocal models with both δ and h. While the first of these limiting processes has been studied in the literature, see for instance [1, 25, 16, 35], we provide analysis here on the other three limits. We note that much of the numerical analysis presented here deal with problems whose exact solutions are considered to be smooth, though one distinct advantage of the nonlocal model is in admitting non-smooth solutions. We remark that by considering smooth solutions, we are able to derive many analytical results more directly with less technical jargon. Much of the conclusions presented here can be extended suitably to cases involving non-smooth solutions using more careful functional analytical tools. We leave such analysis to future works [33] while focusing on getting the main messages across here with relatively simple analysis Geometric discretization. For simplicity, with Ω = (, 1), we consider a uniform mesh (grid) throughout our discussion. For a positive integer N, we set h = 1/(N +1) and let δ = rh+δ for a nonnegative integer r < N and δ [, h). We now introduce grid points on Ω Ω I as {x i = ih} i ΩN where the index set is defined by Ω N = { r,...,, 1,..., N + r + 1}. Denote I j = ((j 1)h, jh) for 1 j r, and I r+1 = (rh, δ). We also define the standard piecewise constant basis functions by (3.1) φ i (x) = { 1 for x (xi 1, x i ), otherwise, for i Ω N, and the standard continuous piecewise linear hat basis functions by (x x i 1 )/h for x (x i 1, x i ), (3.2) φ 1 i (x) = (x i+1 x)/h for x [x i, x i+1 ), for i Ω N., otherwise

5 Different approximations to nonlocal diffusion and linear peridynamic equations Quadrature based finite difference discretization. For a parameter α [, 2], we first use the symmetry of γ δ to write the nonlocal operator L δ as (3.3) L δ u(x) = = = δ (u(x + s) u(x))γ δ (s)ds (u(x s) 2u(x) + u(x + s))γ δ (s)ds u(x s) 2u(x) + u(x + s) s α s α γ δ (s)ds. This simple reformulation is revealing: given a non-negative kernel γ δ, the nonlocal operator L δ is in fact a weighted average of second order difference operators. The fact that the average is taken by integrating over (, δ) means that rather than a finite difference, we in fact have a continuum of differences if γ δ is supported over a continuum region (or rather it is more regular than a singular measure such as a finite combination of Dirac-delta measures and possibly distributional derivatives). Moreover, direct correspondence can also be made between L δ and classical differential operators as in [12] when γ δ is formally taken to be derivatives of Dirac-delta measures. With (3.3), it is then intuitively clear that we may approximate the continuum difference represented by L δ by discrete finite differences through various quadrature approximations (thus the name quadrature based finite difference discretization). To first give a simple class of quadrature based difference approximation for L δ, we consider a class of discrete operators L h δ,, parametrized by a constant α [, 2] given by (3.4) L h δ,u i = r+1 m=1 u i m 2u i + u i+m (mh) α I m s α γ δ (s)ds. i = 1,...N, α [, 2], where {u i } are approximations of {u(x i )}. We note that (3.4) is well-defined only for integrable s α γ δ (s). If γ δ (s) is itself integrable as in many existing studies, then we may take any α in [, 2], otherwise, restrictions on α may be required, for example, if only (2.3) is assumed, then it leaves α = 2 as the only feasible choice for all kernels having finite second order moment. Since (3.4) represents a simple Riemann sum like quadrature of the integral (3.3), the quadrature error is of order O(h) with a fixed horizon δ. We may consider other higher order quadratures like the trapezoidal and Simpson s rules for the integral (3.3), with which we can expect higher orders of accuracy. Equivalently, we may work with other reformulations of (3.3). basis functions {φ 1 i } share the property that r+1 j= φ1 j For example, since continuous piecewise linear (x) = 1, for x (, δ), we have (3.5) L δ u(x) = u(x s) 2u(x) + u(x + s) s α γ δ (s)ds r+1 = m= s α u(x s) 2u(x) + u(x + s) s α φ 1 m(s)s α γ δ (s)ds. Then, by utilizing that (u(x s) 2u(x) + u(x + s))/s α as s for smooth u and α [, 2), we arrive at the following

