analysis and approximation of nonlocal … · and fractional derivative models for anomalous di...

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ANALYSIS AND APPROXIMATION OF NONLOCAL DIFFUSIONPROBLEMS WITH VOLUME CONSTRAINTS

QIANG DU∗, MAX GUNZBURGER† , R. B. LEHOUCQ‡ , AND KUN ZHOU§

11 January 2012

Abstract. A recently developed nonlocal vector calculus is exploited to provide a variationalanalysis for a general class of nonlocal diffusion problems described by a linear integral equation onbounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawnbetween 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 rangingfrom continuum mechanics to graph theory. In particular, it is shown that fractional Laplacianand fractional derivative models for anomalous diffusion are special cases of the nonlocal model fordiffusion we consider. The numerous applications elucidate different interpretations of the operatorand the associated governing equations. For example, a probabilistic perspective explains that thenonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for asymmetric jump process. Sufficient conditions on the kernel of the nonlocal operator and the notionof volume constraints are shown to lead to a well-posed problem. Volume constraints are a proxy forboundary conditions that may not be defined for a given kernel. In particular, we demonstrate fora general class of kernels that the nonlocal operator is a mapping between a constrained subspaceof a fractional Sobolev subspace and its dual. We also demonstrate for other particular kernels thatthe inverse of the operator does not smooth but does correspond to diffusion. The impact of ourresults is that both a continuum analysis and a numerical method for the modeling of anomalousdiffusion on bounded domains in Rn are provided. The analytical framework allows us to considerfinite-dimensional approximations using both discontinuous or continuous Galerkin methods, bothof which are conforming for the nonlocal diffusion equation we consider; error and condition numberestimates are derived.

Key words. nonlocal diffusion, nonlocal operator, fractional Laplacian, fractional operator,fractional Sobolev spaces, vector calculus, anomalous diffusion, finite element methods, nonlocalheat conduction, peridynamics

AMS subject classifications. 26A33, 34A08, 34B10, 35A15, 35L65, 35B40, 45A05, 45K05,60G22, 76R51

1. Introduction. It is well understood that that Fick’s first law, which is a con-stitutive relation for diffusive fluxes, is a questionable model for numerous phenomena;see [11, 43, 44] for discussions and numerous citations to the literature. Equivalently,whenever the associated underlying stochastic process is not given by Brownian mo-tion, the diffusion is deemed anomalous. In particular, anomalous superdiffusionrefers to situations that can be, at times, modeled using fractional spatial derivativesor fractional spatial differential operators [43]. In this paper, we consider an integro-

∗Department of Mathematics, Pennsylvania State University, University Park, PA 16802;[email protected]. Supported in part by the U.S. Department of Energy grant DE-SC0005346and the U.S. NSF grant DMS-1016073.†Department of Scientific Computing, Florida State University, Tallahassee FL 32306-4120;

[email protected]. Supported in part by the U.S. Department of Energy grant number DE-SC0004970 and the U.S. National Science Foundation grant DMS-1013845.‡Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185–1320; rblehou@

sandia.gov. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Mar-tin Company, for the U.S. Department of Energy under contract DE-AC04-94AL85000. Supportedin part by the U.S. Department of Energy grant number FWP-09-014290 through the Office ofAdvanced Scientific Computing Research, DOE Office of Science.§Department of Mathematics, Pennsylvania State University,University Park, PA 16802;

[email protected]. This author is supported in part by the U.S. Department of Energy grantDE-SC0005346 and the U.S. NSF grant DMS-1016073.

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2 Q. DU, M. GUNZBURGER, R. B. LEHOUCQ AND K. ZHOU

differential equation model for anomalous superdiffusion that, among other desirablefeatures, has the fractional Laplacian and fractional derivative models as special cases.To distinguish our model from existing models for superdiffusion and to highlight thatit applies to a wider range of phenomena, we refer to our model as a nonlocal modelfor diffusion.

Let Ω ⊂ Rn denote a bounded, open domain. For u(x) : Ω→ R, define the actionof the linear operator L on the function u(x) as

Lu(x) := 2

∫Ω

(u(y)− u(x)

)γ(x,y) dy ∀x ∈ Ω ⊆ Rn, (1.1)

where the volume of Ω is non-zero and the kernel γ(x,y) : Ω × Ω → R denotes anon-negative symmetric mapping, i.e., γ(x,y) = γ(y,x) ≥ 0. The operator L isdeemed nonlocal because the value of Lu at a point x requires information aboutu at points y 6= x; this should be contrasted to local operators, e.g., the value of∆u at a point x requires information about u only at x. We have that Lu is thespatial contribution in nonlocal diffusion equations (see (1.2a) and Section 3.3 andalso, e.g., [4]) and nonlocal wave equations (see Section B.2). Section 5.3 brieflydiscusses nonlocal advection-diffusion problems.

It is shown in [27] that if γ(x,y) = 12∆yδ(y−x), where δ(·) denotes the Dirac delta

measure and ∆y denotes the Laplacian operator with respect to y, then L = ∆ inthe sense of distributions. Selecting other kernels and under an appropriate rescaling,the operator L is a generalization of the classical Laplacian operator [4, 51] or, moregenerally, the operator ∇· (C ·∇), where C denotes a second-order tensor [27, 34]. InSection 3.2, using a recently developed nonlocal vector calculus [27, 34], the operator Lis recast as the composition of nonlocal divergence and gradient operators in analogyto the composition ∇ · (C · ∇) for second-order elliptic operators.

The operator L and its generalizations arise in many applications such as imageanalyses [19, 32, 33, 39], machine learning [47], nonlocal Dirichlet forms [5, Sec. 3.6],kinetic equations [12, 38], phase transitions [15, 31], nonlocal heat conduction [16], andthe peridynamic model for mechanics [48] and its one-dimensional variants [49, 50] forwhich L arises directly. We briefly discuss some of the above applications and relatedmathematical work in Appendix B. Moreover, in Appendix A, we discuss the closeconnections between our model and existing models for anomalous superdiffusion; inparticular, we show that the fractional Laplacian operator and a fractional derivativeoperator are special cases of the operator L.

Our interest in the operator L is its participation in the time-dependent nonlocalvolume-constrained diffusion problem

ut − Lu = b on Ω, t > 0

u(x, 0) = u0 on Ω ∪ ΩI

Vu = g on ΩI , t > 0

(1.2a)

for the function u(x, t) and its steady-state counterpart−Lu = b on Ω

Vu = g on ΩI ,(1.2b)

where V denotes a linear operator of constraints acting on a volume ΩI that is dis-joint from Ω. Volume constraints are natural extensions, to the nonlocal case, of

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NONLOCAL DIFFUSION PROBLEMS WITH VOLUME CONSTRAINTS 3

boundary conditions for differential equation problems. For example, if V is the iden-tity operator, the last equation in (1.2a) or (1.2b) is a nonlocal “Dirichlet” volumeconstraint. Nonlocal versions of Neumann and Robin boundary conditions are alsonaturally defined; see Section 4.1.