6 6 XIAOCHUAN TIAN AND QIANG DU (3.6) L h δ,1u i = r m=1 u i m 2u i + u i+m (mh) α + u i r 1 2u i + u i+r+1 (r + 1) α h α φ 1 m(s)s α γ δ (s)ds I m I m+1 I r+1 φ 1 r+1(s)s α γ δ (s)ds, i = 1,...N. As to be demonstrated later, the class of schemes (3.6) is more accurate than that corresponding to (3.4) and it is also closely related to finite element approximations. The nonlocal stiffness matrices for above schemes are denoted by A α D, and Aα D,1 for (3.4) and (3.6) respectively, where the subscript {D, 1} refers to the use of piecewise linear interpolation in the difference approximation (in contrast to the subscript {D, } used for (3.4) which corresponds to piecewise constant interpolation) while the superscript refers to the value of α used in (3.5). If δ =, that is δ = rh for an integer r, then the integral over I r+1 in above schemes should be taken as zero. With the above defined discrete nonlocal difference operators, the proposed quadrature based finite difference schemes of (2.2) are (3.7) { L h u i = f δ (x i ) i {1,..., N} u i = i { r,..., 1, } {N + 1,..., N + r + 1} where L h can be either L h δ, or Lh δ,1 as defined above. Let U be a column vector with entries {u i } N i=1, and b be that with entries {b i = f δ (x i )} N i=1, we may rewrite the corresponding linear systems as (3.8) A α D,U = b and A α D,1U = b respectively, where {A α D,, α [, 2]} are nonlocal stiffness matrices associated with (3.4) while {A α D,1, α [, 2)} are for (3.6). To avoid technical complications in numerical analysis, we only work with the cases α = and 1 for schemes defined by (3.6) in the sequel though more general α are considered for that defined by (3.4) Finite element discretization. Similar to [1, 24], we define the energy space and the constrained energy space as S = {u L 2 (Ω Ω I ) γ δ (r)(u(x + r) u(x)) 2 drdx < } Ω B δ () and S = {u S u = in Ω I }. Denote the bilinear form on L 2 (Ω) L 2 (Ω) and the inner product on L 2 (Ω) by B(u, v) := 1 (3.9) (u(x + r) u(x))(v(x + r) v(x))γ δ (r)drdx, 2 Ω δ (3.1) (u, v) Ω := u(x)v(x)dx. then, the weak formulation of (2.2) is given by: finding u S such that v S, (3.11) B(u, v) = (f δ, v) Ω. Ω Let S h S be a family of finite element spaces corresponding to a uniform mesh {x i } parameterized by the mesh size h, as described earlier, with {φ k i }N h i=1 being the nodal basis. Let u h S h be the Galerkin approximation of u given by

7 Different approximations to nonlocal diffusion and linear peridynamic equations 7 (3.12) B(u h, v h ) = (f δ,, v h ) Ω v h S h. Now suppose u h = N h i=1 u iφ k i (x), we pay particular attention to the cases k = and 1 with the case k = corresponding to piecewise constant basis functions (3.1) (if the energy space admits such functions, which is guaranteed if γ δ (r) has finite first order moment), and the case k = 1 corresponding to standard continuous piecewise linear elements with hat basis functions given by (3.2) (which works for γ δ (r) that has finite second order moment). Similar to difference approximations, let U be the column vector composed of the nodal values {u i } N h i=1, and bk being the vector with entries {(f δ, φ k i ) Ω/h} N h i=1 which represent the weighted average of f δ around x i. Then (3.12) gives linear systems A E,k U = b k with {A E,k } 1 k= being the nonlocal stiffness matrices for the finite element approximation. Entries of A E,k are given by {B(φ k i, φk j )/h}. The scaling factor h is needed to have the discrete system being consistent to that obtained from finite difference approximations. Indeed, one has (f δ, φ k i ) Ω/h f δ (x i ) as h. We will examine how properties of A E,k are affected by choices of the nonlocal kernels and discrete spaces and compare with A α D, and Aα D,1 generated by the quadrature based finite difference methods Nonlocal stiffness matrices. When comparing finite difference methods with finite element methods, we distinguish two cases for simplicity: in one case the horizon is no more than h while δ = rh for some integer r > 1 in the second case. We let {a ij } represent entries of nonlocal stiffness matrices. Their specific forms are provided as a reference. Case 1: δ h, first, entries of A α D, are given by: 2 δ h α s α γ δ (s)ds, i = j (3.13) a ij = 1 h α s α γ δ (s)ds, i j = 1, otherwise. As for entries of A α D,1, we only discuss the cases α = or 1 and we note that (3.14) A α D,1 = A 1+α D, for α =, 1 so that their expressions can be found in (3.13). Next, for entries of A E,, we have (3.15) A E, = A D,1 = A 1 D,. Finally, entries of A E,1 are given by: 2 δ h 2 s 2 γ δ (s)ds 1 h 3 s 3 γ δ (s)ds, (3.16) a ij = 2 3h 3 i = j s 3 γ δ (s)ds 1 h 2 s 2 γ δ (s)ds, i j = 1 1 6h 3 s 3 γ δ (s)ds, i j = 2, otherwise.