The nonlocal vector calculus of [27] makes transparent the analogies we drawbetween the nonlocal problems (1.2a) and (1.2b) and parabolic and elliptic boundary-value problems, respectively, involving second-order scalar elliptic operators. Forexample, we will draw several analogies between (1.2a) and the classical diffusionproblem

ut −∇ · (C · ∇u) = b on Ω

u(x, 0) = u0 on Ω

Bu = g on ∂Ω

(1.3a)

as well as between (1.2b) and the (generalized) Poisson problem−∇ · (C · ∇u) = b on Ω

Bu = g on ∂Ω,(1.3b)

where C : Rn → Rn×n denotes a second-order tensor and B denotes a linear operatoracting on the boundary ∂Ω of the volume Ω.

At first glance, the differences between the volume-constrained problems (1.2a)and (1.2b) and the boundary-value problems (1.3a) and (1.3b), respectively, are ob-vious. First, the former pair involves the integral operator L whereas the latter pairinvolves a second-order spatial differential operator. Second, in (1.2a) and (1.2b),constraints are imposed on the solution over a nonzero volume ΩI that is not neces-sarily located near or at the boundary of Ω; on the other hand, in (1.3a) and (1.3b),constraints are imposed precisely at the bounding surface ∂Ω. These distinctions areessential in characterizing the differences in the properties of, e.g., problems (1.2b)and (1.3b) and of their solutions. In Section 1.1, we discuss why volume constraintsare not only useful, but are necessary to treat certain classes of nonlocal problemshaving the form (1.2a) or (1.2b).

Section 2 reviews classical diffusion problems and, in particular, the notions ofa local flux, local balance laws, local diffusion problems, and the well posedness ofsteady-state local diffusion problems. This discussion serves to set up Sections 3 and 4in which analogous notions for nonlocal volume-constrained problems are considered.In Section 3.1, the notion of a nonlocal flux is discussed. That discussion is crucialto Section 3.3 in which nonlocal balance laws are posed from which nonlocal diffusionproblems of the type (1.2a) are derived; the latter derivation depends on results fromthe nonlocal vector calculus of [27] that is briefly reviewed in Section 3.2. The materialin Sections 2 and 3 render transparent the correspondences between local and nonlocaldiffusion problems, with a central correspondence between the boundary-constraintoperator B in the local case and the volume-constraint operator V in the nonlocalcase.

Section 4 contains the primary contribution of our paper; there, the well posednessof steady-state volume-constrained diffusion problems of the type (1.2b) is demon-strated by exploiting the nonlocal vector calculus reviewed in Section 3.2. The notionof volume-constrained problems enables us to formulate and solve diffusion problemsin situations where boundary conditions are not well defined, e.g., diffusive regimeswhere the Fourier symbol of the self-adjoint diffusive operator is of non-negative order

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less than or equal to 1/2. Such diffusion problems, e.g., the volume-constrained prob-lem (1.2b), allow discontinuous functions as solutions, given appropriate conditions onthe kernel γ. Also, because the fractional Laplacian operator and a fractional deriva-tive operators are special cases of the operator L (see Section A), the well posednessof volume-constrained problems on bounded domains in Rn posed in terms of theseoperators also readily follows, a previously unavailable result.

In Section 5, the well-posedness for nonlocal evolution problems such as (1.2a)and a nonlocal wave equation are briefly discussed and a nonlocal model for advection-diffusion problems involving the operator L is presented. Also, a brief discussion ofvanishing nonlocality is given that demonstrates that, in the limit when the supportof the kernel γ decreases, the classical local diffusive operator is recovered. Anothercontribution of our paper is the analyses of the convergence and of error and conditionnumber estimates for finite-dimensional discretizations of nonlocal volume-constrainedproblems; these analyses are given in Section 6 where the focus is on finite elementmethods. The finite element methods considered are conforming and, for appropriatekernels γ, allow for the use of piecewise polynomials that are not required to becontinuous across element faces, e.g., discontinuous Galerkin methods are conformingin those cases. This is in stark contrast to discontinuous Galerkin methods for thediscretization of, e.g., (1.3b) for which they are nonconforming [6].

An inspiration for this paper is to extend results of [29, 51] and [4, Chap. 1-3]to the volume-constrained problem (1.2b) on bounded domains. The well-posednessresults derived in this paper extend the result established in [34] on a bounded domainfor a class of kernels γ and volume constraint operators that lead to an operator Lwhose inversion does not regularize the data. This latter result was extended in[2] to another class of volume constraints. In [4, Chapter 1] the well posedness ofthe Cauchy problem for the nonlocal diffusion equation (1.2a) is considered whereas[4, Chapter 2–3] consider a special choice of volume constraints for the case of γ aradial function, i.e., γ(x,y) = γ(y − x), with γ(0) > 0 and

∫Rn γ(z) dz = 1. Such

conditions on γ imply that L is a mapping from L2(Rn) to L2(Rn); conditions suchthat inversions of L smooth the data are not considered.

1.1. The need for volume constraints. Consideration of the need for impos-ing volume constraints for problems involving the nonlocal operator L requires us todiscuss in more detail and with greater precision the differences between problemsinvolving L and those involving second-order elliptic partial-differential operators.First, we recall that if u = 0 on ∂Ω, i.e., if in (1.3b) g = 0 and B is the identityoperator, and if appropriate conditions on Ω and C are assumed, then, given datab ∈ H−1(Ω), a weak formulation of the boundary-value problem (1.3b) is well posedin H1

0 (Ω), i.e., there exists a unique solution u ∈ H10 (Ω) and moreover that solution

depends continuously on the data b. Alternately, we have that ∇·(C ·∇) is a boundedoperator from H1

0 (Ω)→ H−1(Ω) having a bounded inverse.

In contrast, if g = 0 and V is the identity operator in (1.2b), i.e., if u = 0 onΩI , then, for appropriate choices for the kernel γ, we demonstrate, in Section 4, thata variational formulation of the volume-constrained problem (1.2b) is well posed inthe space Hs

c (Ω ∪ ΩI) for 0 < s < 1, provided the given data b belongs to the dualspace of Hs

c (Ω ∪ ΩI), where Hsc (Ω ∪ ΩI) is the subspace of the fractional Sobolev

space Hs(Ω ∪ ΩI) constrained to satisfy the volume constraint u = 0 on ΩI . We evendemonstrate, for a particular class of kernels γ, that the variational formulation iswell posed in L2

c(Ω ∪ ΩI), provided the given data b belongs to L2c(Ω ∪ ΩI) as well.