8 8 XIAOCHUAN TIAN AND QIANG DU Case 2: δ = rh, and let m = j i, entries of A α D, are given by: 2 r s h α l=1 Il α l α γ δ(s)ds, m = (3.17) a ij = 1 s h Im α α m α γ δ(s)ds, 1 m r, otherwise. Then, entries of A α D,1 for α = or 1 are given by: 2 ( 1 δ h α s α γ δ (s)ds + 1 h h h 1+α s 1+α γ δ (s)ds ), m = 1 (3.18) a ij = h α φ 1 m(s)s α γ δ (s)ds, 1 m r 1 I m I m+1 1 h α φ 1 r(s)s α γ δ (s)ds, m = r I r, otherwise. Next, for entries of A E,, we have (3.19) A E, = A D,1. Finally, entries of A E,1 are given by a ii = h ( s h 3 + 2s2 h 2 )γ δ(s)ds + 2h and for m = j i = 1, a ij = I m+2 h + 3h γ δ (s)ds, ( 2s3 3h 3 s2 h 2 )γ δ(s)ds + 2h 2h h 2h h ( s3 3h 3 2s2 h 2 ( s3 6h 3 3s2 2h 2 + 9s 2h 25 6 )γ δ(s)ds s h 4 3 )γ δ(s)ds ( s3 2h 3 + 5s2 2h 2 7s 2h )γ δ(s)ds 3h γ δ (s)ds. The expressions for a ij with 2 m = j i r + 1 are more involved: ( s 3 (m 2)s2 a ij = + I m 1 6h3 2h 2 (m2 4m + 4)s + m3 6m m 8 2h 6 ( s 3 (3m 2)s2 + I m 2h3 2h 2 + (3m2 4m)s 3m3 6m ) γ δ (s)ds 2h 6 ( s 3 (3m + 2)s2 + + I m+1 2h3 2h 2 (3m2 + 4m)s + 3m3 + 6m 2 4 ) γ δ (s)ds 2h 6 ( s 3 (m + 2)s2 + 6h3 2h 2 + (m2 + 4m + 4)s m3 + 6m m + 8 2h 6 ) γ δ (s)ds ) γ δ (s)ds. We note that for m > r+1, integrals over I m are set to be zero in the above expressions.

9 Different approximations to nonlocal diffusion and linear peridynamic equations 9 4. Properties of the discrete schemes. We now present some basic properties of discrete schemes in analogous to results widely known for local problems Properties of nonlocal stiffness matrices. Let us first study the nonlocal stiffness matrices. Lemma 4.1. For quadrature based finite difference schemes (3.7) with operators in (3.4) and (3.6), and the finite element discretization (3.12) with either piecewise constant or continuous linear elements, the nonlocal stiffness matrices are all symmetric, positive definite matrices. Proof. The symmetry is obvious. The positive definiteness of nonlocal stiffness matrices of the finite element methods is a consequence of the coercivity of the bilinear form B [1, 24]. Meanwhile, A α D, and Aα D,1 can all be viewed as nonnegative linear combinations of the stiffness matrices associated with the 2nd order center difference approximations which thus must remain positive definite. Lemma 4.2. For quadrature based finite difference schemes (3.7) with operators in (3.4) and (3.6), and finite element discretization (3.12), 1) the nonlocal stiffness matrices A α D,, Aα D,1, and A E, = A D,1 are all M-matrices; 2) for the case δ (, h], A α D,, Aα D,1, and A E, are all scalar multiples of the tridiagonal matrix A associated with the second order centeral difference operator with 2/h 2 as diagonal entries and 1/h 2 as off-diagonal entries, in particular, ( ) δ (4.1) A α D, = h 2 α s α γ δ (s)ds A, A 1 D,1 = A 2 D, and A E, = A D,1 = A 1 D,. Thus, cond(a α D, ) = cond(a D,1 ) = cond(a1 D,1 ) = cond(a E,) = sin 2 ( hπ 2 ) = O(h 2 ). Proof. Direct inspection shows that A α D, and A E, have positive diagonal and non-positive off-diagonal entries. Moreover, any of their row sums is either zero or is the negative of some partial sum of of the off-diagonal entries, hence we get M- matrices. For the case δ h, the conclusions in 2) are obvious following from the well-known spectral estimates of the tridiagonal matrix A. Lemma 4.3. For the finite element nonlocal stiffness matrix A E,1, we have 1) any off-diagonal entry not adjacent to the diagonal is non-positive; 2) A E,1 is an M-matrix iff all entries adjacent to the diagonal are non-positive; 3) for δ h, A E,1 is an M-matrix. 4) for L 1 integrable kernel γ δ with a given fixed δ >, for small enough h, the entries adjacent to the diagonal are positive, thus A E,1 is not an M-matrix in this case. Proof. We note that for m = i j 2, the basis functions φ 1 i and φ1 j do not have overlapping support, thus the entries {a ij } of A E,1 satisfy a ij = 1 (φ 1 i (x + r) φ 1 i (x))(φ 1 2 j(x + r) φ 1 j(x))γ δ (r)drdx = 1 2 Ω Ω δ δ (φ 1 i (x + r)φ 1 j(x) + φ 1 j(x + r)φ 1 i (x))γ δ (r)drdx, for m 2. Moreover, A E,1 has positive diagonal entries, and any row sum of A E,1 is either zero or is the negative of some partial sum of the off-diagonal entries, we have 1) and thus 2). For 3), the entries adjacent to the diagonal are given by s 2 3h 3 (2s 3h)γ δ(s)ds s 2 h 3 (s h)γ δ(s)ds

10 1 XIAOCHUAN TIAN AND QIANG DU with δ h. Thus, A E,1 is an M-matrix. On the other hand, for 4), based on the explicit expressions of the entries of stiffness matrix given earlier, we note that for i j = 1, a ij contains the term 1 δ 3 3h γ δ(s)ds. With L 1 integrable kernel γ δ and a fixed δ >, as h, this term converges to a positive constant 1 δ 3 γ δ(s)ds >. Meanwhile, it is easy to see that all the other terms are bounded in absolute value by a constant multiple of 3h γ δ (s)ds which goes to as h. So for small h, the sign of a ij is strictly positive, and this implies 4). Comparing different nonlocal stiffness matrices, a few remarks are in order. First, we know that for the classical Poisson equation (2.5), standard centeral finite difference, finite volume and continuous piecewise finite element discretizations lead to an identical stiffness matrix (given by the tridiagonal A), with the difference being the right hand side vectors corresponding to b, b and b 1 respectively. Such a feature no longer holds in general for nonlocal models and we encounter generally different nonlocal stiffness matrices. However, the nonlocal stiffness matrix A E, generated by the piecewise constant finite element and the matrix A D,1 generated by a quadrature finite difference scheme do remain the same in both cases considered above. Secondly, for δ h, we see that the condition numbers of nonlocal stiffness matrices are on the order of O(h 2 ) uniformly with respect to δ, which is consistent with the bounds derived in [1, 1, 35]. The limit when δ is examined more closely later Discrete maximum principle. Since the kernels γ δ of the nonlocal operators L δ are assumed to be nonnegative and symmetric in this work, it is easy to see that the nonlocal equation (2.2) satisfies maximum principle. Based on earlier discussions on the nonlocal stiffness matrices, we can investigate if such a property is preserved in the discrete schemes. We note that the discrete maximum principle can be readily used to establish the stability of the discrete schemes. As it turns out, our finite difference discretization and finite element discretization using piecewise constant basis always preserve discrete maximum principle (DMP). Meanwhile, for finite element discretization under piecewise linear function basis, the discrete maximum principle may not always hold for general kernels. Theorem 4.4. The quadrature based difference schemes (3.7) with operators in (3.4) and (3.6), and the finite element discretization (3.12) with piecewise constants always satisfy the DMP: f δ = f δ (x) for x Ω u i max(u j, j I b ) (i = 1,..., N) or f δ = f δ (x) for x Ω u i min(u j, j I b ) (i = 1,..., N) where I b = { r,..., 1, } {N + 1,..., N + r + 1}. Moreover, the finite element discretization (3.12) with piecewise linear basis also satisfies the DMP for δ h. Proof. These properties follow directly from lemmas 4.1 and 4.2 which imply that (A α D, ) 1, (A α D,1 ) 1 and (A E, ) 1 are non-negative matrices. Similarly, by lemma 4.3 (A E,1 ) 1 is non-negative when δ h. The DMP then follows. A consequence of the DMP is the stability of finite difference approximations. Theorem 4.5. For the quadrature based finite difference discretization (3.7) with L h defined by operators in (3.4) and (3.6), (L h ) 1 is bounded uniformly in h for a given δ. The h-independent bound of (L h ) 1 is also uniform in δ as δ. Proof. By the results of lemma 4.1 and theorem 4.4, we have that ( L h ) 1 is

11 Different approximations to nonlocal diffusion and linear peridynamic equations 11 nonnegative. Let us consider first the operator corresponding to (3.4). Define (4.2) C r m,α(h) = r+1 then we have C r,α(h) C δ and 2(jh) 2m+2 α (2m + 2)! L h δ, w h δ = (1, 1,..., 1) T, where w h δ = I j s α γ δ (s)ds, m, C r,α 1 δ(1 + δ) x) + (h)x(1 C,α r (h), and w h δ (x i) for i { r,..., 1, } {N + 1,..., N + r + 1}. By the DMP, ( L h δ,) 1 w h δ 1 + 4δ(1 + δ) 1 + 4δ(1 + δ) 4C,α r (h). 