Alternately, we have that L is a bounded operator from Hsc (Ω ∪ ΩI) to its dual space

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or from L2c(Ω ∪ ΩI)→ L2

c(Ω ∪ ΩI), as the kernel warrants, and that L has a boundedinverse. In particular, the solution operator for (1.2b) regularizes, i.e., smooths, thedata b to a lesser extent compared to the solution operator for (1.3b) and, underappropriate conditions on γ, the solution is no smoother than the data. The latteroccurs, for example, when L is a Hilbert-Schmidt operator.

The fact that weak formulations of the volume-constrained problem (1.2b) are wellposed in subspaces of Hs(Ω ∪ ΩI) for s ∈ [0, 1/2] deserves further comment. First,why is treating such a case important? The answer is that it is precisely these spacesthat contain functions with jump discontinuities. Thus, if one wants to admit solutionsthat have jump discontinuities, one has to work with spaces such as Hs(Ω ∪ ΩI) fors ∈ [0, 1/2]. We next ask: could we instead impose a surface constraint, e.g., couldwe consider the problem

−Lu = b on Ω

u = 0 on ∂Ω.

We have that, for appropriate kernels, Lu is well defined for u ∈ Hs(Ω ∪ ΩI) fors ∈ [0, 1). However, the restriction of a function u ∈ Hs(Ω ∪ ΩI) onto ∂(Ω ∪ ΩI) isnot defined for s ∈ [0, 1/2], i.e., the trace of such u is not defined. This means thatif s ∈ [0, 1/2], we cannot impose constraints on u restricted to ∂(Ω ∪ ΩI). However,a volume constraint for which the operator V is the restriction operator onto ΩIis well-defined for all s ∈ [0, 1) and beyond. Thus, we conclude that for nonlocaloperator equations posed on bounded domains, the application of volume constraintsis necessitated for operators that are bounded acting on Hs(Ω ∪ ΩI) functions withs ∈ [0, 1/2].

For example, the well-posedness of (1.3b) when ∇ · ∇ is replaced by ∆2s (thefractional Laplacian) is not discussed whenever s ∈ [0, 1/2] because of the lack of well-defined traces of functions belonging to Hs(Ω). More generally, the well-posednessand tractable numerical methods for pseudo-differential or fractional self-adjoint dif-ferential operators on bounded domains when the degree of the (Fourier) symbol liesin s ∈ (0, 1/2] are not available. In contrast, the volume-constrained, nonlocal, prob-lem (1.2b) is, for appropriate kernels γ, well posed for s ∈ [0, 1) and finite elementdiscretizations are easily defined from the variational characterizations possible forsuch problems. Thus, we see that in such cases, volume constraints are expedient.As further examples, we note that the use of volume constraints for equations in-volving the operator L immediately removes the limitation encountered in [7, 30] toonly consider a fractional dispersion equation and kernels γ, respectively, where thesolutions are in Hs

0(Ω) for s ∈ (1/2, 1), and to present a numerical method for thesolution of the fractional Laplacian operator equations on bounded domains in lieu ofthe random walk approximation used in [52].

2. The classical local differential equation setting for diffusion. We re-view well-known notions related to diffusion in the classical differential equation set-ting, starting with the notion of a local flux. This review serves as a template forthe discussion of similar notions in Sections 3 and 4 for nonlocal diffusion and alsoprovides comparisons and analogies between the local and nonlocal cases.

2.1. Local fluxes. Let Ω1 ⊂ Rn and Ω2 ⊂ Rn denote two disjoint open regions.If Ω1 and Ω2 have a nonempty common boundary ∂Ω12 = Ω1 ∩ Ω2, then, for a

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sufficiently smooth1 vector function q(x), the expression∫∂Ω12

q · n1 dA (2.1)

represents the classical local flux out of Ω1 into Ω2, where n1 denotes the unit normalon ∂Ω12 pointing outward from Ω1 and dA denotes a surface measure in Rn; q · n1 isreferred as the flux density along ∂Ω12. The flux, then, conveys a notion of directionout of and into a region and is a proxy for the interaction between Ω1 and Ω2. It isimportant to note that the flux from Ω1 into Ω2 occurs across their common boundaryand that if the two disjoint regions have no common boundary, then the flux fromone to the other is zero. The classical flux (2.1) is then deemed to be local becausethere is no interaction between Ω1 and Ω2 when separated by a finite distance. Theclassical flux satisfies the action-reaction principle∫

∂Ω12

q · n1 dA+

∫∂Ω21

q · n2 dA = 0, (2.2)

where, of course, ∂Ω12 = ∂Ω21 and n2 = −n1 denotes the unit normal on ∂Ω12

pointing outward from Ω2. In words, the flux∫∂Ω12

q · n1 dA from Ω1 into Ω2 across

their common boundary ∂Ω12 is equal and opposite to the flux∫∂Ω21

q · n2 dA fromΩ2 into Ω1 across that same surface.

2.2. Local diffusion. Let Ω denote a bounded, open set in Rn. Then, classicalbalance laws have the form

d

dt

∫Ω

u dx =

∫Ω

b dx−∫∂Ω

q · n dA ∀ Ω ⊆ Ω, t > 0, (2.3)

where n denotes the unit normal on ∂Ω pointing outwards from Ω, b denotes thesource density for u in Ω, and q ·n denotes the flux density along ∂Ω corresponding tou. In words, (2.3) states that the temporal rate of change of the quantity

∫Ωu(x, t) dx

is given by the amount of u created within Ω by the source b minus the flux ofu out of Ω through its boundary ∂Ω. If b = 0 in Ω and q · n = 0 on ∂Ω, then∫

Ωu(x, t) dx =

∫Ωu(x, 0) dx, i.e., if there are no sources of u within Ω and there is no

flux of u out of Ω, then∫

Ωu(x, t) dx, the quantity of u in Ω, is conserved.

Applying the Gauss theorem, we have from (2.3) that

d

dt

∫Ω

u dx =

∫Ω

b dx−∫

Ω

∇ · q dx ∀ Ω ⊆ Ω, t > 0. (2.4)

Because Ω ⊆ Ω is arbitrary, (2.4) leads to the field equation

ut +∇ · q = b ∀x ∈ Ω, t > 0. (2.5)

Classical diffusion arises when the relation between q and u is given by Fick’s firstlaw q = −C · ∇u, where C(x) denotes a symmetric, positive definite second-ordertensor. Substitution into (2.5) yields the classical diffusion equation

ut −∇ · (C · ∇u) = b ∀x ∈ Ω, t > 0. (2.6a)

1The components of q belonging to H1(Ω) is sufficient but one can have even weaker spaces.