4C δ so ( L h δ, ) 1 is uniformly bounded in h. Similarly, we may define (4.3) C r α(h) = r m= = C δ + (mh) 2 α φ 1 m(s)s α γ δ (s)ds I m I m+1 I r+1 ((r + 1)h) 2 α φ 1 j+1(s)s α γ δ (s)ds (I h (s 2 α ) s 2 α )s α γ δ (s)ds C δ where I h (s 2 α ) is the piecewise linear interplant of s 2 α with respect to the mesh. By replacing C r,α(h) with C r α(h) in the definition of w h δ, we see that ( Lh δ,1 ) 1 is also uniformly bounded in h for fixed δ. Moreover, since C δ C as δ, we also have the bound being independent of δ as δ.. The uniform bounds above give the L stability of nonlocal quadrature based finite difference approximations and the piecewise constant finite element approximation, which is needed later in convergence analysis. We note that for the finite element discretization (3.12) with continuous piecewise linear basis, the DMP does not hold in general as the corresponding nonlocal stiffness matrix fails to be an M-matrix Local limits of discrete schemes. To link with the later analysis on the behavior of discrete schemes as the horizon goes to zero, we are interested in analyzing the limit δ for fixed h. In this case, all nonlocal stiffness matrices are of the forms studied in the case 1 of section 3.4. Thus, except for the finite element piecewise linear approximations, all other schemes give scalar multiples of the tridiagonal matrix A as shown in (4.1) of lemma 4.2. These scalars are given by either C δ or for α [, 2), ( ) δ C δ h 2 α s α γ δ (s)ds = h 2 α s α γ δ (s)ds s 2 γ δ (s)ds = h 2 α C δ s 2 α, for some s (, δ), which, as δ, goes to infinity for fixed h, since C δ C >. Thus, we have Theorem 4.6. For quadrature based finite difference schemes (3.7) with operators in (3.4) and (3.6), and finite element discretization (3.12), as δ, we have

12 12 XIAOCHUAN TIAN AND QIANG DU 1) A 2 D, = A1 D,1 = C δa CA so the corresponding discrete schemes converge to the standard 2nd order centeal finite difference approximation of the local limit. Similarly, A E,1 CA so the corresponding discrete schemes converge to the standard continuous piecewise linear approximation of the local limit; 2) the solutions of (3.7) with operators in (3.4) for α [, 2) and in (3.6) for α =, together with piecewise constant element solutions of (3.12), all converge to zero. We note also that the only difference between various local limits of the schemes considered in 1) of theorem 4.6 are the right hand side vectors given by either point wise values or weighted averages using finite element basis functions as weights. The above theorem indicates the limits of those discrete schemes proposed for nonlocal problems may not always yield convergent discrete schemes of the correct continuum local limit if we fix the mesh size while letting the horizon δ. 5. Convergence of discrete schemes to nonlocal problems with fixed horizon. In this section, we show the convergence of discrete schemes presented earlier to nonlocal model (2.2) as h with δ and γ δ (s) being given Quadrature based finite difference discretization. Let us consider the truncation error of the discrete operator L h. We begin with the following lemma. Lemma 5.1. Suppose that a function G = G(x) has a bounded derivative G on [, δ] and a function g is nonnegative and integrable on [, δ], then (5.1) r+1 G(s)g(s)ds = G(jh) g(s)ds + O(h), as h. I j Proof. It is easy to see that as h, r+1 r+1 G(s)g(s)ds G(jh) g(s)ds I j G(s) G(jh) g(s)ds ch I j g(s)ds for some constant c depending on G which gives (5.1). For the quadrature based finite difference discretization given by (3.4), we may take the G and g in (5.1) as (5.2) G(s) = [u(x i + s) + u(x i s) 2u(x i )]/s α, g(s) = s α γ δ (s) i = 1,..., N. Using Taylor expansion, it is easy to see that if u is uniformly Holder continuous with exponent α, then for α 1, we have G (s) uniformly bounded. Moreover, if u (3) is bounded, then for α = 2, we also have G (s) uniformly bounded. Then by lemma 5.1, the consistency (truncation) error of the quadrature based finite difference discretization defined by operators in (3.4) is O(h) if s α γ δ (s) is in L 1. Theorem 5.2. Let the solution u of the nonlocal problem be smooth, and either u is uniformly Holder continuous with exponent α 1, s α γ δ (s) is bounded in L 1, or u (3) is uniformly bounded and α = 2, then for the quadrature based finite difference discretization (3.7) with operator defined by (3.4), the consistency error satisfies (5.3) max 1 i N L δu(x i ) L h δ,u(x i ) = O(h), as h. So the error of the difference solution is also order O(h), i.e., u i u(x i ) = O(h). Proof. The result follows from the truncation error analysis and the stability given in theorem 4.5.

13 Different approximations to nonlocal diffusion and linear peridynamic equations 13 We see that the discretization based on (3.4) is a first order method. This is not surprising as the difference scheme is obtained via a simple Riemann sum. To improve the accuracy of convergence, we may consider other quadratures for (5.1). The scheme using (3.6) is one way to improve the accuracy. Instead of the Riemann sum using the piecewise constant approximation of G, it is based on a trapezoidal rule using a piecewise linear interpolation, denoted by I h G. The following lemma discusses the higher order of accuracy if we approximate G by I h G. Lemma 5.3. Suppose that G is bounded on [, δ], g is nonnegative and integrable on [, δ], then the discretization (5.4) r+1 G(s)g(s)ds = I h G(s)g(s)ds + O(h 2 ). I j Proof. Direct calculation shows r r+1 G(s)g(s)ds I h G(s)g(s)ds I j Ij h 2 2 max G (s) g(s)ds = O(h 2 ). s Now consider again G and g given by (5.2), we get the schemes corresponding to (3.6). Moreover, for α = or 1 with u (2+α) uniformly bounded and s 1+α γ δ (s) bounded in L 1, these schemes are second order accurate approximations that satisfy the DMP and thus are also numerical stable. Notice that in order to a well-defined numerical scheme, we need G() = and we also need the smoothness of G(s) in s as s, both of which are valid for α = or 1. We provide the following theorem related to schemes (3.6) whose proof is similar to that of (5.2) and is omitted. Theorem 5.4. For the quadrature based difference schemes (3.7) with operators in (3.6), if u (2+α) are uniformly bounded and s 1+α γ δ (s) are bounded in L 1 corresponding to α = or 1, then we have the nonlocal stiffness matrices being M-matrices and the schemes convergent with error being O(h 2 ), i.e., u i u(x i ) = O(h 2 ). For error analysis of finite element discretization, we expect that the piecewise constant approximation is of first order in L 2 with γ δ (s) bounded in L 1, such results can be derived under much less regularity assumptions on the exact solution (say u is in H 1 ), while the continuous piecewise linear approximation is of first order in the energy space for more general kernels. We refer to [1] for details. 6. Convergence of discrete schemes to local problems with vanishing δ and h. In some of the practical simulations of PD models, the grid size has been coupled with the horizon. While physical considerations might be behind such a coupling, an added benefit of making this choice is that the growth of interacting neighboring grid points or elements can be properly controlled in the numerical simulations. It is known that the nonlocal model (2.2) converges to the local model (2.5) as δ (see more general discussions in [1, 24]). Yet, we see that the local limits of discrete nonlocal schemes may or may not correspond to convergent discrete schemes of the correct local equation. Thus, a natural and important question is that when both h and δ approach zero, what is the limiting behavior of the numerical solution. Here, we consider two cases that δ h and δ = rh for a fixed integer r > 1 respectively. In both cases, when h, δ also tends to zero. We show that the limiting behavior of numerical approximations can be very complex and is very much dependent on

14 14 XIAOCHUAN TIAN AND QIANG DU the schemes used. Specifically, for our problem, we show that for the finite difference discretization (3.4), only the case of α = 2 leads to the solution of the correct local equation in general. Similar conclusion can be drawn for (3.6) with α = 1. Meanwhile, for finite element discretization, the case with continuous piecewise linear basis gives the correct local limit but not the case with piecewise constant basis. What is intriguing is that in all these cases, the limits often exist and they satisfy local differential equations that are different from the correct local limit. Practically speaking, this means that while numerical convergence might be observed, it is possible that a wrong limiting solution is obtained Finite difference in local limit. First, let us examine the finite difference discretization (3.4). Given a smooth function u, we see that for finite difference discretization that : (6.1) L h u(x i ) = Cm,α(h)u r (2m+2) (x i ), m= where the coefficients are given by (4.2). While the limit of {C r,α(h)} as h has been examined for a given δ, some additional properties of {C r m,α(h)} for more general cases are in order. Lemma 6.1. Given the coefficients defined in (4.2), we have 1) C r m,α(h) is a strictly decreasing function of α, in particular, 2) for m 1, C r m,α(h) C r m,2(h) = C r m,α(h) r+1 δ 2m (2m + 2)! Cr,α(h) 2(jh) 2m s 2 γ δ (s)ds. (2m + 2)! I j ((r + 1)h)2m C r (2m + 2)!,α(h). Proof. We note that 1) follows from s jh in I j and 2) is a consequence of jh δ for any j and δ (r + 1)h. A consequence of the above lemma is that Theorem 6.2. Let r := δ/h be fixed and suppose that C r,α(h) C r α as h for some constant C r α (, ), then L h = L h δ, u(x i), as defined in (3.4), converges to C r αu (x i ) as h. Moreover, if C r,α(h) C r α = O(h β ) for some constant β >, then the truncation error is of order h min(β,2) for a smooth solution u. Proof. Under the condition on C r,α(h), we see that for h, C r,α(h) is uniformly bounded, so that Cm,α(h)u r (2m+2) (x i ) = O(h 2 ), m=1 based on 2) of lemma 6.1. This together with C,α(h) r Cα r = O(h β ) lead to the conclusion of the theorem. The above theorem further implies that Corollary 6.3. Let r := δ/h be fixed then as h the difference approximation defined by (3.4) with α = 2 converges to the solution of the local limit (2.5).