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It is well known that (2.6a) does not uniquely determine u so that one must alsorequire u to satisfy an initial condition

u(x, 0) = u0(x) ∀x ∈ Ω (2.6b)

and a boundary condition

Bu = g ∀x ∈ ∂Ω, t > 0, (2.6c)

where B denotes an operator acting on functions defined on ∂Ω. Common choicesinclude Bu = u, Bu = (C · ∇u) · n, or Bu = (C · ∇u) · n + ϕu (with ϕ(x, t) ≥ 0) foru belonging to appropriate spaces, applied on all of ∂Ω, giving the classical Dirichlet,Neumann, and Robin problems, respectively. One can also have mixed boundaryconditions for which two or more of these choices are applied on disjoint, covering partsof ∂Ω. In (2.6), b(x, t) : Ω× (0, T )→ R, u0(x) : Ω→ R, and g(x, t) : ∂Ω× (0, T )→ Rare given functions. The system (2.6) characterizes diffusion because if b = 0 andg = 0 and for the choices for B, we have that, using Green’s first identity,

1

2

d

dt

∫Ω

u2 dx+

∫Ω

∇u · (C · ∇u) dx ≤ 0

so that the rate of decay of ‖u‖L2(Ω) depends upon spatial variations. For Dirichletor Neumann boundary conditions, this relation holds as an equality.

2.3. Steady-state local diffusion with boundary constraints. Steady-statediffusion occurs when ut = 0 in (2.6a). We then have that the initial-boundary valueproblem (2.6) reduces to the elliptic boundary-value problem (1.3b), where, of course,now b and g do not depend on t.

The variational analysis for steady-state diffusion starts by considering the solu-tion of the minimization problem

minimize1

2

∫Ω

∇u · (C · ∇u) dx +1

2

∫∂Ωr

ϕu2 dA−∫

Ω

ub dx−∫∂Ωn

ugn dA

−∫∂Ωr

ugr dA

subject to u = gd on ∂Ωd

and, if ∂Ω = ∂Ωn,

∫Ω

u dx = cn,

(2.7)

where ∂Ωd, ∂Ωn, and ∂Ωr are the disjoint, covering parts of the boundary ∂Ω on whichDirichlet, Neumann, and Robin boundary conditions are applied, respectively; ϕ > 0,b, gd, gn, and gr denote given functions and cn a given constant. The Euler-Lagrangeequations corresponding to the minimization problem (2.7) lead to the boundary-valueproblem

−∇ · (C · ∇u) = b in Ω (2.8a)

(C · ∇u) · n = gn on ∂Ωn (2.8b)

(C · ∇u) · n + ϕu = gr on ∂Ωr (2.8c)

u = gd on ∂Ωd (2.8d)∫∂Ω

u dA = cn if ∂Ω = ∂Ωn, (2.8e)

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where, if ∂Ω = ∂Ωn, the data b and gn are required to satisfy the compatibilitycondition ∫

Ω

b dx +

∫∂Ω

gn dA = 0. (2.9)

Note that (2.8d) and (2.8e) are essential boundary conditions that must be imposedon candidate solutions of the minimization problem (2.7) whereas (2.8b) and (2.8c)are natural boundary conditions that need not be explicitly imposed on candidatesolutions. Note also that in case ∂Ω = ∂Ωn, (2.9) is necessary to show the existenceof a solution of (2.8) whereas (2.8e) is sufficient to ensure that that solution is unique.

The above formal procedures are made precise and well posedness results areobtained for (2.8) by choosing appropriate function spaces for the data and solution,then defining the symmetric bilinear form a(u, v) :=

∫Ω∇v · (C ·∇u) dx+

∫∂Ωr

ϕuv dA

and the linear functional l(v) :=∫

Ωvb dx +

∫∂Ωn

vgn dA +∫∂Ωr

vgr dA, and theninvoking the Lax-Milgram theorem. Necessary hypotheses are that the bilinear formis continuous and coercive and the linear functional is continuous which, for (2.8), holdtrue; see, e.g., [18]. For example, if ∂Ω = ∂Ωd and gd = 0, i.e., for the homogeneousDirichlet problem, we have that u ∈ H1

0 (Ω) and b ∈ H−1(Ω) whereas if ∂Ω = ∂Ωn,gn = 0, and cn = 0, i.e., for the homogeneous Neumann problem, we have thatu ∈ H1

c (Ω) := v ∈ H1(Ω) :∫

Ωv dx = 0 and b ∈ (H1

c (Ω))∗ where the latter spacedenotes the dual of H1

c (Ω).

3. Nonlocal fluxes, a nonlocal vector calculus, and nonlocal diffusion.In this section and in Section 4, we parallel, for the nonlocal case, the presentationgiven in Section 2 for the local diffusion case; in particular, Sections 3.1 and 3.3 mimicSections 2.1 and 2.2, respectively, whereas Section 4 mimics Section 2.3. Recall that,in Sections 2.2 and 2.3, elements of the classical vector calculus for differential opera-tors were invoked; analogously, in Sections 3.3 and 4, elements of a vector calculus fornonlocal operators are invoked. Thus, the nonlocal vector calculus, which is a gen-eralization of the classical vector calculus to nonlocal operators, enables us to studynonlocal diffusion problems in a manner analogous to how classical diffusion is stud-ied. Because the nonlocal calculus may not be familiar to the reader, we provide, inSection 3.2, a brief review of the nonlocal vector calculus developed in [27].

The clarity achieved through the use of the nonlocal vector calculus to enable adevelopment of nonlocal diffusion that mimics in every way that for local diffusionbenefits both mathematical analyses and physical interpretations, as is demonstratedin the remainder of the paper.

3.1. Nonlocal fluxes. The key to understanding (1.2a) as a model for nonlocaldiffusion is the identification of a nonlocal flux. This enables us to state a nonlocalbalance law that postulates that the rate of change of an extensive quantity over someregion is equal to the production of that quantity in that region minus the flux of thesame quantity out of that region. Here, we review the discussion about nonlocal fluxesgiven in [27] where a detailed discussion is provided. In the discussion, ψ(x,y) maybe a scalar or vector or tensor function. For the sake of brevity, in this subsectionand in Section 3.2, we suppress explicit reference to the time dependence of variables.