15 Different approximations to nonlocal diffusion and linear peridynamic equations 15 Consequently, the difference scheme with α = 2 gives a convergent scheme in both nonlocal and local regimes. Proof. We note that by assumption of the kernel γ δ, we have C r,2(h) = C δ = s 2 γ δ (s)ds C, as δ. The conclusion then follows from theorems 6.2 and 4.5. While the case α = 2 offers a consistent approximation in both nonlocal and local regimes, we see that this fails to hold for more general α 2. Indeed, by 1) of lemma 6.1, for fixed r, we have C,α(h) r > C,2(h). r If the strict inequality holds in the limit h, we see that the difference approximation defined by (3.4) with α 2 would converge to the solution of a different local limit with a diffusion coefficient C,α r > C. Indeed, we illustrate such possibilities with a simple constant kernel γ δ = 3δ 3 which corresponds to the diffusion coefficient C = 1 in the local limit. Let us examine a few special cases, first, we consider δ h. By the explicit construction given in (3.13), we get with α 2 that C r,α(h) = 3 α + 1 ( ) 2 α h 3 δ α + 1 > 1. We see that in this case, the finite difference approximation would either converge to zero (if δ/h ) or converge to a local equation with a diffusion coefficient strictly larger than C = 1. Next, we consider the case δ = rh for some fixed integer r > 1. Then C r,α(h) is a constant independent of h, that is, C,α(h) r = 3 1 α + 1 r 3 r j 2 α (j α+1 (j 1) α+1 ) > C,2(h) r = C = 1, which again implies that the difference approximation defined by (3.4) would converge to the solution of a different local limit with a diffusion coefficient larger than C = 1. However, we see that, independent of h, C r,α(h) C r,2(h) = C = 1, as r, so we may expect the convergence of difference approximations to the solution of the correct local solution if the number of grid points increases faster than the horizon decreases, this is in agreement with related numerical experiments reported in [6]. In summary, we see for the special constant kernel γ δ, finite difference solutions given by (3.4) with α 2 have a convergent local limit, yet it might not be the solution of the correct local limit for the simple constant kernel as h when either δ h or δ/h is a fixed integer. Moreover, the limiting solutions can be quite different for schemes using different parameters (such as the constant α). Similar conclusions can be expected for other choices of nonlocal interaction kernels. In section 7 we will plot Cα r for some special kernels and see clearly that Cα r varies with α and r. Thus, we see the risk in using a mesh dependent horizon parameter in the numerical simulations of nonlocal models as the local limiting behavior might be rather unpredictable, even though the schemes can provide good convergence properties in the nonlocal regime. Similar discussions can be made for finite difference solutions given by (3.6). Note that for δ h, based on (3.14), we have A D,1 = A1 D, and A1 D,1 = A2 D, thus the local

16 16 XIAOCHUAN TIAN AND QIANG DU limit is the same as that discussed above. In particular, we expect the convergence of (3.6) to the correct local limit for α = 1 while to a different limit for α =. For δ = rh with some fixed integer r > 1, we evaluate the difference stencil for a smooth function u. Using Taylor expansion, the coefficient of the leading order term is then given by Cα(h) r as defined in (4.3). This implies: Theorem 6.4. For δ = rh with some fixed integer r > 1, solutions of the quadrature based finite difference scheme (3.7) with operators given in (3.6) converges to the solution of the local limit (2.5) as h with the correct diffusion coefficient C if α = 1. On the other hand, for α = and C(h) r as defined in (4.3), the difference solutions converge to the local limit solution of (2.5) with the diffusion coefficient (6.2) Cr = lim h C r (h) = C + lim h which equals to C if and only if (6.3) lim h rh rh (I h (s 2 ) s 2 )γ δ (s)ds C (I h (s 2 ) s 2 )γ δ (s)ds =. Proof. By (4.3), we see that for α = 1, s = I h (s), so that the leading order coefficient is exactly C δ thus leading to convergence. On the other hand, if (6.3) holds, then for α =, the leading order coefficient of the Taylor expansion goes to C as h. Moreover, the coefficients of the higher order terms are all going to zero uniformly, or rather the truncation error approaches to zero uniformly as δ. Thus, by the theorem 4.5 we have the desired convergence result. We note that (6.3) does not hold in general. For example, take γ δ = 3δ 3 so that C δ = C = 1, then direct calculation shows that C(h) r = C r = 1 + 1/(2r 2 ) > 1. In fact, similar observations hold for the following popular choices of the kernel. Lemma 6.5. For δ = rh with fixed r, we have 1) given a non-increasing function γ δ = γ δ (s) in [, δ), C r (1 + 1/(8r 3 ))C δ. 2) if γ δ (s) = γ 1 (s/δ)/(δs 2 ) for some nonnegative function γ 1 (s) on (, 1) with 1/(2r) γ 1 (s)ds = τ, and 1 γ 1 (s)ds = 1, for some τ >, then C = 1 and C r 1 + τ. Proof. For 1), we notice that rh (I h (s 2 ) s 2 )γ δ (s)ds h/2 = 1 8r 3 (I h (s 2 ) s 2 )γ δ (s)ds t 2 γ δ ( t 2r )dt 1 8r 3 h/2 s 2 γ δ (s)ds t 2 γ δ (t)dt = C δ 8r 3. For 2), by the scaling it is easy to see C = 1. Similar as the above, we have rh (I h (s 2 ) s 2 )γ δ (s)ds h/2 s 2 γ δ (s)ds = h/2 1 δ γ 1( s δ )ds = τ >. We see that (6.3) does not hold for kernels considered in the above lemma and the corresponding local limits have coefficients different from the desired one.