For any point x ∈ Rn, we identify∫Ω

ψ(x,y) dy ∀ Ω ⊆ Rn (3.1)

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as the nonlocal flux density at x into Ω. We have that the following three statementsare equivalent:

• ψ(x,y) is an antisymmetric function, i.e., ψ(x,y) = −ψ(y,x); (3.2a)

• there are no self-interactions, i.e.,

∫Ω

∫Ω

ψ(x,y) dy dx = 0 ∀ Ω ⊆ Rn; (3.2b)

• for two regions Ω1,Ω2 ⊂ Rn, both having nonzero volume, we have

the nonlocal action-reaction principle∫Ω1

∫Ω2

ψ(x,y) dy dx +

∫Ω2

∫Ω1

ψ(x,y) dy dx = 0 ∀Ω1,Ω2 ⊂ Rn. (3.2c)

With (3.1) denoting a nonlocal flux density, for any two open regions Ω1 ⊆ Rnand Ω2 ⊆ Rn, both having nonzero volume, we identify∫

Ω1

∫Ω2

ψ(x,y) dydx (3.3)

as a scalar interaction or nonlocal flux from Ω1 into Ω2. Because we require no selfinteractions, i.e., that the flux from a region into itself vanishes so that (3.2b) holdswhenever Ω1 = Ω2, we have that ψ : (Ω1 ∪ Ω2)× (Ω1 ∪ Ω2)→ R is an antisymmetricfunction. By (3.1),

∫Ω2ψ(x,y) dy is the nonlocal flux density at a point x ∈ Ω1 into

the region Ω2. As is the case for the local flux density q ·n1, the nonlocal flux density∫Ω2ψ(x,y) dy is related to an intensive variable through a constitutive relation; see

Section 3.3.Based on the above discussion, we see that (3.2c) is the nonlocal analogue of

(2.2). In words, (3.2c) states that the flux (or interaction) from Ω1 into Ω2 is equaland opposite to the flux (or interaction) from Ω2 into Ω1. The flux is nonlocal because,by (3.2c), the interaction may be nonzero even when the closures of Ω1 and Ω2 have anempty intersection. This is in stark contrast to local interactions for which we haveseen that the interaction between Ω1 and Ω2 vanishes if their closures have emptyintersection, i.e., if they have no common boundary.

3.2. Elements of a nonlocal vector calculus. In [27], a nonlocal vector cal-culus is developed; here, we briefly review those aspects of that calculus that areuseful in the sequel.

Given the vector mappings ν(x,y),α(x,y) : Rn×Rn → Rk with α antisymmetric,i.e., α(y,x) = −α(x,y), the action of the nonlocal divergence operator D on ν isdefined in [27] as2

D(ν)(x) :=

∫Rn

(ν(x,y) + ν(y,x)

)·α(x,y) dy for x ∈ Rn, (3.4)

where D(ν) : Rn → R. The nonlocal divergence operator D is motivated by equatingψ(x,y) introduced in §3.1 with the integrand of (3.4); for a full justification of thischoice, see [27]. Here we just mention that this choice results from an application ofthe Schwarz kernel theorem after making the natural assumptions that the nonlocaldivergence operator should be a linear operator and that the integral of the nonlocaldivergence of a vector over any domain should equal the flux out of that domain; the

2In [27], a more general expression for the nonlocal divergence operator is derived; however, thespecialized definition (3.4) suffices for the treatment of nonlocal diffusion given in this paper.

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latter assumption is motivated by the fact that the classical differential divergenceoperator may be so defined.

Given the mapping u(x) : Rn → R, the adjoint operator D∗ corresponding to D isthe operator whose action on u is given by

D∗(u)(x,y) = −

(u(y)− u(x)

)α(x,y) for x,y ∈ Rn, (3.5)

where D∗(u) : Rn×Rn → Rk. With D∗ denoting the adjoint of the nonlocal divergenceoperator, we view −D∗ as a nonlocal gradient. The mapping D(ν) is scalar-valuedin analogous fashion to the local differential divergence of a vector function and themapping D∗(u) is vector-valued in analogous fashion to the local differential gradientof a scalar function u(x).

From (3.4) and (3.5), one easily deduces that if Θ(x,y) = Θ(y,x) denotes asecond-order tensor satisfying Θ = ΘT , then

D(Θ · D∗u)(x) = −2

∫Rn

(u(y)− u(x)

)α(x,y) ·

(Θ(x,y) ·α(x,y)

)dx for x ∈ Rn,

where D(Θ · D∗u) : Rn → R. Comparing with (1.1), we have that

Lu = −D(Θ · D∗u

)with γ = α · (Θ ·α). (3.6)

Thus, the operator L is a composition of nonlocal divergence and gradient operators sothat if Θ is the identity tensor, L can be interpreted as a nonlocal Laplacian operator.Because D and D∗ are adjoint operators, if Θ is also positive definite, the operator−L is non-negative; see [32, Proposition 2.1] and also [12, 27, 34] for discussion whenΘ is the unit tensor.

Given an open subset Ω ⊂ Rn, the corresponding interaction domain is definedas

ΩI := y ∈ Rn \ Ω such that α(x,y) 6= 0 for some x ∈ Ω (3.7)

so that ΩI consists of those points outside of Ω that interact with points in Ω. Notethat the situation ΩI = Rn \Ω is allowable as is Ω = Rn in which case ΩI = ∅. Then,corresponding to the divergence operator D

(ν)

: Rn → R defined in (3.4) we definethe action of the nonlocal interaction operator N (ν) : Rn → R on ν by

N(ν)(x) := −

∫Ω∪ΩI

(ν(x,y) + ν(y,x)

)·α(x,y) dy for x ∈ ΩI . (3.8)

In [27], based on a discussion along the lines of that in Section 3.1, it is shown that∫ΩIN (ν) dx can be viewed as a nonlocal flux out of Ω into ΩI . The main difference

between the local and nonlocal cases is that, in the former case, the flux out of adomain Ω is given by the boundary integral

∫∂Ω

q ·n dA whereas, in the nonlocal case,that flux is given by the volume integral

∫ΩIN (ν) dx.

With D and N defined as in (3.4) and (3.8), respectively, we have the nonlocalGauss theorem ∫

Ω

D(ν) dx =

∫ΩI

N (ν) dx. (3.9)

Next, let u(x) and v(x) denote scalar functions. Then, it is a simple matterto show that the nonlocal divergence theorem (3.9) implies the generalized nonlocal

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Green’s first identity∫Ω

vD(Θ·D∗u) dx−∫

Ω∪ΩI

∫Ω∪ΩI

(D∗v)·(Θ·D∗u) dydx =

∫ΩI

vN (Θ·D∗u) dx. (3.10)

For details concerning the nonlocal calculus, see [27], where one also finds furtherresults for the nonlocal divergence operator D, including a nonlocal Green’s secondidentity, as well as analogous results for nonlocal gradient and curl operators. Inaddition, in [27], further connections are made between the nonlocal operators andthe corresponding local operators. For example, it is shown there that for the specialkernel α(x,y) = −∇yδ(y − x), where δ(·) denotes the Dirac delta measure and ∇y

denotes the differential gradient with respect to y, one has that

D(ν) = ∇ · (ν(x,x)) = ∇ ·(ν(x,y)

∣∣y=x

),

∫Rn

D∗(u) dy = −∇u(x),

and ∫ΩI

vN (ν) dx =

∫∂Ω

v(x)ν(x,x) · n dA ∀ v ∈ C∞0 (Rn).