17 Different approximations to nonlocal diffusion and linear peridynamic equations Finite element in local limit. Now we consider finite element methods. Numerical experiments in [9] indicate that when the ratio of horizon radius and the grid size is fixed as a constant, piecewise constant approximations fail to converge in general, while piecewise linear approximations converge to the desired limit 1. We now provide some in-depth analysis. First, the piecewise constant element solution with δ h would converge to the local second order equation with a diffusion coefficient C = lim h δ h sγ δ (s)ds lim δ C δ = C. Thus, similar to the finite difference with α = 1, we generally do not expect convergence to the solution of the correct local equation. Theorem 6.6. For the case δ h, the piecewise constant finite element approximation converges to the solution of the local limit (2.5) as h with the diffusion coefficient C C. For the case where δ = rh with a fixed integer r > 1, by the equivalence of the nonlocal stiffness matrices between the piecewise constant finite element approximation and that for the difference approximation (3.6) with α =, we get: Theorem 6.7. In the case δ = rh for some fixed integer r > 1, the piecewise constant finite element approximation converges to the solution of the local limit (2.5) as h with the correct diffusion coefficient C iff (6.3) holds. Next, for the continuous piecewise linear element case with δ h, we see from the nonlocal stiffness matrix that it is a combination of the standard three-point second order centeral difference with a five-point fourth order finite difference of the form: C δ u(x j+1 ) 2u(x j ) + u(x j 1 ) h 6 h 2 ( ) s 3 u(xj+2 ) 4u(x j+1 ) + 6u(x j ) 4u(x j 1 ) + u(x j 2 ) γ δ (s)ds h 4. Since hs3 γ δ (s)ds hδc δ, as δ, h, we see that the correct local limit is always assured. This gives the following theorem. Theorem 6.8. For the case δ h, the continuous piecewise linear finite element approximation gives a consistent difference approximation to the local limit (2.5) with the correct diffusion coefficient C as h. Finally, for the piecewise linear finite element scheme and δ = rh with some fixed integer r > 1, we may examine the difference stencil: r+1 a ii u(x i ) + (a i,i+m u(x i+m ) + a i,i m u(x i m )), m=1 with (a ij ) = A E,1 being entries of the nonlocal stiffness matrix A E,1. Again, to leading order of expansion, by the symmetry of A E,1, we get the coefficient of u (x i ) as (6.4) r+1 m=1 a i,i+m (mh) 2 = 1 h r+1 m=1 B(φ 1 i+m, φ 1 i )(mh) 2 = 1 2h B(I h((x x i ) 2 ), φ 1 i ) = C δ + 1 2h B(I h((x x i ) 2 ) (x x i ) 2, φ 1 i ). 1 Similar observations were made by R. Lehoucq and M. Parks in private communications.

18 18 XIAOCHUAN TIAN AND QIANG DU where I h is the continuous piecewise linear interpolation operator with respect to the mesh and we have used the observation that B((x x i ) 2, φ 1 i ) = ( L δ (x x i ) 2, φ 1 i ) = 2C δ (1, φ 1 i ) = 2hC δ. Let w(x) = (x x i ) 2, w h be the piecewise linear finite element approximation of w with its values outside Ω matching with w. Then, B(I h (w) w, φ 1 i ) = B(I h(w) w h, φ 1 i ). So the term remains to be estimated is equivalent to B(I h (w) w h, φ 1 i ). We first recall that for the 1d second order differential operator, it is a well-known textbook fact that the continuous piecewise linear finite element solution is precisely the linear interpolant. This property is not expected to remain true in general in our nonlocal setting. However, we have a perhaps surprising property which meets our needs here. Lemma 6.9. For the bilinear form defined in (3.9), we have (6.5) B(u, φ 1 i ) = B(I 1 hu, φ 1 i ), i for any quadratic function u. Proof. Since B is bilinear, we only need to prove the equation for u(x) = x 2. Now we know that the error function e u = u Ih 1 u is piecewise quadratic of the form: e u (x) = u(x) I 1 hu(x) = (x x j 1 )(x x j ), x [x j 1, x j ], Thus, on a uniform mesh, e u is periodic with period h, and symmetric with respect to each mesh grid x j and the mid-point of the mesh element x j + h/2. Then, (6.6) B(e u, φ 1 i ) = (L δ e u, φ 1 i ) xi+1 ( ) = φ 1 i (x) (e u (x + s) + e u (x s) 2e u (x))γ δ (s)ds dx = x i 1 ( x i+1 γ δ (s) x i 1 ) φ 1 i (x)(e u (x + s) + e u (x s) 2e u (x))dx ds. We now show that β(s) = xi x i 1 φ 1 i (x)(e u (x + s) + e u (x s))dx is independent of s. Indeed, since e u is h-periodic, we have β(s) = h x h (e u(x + s) + e u (x s))dx. Now, we first do a change of variable y = x in the second integral to get β(s) = h x h h e y h u(x+s)dx+ h e x u( y s)dy = h e y u(x+s)dx h h e u(y+s)dy where in the second integral of the second equality, we have used the fact that e u is even. Apply a change of variable y = x h to the last integral, we get for any s, β(s) = h x h h e h x u(x + s)dx + h e u(x + s)dx = h e u (x + s)dx = h e u (x)dx,