3.3. Nonlocal diffusion. Let Ω denote a bounded, open set in Rn. Nonlocalbalance laws have the form

d

dt

∫Ω

u dx =

∫Ω

b dx−∫

ΩI

N (ν) dx ∀ Ω ⊆ Ω, t > 0, (3.11)

where b(x, t) denotes the source density for u in Ω and ΩI ⊆ ΩI denotes the interaction

region corresponding to Ω. In words, (3.11) states that the temporal rate of change

of the quantity∫

Ωu(x, t) dx is given by the amount of u created within Ω ⊆ Ω by the

source b minus the nonlocal flux of u out of Ω into ΩI . If b ≡ 0 and N (ν) ≡ 0 in ΩI ,then

∫Ωu(x, t) dx =

∫Ωu(x, 0) dx, i.e., just as in the local case, if there are no sources

of u within Ω and there is no flux of u out of Ω, then∫

Ωu(x, t), the quantity of u in

Ω, is conserved.Applying the nonlocal Gauss’ theorem3 (3.9) to the last term in (3.11) results in

d

dt

∫Ω

u dx =

∫Ω

b dx−∫

Ω

D(ν) dx ∀ Ω ⊆ Ω, t > 0. (3.12)

Because Ω ⊆ Ω can be chosen arbitrarily, the balance law (3.12) implies the nonlocalfield equation

ut +D(ν) = b ∀x ∈ Ω, t > 0. (3.13)

Nonlocal diffusion arises when, in analogy to local diffusion,4

ν = Θ · (D∗u), (3.14)

3Recall that it is exactly at the analogous point in Section 2.2, i.e., at (2.4), that we invoked theclassical Gauss’s theorem.

4Recall that D∗, being the adjoint of the nonlocal divergence, represents the negative of a nonlocalgradient. This accounts for the absence of the minus sign when compared to the local relationq = −C · ∇u.

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where Θ(x,y) denotes a symmetric, positive definite (in the matrix sense) second-order tensor having elements that are symmetric functions of x and y. The relation(3.14) represents a nonlocal Fick’s first law. Substitution of (3.14) into (3.13) leadsto the nonlocal diffusion equation

ut +D(Θ · D∗u) = b ∀x ∈ Ω, t > 0. (3.15a)

We append the initial condition

u(x, 0) = u0(x) ∀x ∈ Ω (3.15b)

and the volume constraint

Vu = g ∀x ∈ ΩI , t > 0 (3.15c)

to (3.13). Examples of the operator V are given in Section 4. The system (3.15) isthe nonlocal analog of the local differential system (2.6). From (3.6), we see that(3.15) is exactly the problem (1.2a) so that the latter is indeed a nonlocal diffusionproblem. The system (3.15) represents diffusion because if b = 0 and g = 0, then, forthe volume constraints considered in this paper,

1

2

d

dt

∫Ω

u2 dx +

∫Ω∪ΩI

∫Ω∪ΩI

D∗u ·(Θ · D∗u

)dy dx = 0.

This relationship is derived by multiplying (3.13) by u, integrating the result over Ω,and using the nonlocal Green’s first identity (3.10). In analogy to local diffusion asexplained at the end of Section 2.2, the rate of decay of ‖u‖L2(Ω) also depends uponspatial variations.

4. Steady-state nonlocal volume-constrained diffusion problems. As wedid in Section 2.3 for classical local diffusion, in this section we study the steady-statenonlocal diffusion problem. In Section 4.1, we discuss general problems involvingmixed inhomogeneous “Dirichlet”, “Neumann”, and “Robin”-type volume constraints;the correspondences between the discussions of local and nonlocal steady-state diffu-sion found in Sections 2.3 and 4.1, respectively, are transparent. In Sections 4.2–4.4in which the well-posedness of nonlocal volume-constrained problems is considered,we specialize to homogeneous “Dirichlet” and “Neumann”-type problems.

4.1. Nonlocal variational problems and volume-constrained problems.Given an open region Ω ⊂ Rn and the corresponding interaction domain ΩI definedin (3.7), let ΩI = ΩId ∪ ΩIn ∪ ΩIr with ΩId, ΩIn, and ΩIr mutually disjoint andwith at most two empty. We then define the energy functional

E(u; b, gn, gr) :=1

2

∫Ω∪ΩI

∫Ω∪ΩI

D∗(u)(x,y) ·(Θ(x,y) · D∗(u)(x,y)

)dy dx

+1

2

∫ΩIr

ϕ(x)u2(x) dx

−∫

Ω

b(x)u(x) dx−∫

ΩIn

gn(x)u(x) dx−∫

ΩIr

gr(x)u(x) dx

=1

2

∫Ω∪ΩI

∫Ω∪ΩI

(u(y)− u(x)

)2γ(x,y) dy dx +

1

2

∫ΩIr

ϕ(x)u2(x) dx

−∫

Ω

b(x)u(x) dx−∫

ΩIn

gn(x)u(x) dx−∫

ΩIr

gr(x)u(x) dx,

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where ϕ(x) > 0, b(x), gn(x), and gr(x) are given functions defined on ΩIr, Ω, ΩIn,and ΩIr, respectively, and γ(x,y) = α(x,y) ·

(Θ(x,y) · α(x,y)

). Consider the

constrained minimization problem

minE(u; b, gn, gr) subject to Ec(u; g) = 0, (4.1)

where Ec(u; g) denotes a constraint functional. For example, let

Ec(u; g) = Edc (u; gd) :=

∫ΩId

(u− gd)2 dx if ΩId 6= ∅, (4.2)

where gd(x) is a given function defined on ΩId. Note that Edc (u; gd) = 0 impliesthat u(x) = gd(x) a.e. in ΩId, i.e., we have a “Dirichlet”-type condition applied overthe interaction domain ΩId ⊆ ΩI having positive measure. On the other hand, ifΩId ∪ ΩIr = ∅, let

Ec(u; g) = Enc (u; cn) :=(cn −

∫Ω∪ΩI

u dx)2

when ΩIn = ΩI (4.3)

for a given constant cn so that Enc (u; cn) = 0 implies that∫

Ω∪ΩIu dx = cn. Proceeding

formally with the use of standard techniques from the calculus of variations, we obtainthe first-order necessary conditions corresponding to the minimization problem (4.1)given by∫

Ω∪ΩI

∫Ω∪ΩI

D∗(u)(x,y) ·(Θ(x,y) · D∗(v)(x,y)

)dy dx

+

∫ΩIr

ϕ(x)u(x)v(x) dx

=

∫Ω

b(x)v(x) dx +

∫ΩIn

gn(x)v(x) dx +

∫ΩIr

gr(x)v(x) dx,

(4.4)

where test functions v(x) satisfy the constraint Ec(v; 0) = 0, e.g., for the caseEdc (v; 0) = 0, we have that v = 0 a.e. in ΩId. Note that if ΩI = ΩIn, i.e., ifΩId ∪ ΩIr = ∅, then by setting v(x) = 1 in (4.4) we conclude that the data b and gnare required to satisfy the compatibility condition∫

Ω

b dx +

∫ΩI

gn dx = 0 when ΩI = ΩIn. (4.5)

By applying the nonlocal Green’s first identity (3.10) to (4.4), we obtain, becausev(x) = 0 a.e. in ΩId,∫

Ω

vD(Θ · D∗u) dx+

∫ΩIr

ϕuv dx−∫

ΩIn∪ΩIr

vN (Θ · D∗u) dx

=

∫Ω

bv dx +

∫ΩIn

gnv dx +

∫ΩIr

grv dx.

Because v(x) is arbitrary in Ω ∪ ΩIn ∪ ΩIr, we then obtain that solutions of the

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minimization problem (4.1) satisfy5

−L(u) = D(Θ · D∗u

)= b on Ω (4.6a)

−N(Θ · D∗u

)= gn on ΩIn (4.6b)

−N(Θ · D∗u

)+ ϕu = gr on ΩIr (4.6c)

u = gd on ΩId (4.6d)∫Ω∪ΩI

u dx = cn if ΩI = ΩIn. (4.6e)

Because we have associated N (·) with the nonlocal flux out of Ω into ΩI , we re-fer to (4.6b) and (4.6c) as nonlocal “Neumann” and nonlocal “Robin” volume con-straints whereas, of course, (4.6d) is a nonlocal “Dirichlet” volume constraint. Thus, ifΩI = ΩId, we have the nonlocal “Dirichlet” problem (4.6a) and (4.6d). If ΩI = ΩIr,we have the nonlocal “Robin” problem (4.6a) and (4.6c). If ΩI = ΩIn, we havethe nonlocal “Neumann” problem (4.6a), (4.6b), and (4.6e);6 in this case, the com-patibility condition (4.5) on the data is needed to ensure the existence of a solutionwhereas (4.6e) is a constraint that ensures the uniqueness of that solution. See [27]for a related discussion.

The two choices (4.2) and (4.3) for the constraint operator Ec(u; g) in the vari-ational principle (4.1), or, equivalently, (4.6d) and (4.6e) in the nonlocal volume-constrained problem (4.6), respectively, are essential to the variational principle (4.1),i.e., they must be imposed on candidate minimizers. On the other hand, (4.6b) and(4.6c) are natural to the variational principle (4.1), i.e., they do not have to be im-posed on candidate minimizers. Also, note that the constraints Edc (·; ·) and Enc (·; ·)are quite different; Edc (u; 0) involves the interaction domain ΩI and the the integral ofthe square of u over that domain whereas Enc (u; 0) involves the square of the integralof u over the domain Ω ∪ ΩI . This leads to distinct forms for the constraints ap-pearing in the nonlocal volume-constrained problem (4.6); the constraint (4.6d) holdspointwise almost everywhere in the subdomain ΩId whereas (4.6e) is a single integralconstraint.

Equations (4.2) and (4.3) may not be the only choices for Ec(u; g). Here, ingeneral, we only assume that Ec(·; 0) denotes a bounded, quadratic functional on aHilbert space which is defined in Section 4.2.

In the sequel, for the sake of brevity, we at times confine the discussion to thehomogeneous nonlocal “Dirichlet” problem

D(Θ · D∗u

)= b on Ω (4.7a)

u = 0 on ΩI (4.7b)

and the homogeneous nonlocal “Neumann” problem

D(Θ · D∗u

)= b on Ω (4.8a)

N(Θ · D∗u

)= 0 on ΩI (4.8b)∫

Ω∪ΩI

u dx = 0. (4.8c)

5The correspondence between (2.8) and (4.6) is obvious. We only point out that the apparentsign differences between (2.8a)–(2.8c) and (4.6a)–(4.6c), respectively, result because D∗ denotes thenegative of a nonlocal gradient operator.

6This is equivalent to the approach taken in [4, Chapter 3] where the constraint∫Rn\Ω

(u(y) −

u(x))γ(x,y) dy = 0 is prescribed for the problem

∫Rd

(u(y)− u(x)

)γ(x,y) dy = b for x ∈ Ω.

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4.2. The kernel. We assume that the domain Ω is bounded with piecewisesmooth boundary and satisfies the interior cone condition. For simplicity, we alsoassume that both ΩI and Ω ∪ ΩI have the same properties.

Given positive constants γ0 and ε, we first assume that the symmetric kernel γsatisfies, for all x ∈ Ω ∪ ΩI ,

γ(x,y) ≥ 0 ∀y ∈ Bε(x) with γ(x,y) ≥ γ0 > 0 ∀y ∈ Bε/2(x) (4.9a)

γ(x,y) = 0 ∀y ∈ (Ω ∪ ΩI) \Bε(x), (4.9b)

where Bε(x) := y ∈ Ω ∪ ΩI : |y − x| ≤ ε. Obviously, (4.9) implies that althoughinteractions are nonlocal, they are limited to a ball of radius ε.

The smoothing effected by the inversion of −L = D(ΘD∗(·)

)depends upon the

regularity associated with γ = α · (Θ · α). We consider the following two specialcases.Case 1. There exist s ∈ (0, 1) and positive constants γ∗ and γ∗ such that, for allx ∈ Ω,

γ∗|y − x|n+2s

≤ γ(x,y) ≤ γ∗

|y − x|n+2sfor y ∈ Bε(x). (4.10)

An example for this case is given by the kernel in (A.2) or, more generally, by

γ(x,y) =σ(x,y)

|y − x|n+2s(4.11)

with σ(x,y) bounded from above and below by positive constants; in particular, ifσ is not a radial function, i.e., if σ(x,y) 6= σ(|x − y|), then (4.11) can account forinhomogeneous media properties.Case 2. There exist positive constants γ1 and γ2 such that

γ1 ≤∫

(Ω∪ΩI)∩Bxε

γ(x,y) dy ∀x ∈ Ω (4.12a)∫Ω∪ΩI

γ2(x,y) dy ≤ γ22 ∀x ∈ Ω. (4.12b)

Examples for this case are provided by the kernels used in Refs. [2, 3, 4]We remark that a complete classification of kernels is not our goal; rather, we

treat a sufficiently broad class, as given by the above cases, that are of substantialmathematical and practical interest.

4.3. Equivalence of spaces. For the sake of brevity, in the remainder of thissection we consider only the homogeneous nonlocal “Dirchlet” and “Neumann” prob-lems (4.7) and (4.8), respectively.

We define the nonlocal energy norm, nonlocal energy space, and nonlocal volume-constrained energy space by

|||u||| :=(E(u; 0, 0, 0)

)1/2=

(1

2

∫Ω∪ΩI

∫Ω∪ΩI

D∗(u)(x,y) ·(Θ(x,y) · D∗(u)(x,y)

)dy dx

)1/2 (4.13a)

V (Ω ∪ ΩI) :=u ∈ L2(Ω ∪ ΩI) : |||u||| <∞

(4.13b)

Vc(Ω ∪ ΩI) := u ∈ V (Ω ∪ ΩI) : Ec(u; 0) = 0 , (4.13c)

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respectively, where the constraint equation Ec(u; 0) = 0 is given by (4.7b) and (4.8c)for the nonlocal “Dirchlet” and “Neumann” problems, respectively, and where we haveused the fact that ΩIr = ∅ for both (4.7) and (4.8). We also define |||u|||V ∗c (Ω∪ΩI) tobe the norm for the dual space V ∗c (Ω ∪ ΩI) of Vc(Ω ∪ ΩI) with respect to the standardL2(Ω ∪ ΩI) duality pairing.

The precise assumptions made about the constraint functional are that Ec(·; 0) isa bounded, quadratic functional on V (Ω ∪ ΩI) ∩ L2(Ω ∪ ΩI), that for some constantc

Ec(u; 0) ≤ c(|||u|||2 + ‖u‖2L2(Ω∪ΩI)

)∀u ∈ V (Ω ∪ ΩI), (4.14)

and that Ec(u; 0) is continuous with respect to the norm (|||u|||2 + ‖u‖2L2(Ω∪ΩI))1/2.

Moreover, we assume that the intersection of the set of constant-valued functions withthe set of functions satisfying Ec(u; 0) = 0 is u ≡ 0. Clearly, Ec(u; g) as defined byeither (4.2) or (4.3) satisfy these assumptions.

We now proceed to show that for Case 1, the nonlocal energy space V (Ω ∪ ΩI)is equivalent to the fractional-order Sobolev space Hs(Ω ∪ ΩI) whereas for Case 2,the nonlocal energy space is equivalent to L2(Ω ∪ ΩI). These equivalences imply thatthe quotient space Vc(Ω ∪ ΩI) is a Hilbert space equipped with the norm |||u|||. As aresult, the nonlocal volume-constrained problems (4.7) and (4.8) are well posed; seeSection 4.4.

For s ∈ (0, 1), the standard fractional-order Sobolev space is defined as

Hs(Ω ∪ ΩI) :=u ∈ L2(Ω ∪ ΩI) : ‖u‖L2(Ω∪ΩI) + |u|Hs(Ω∪ΩI) <∞

, (4.15)

where

|u|2Hs(Ω∪ΩI) :=

∫Ω∪ΩI

∫Ω∪ΩI

(u(y)− u(x)

)2|y − x|n+2s

dydx.

Moreover, define the subspaces

Hsc (Ω ∪ ΩI) := u ∈ Hs(Ω ∪ ΩI) : Ec(u; 0) = 0 (4.16)

and

L2c(Ω ∪ ΩI) :=

u ∈ L2(Ω ∪ ΩI) : Ec(u; 0) = 0

. (4.17)

4.3.1. Case 1. The following two lemmas are used to demonstrate that, for thiscase, the spaces Vc(Ω ∪ ΩI) and Hs

c (Ω ∪ ΩI) are continuously embedded within eachother.

Lemma 4.1. Let the function γ satisfy (4.9) and the lower bound of (4.10). Then,

|u|2Hs(Ω∪ΩI) ≤ γ−1∗ |||u|||2 + 4|Ω ∪ ΩI |ε−(n+2s)‖u‖2L2(Ω∪ΩI).

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Proof. We have

|u|2Hs =

∫Ω∪ΩI

∫Bε(x)∩(Ω∪ΩI)

(u(y)− u(x)

)2|y − x|n+2s

dydx

+

∫Ω∪ΩI

∫Ω∪ΩI\Bε(x)

(u(y)− u(x)

)2|y − x|n+2s

dydx

≤ γ−1∗ |||u|||2 + 2ε−(n+2s)

∫Ω∪ΩI

∫Ω∪ΩI

(u2(x) + u2(y)

)dydx

= γ−1∗ |||u|||2 + 4|Ω ∪ ΩI |ε−(n+2s)‖u‖2L2(Ω∪ΩI).

Lemma 4.2. Let the function γ satisfy (4.9) and the upper bound of (4.10).Then,

|||u|||2 ≤ γ∗|u|2Hs(Ω∪ΩI).

Proof. The result directly follows from∫Ω∪ΩI

∫Ω∪ΩI

D∗u ·(Θ(x,y) · D∗u

)dy dx ≤ γ∗

∫Ω∪ΩI

∫Ω∪ΩI

(u(y)− u(x)

)2|y − x|n+2s

dydx.

The following is the first of two nonlocal Poincare-type inequalities presented inthis paper. The inequality established in the next result depends crucially upon thecompact embedding of the fractional space Hs(Ω ∪ ΩI) into L2(Ω ∪ ΩI).

Lemma 4.3. [Nonlocal Poincare inequality I] Let the function γ satisfy (4.9)and (4.10). Then, there exists a positive constant C such that

‖u‖2L2(Ω∪ΩI) ≤ C|||u|||2 ∀u ∈ Vc(Ω ∪ ΩI). (4.18)

Proof. We exploit the standard technique for establishing a Poincare type inequal-ity by implying a contradiction. Assume there exists a sequence uk ∈ Vc(Ω ∪ ΩI)where ‖uk‖2L2(Ω∪ΩI) = 1 for all k such that 1 > k |||uk|||. By Lemma 4.1, we have

‖uk‖2Hs(Ω∪ΩI) < 4|Ω ∪ ΩI |ε−(n+2s) + 1

for sufficiently large k. Because the embedding Hs(Ω ∪ ΩI) → L2(Ω ∪ ΩI) is compactand Hs(Ω ∪ ΩI) is a Hilbert space, there exists a subsequence ukj of uk and anelement u ∈ Hs(Ω ∪ ΩI) such that ukj → u strongly in L2(Ω ∪ ΩI) so that

‖u‖L2(Ω∪ΩI) = 1. (4.19)

By Lemma 4.2, we have that, for any v ∈ Hs(Ω ∪ ΩI), v also belongs to V (Ω ∪ ΩI).By Fatou’s lemma on the convergence of integrals of function sequences that areconvergent almost everywhere,

limkj→∞

|||ukj ||| = |||u||| = 0

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because 1 > k |||uk|||. The definition of ||| · ||| implies that u is a constant. Moreover,

we have the convergence of ukj to u with respect to the norm(|||u|||2+‖u‖2L2(Ω∪ΩI)

)1/2and, by the continuity assumption on Ec, we obtain that

Ec(u; 0) = limkj→∞

Ec(ukj ; 0) = 0

so that u = 0. However, this contradicts (4.19) so that the conclusion (4.18) nowfollows.

Lemmas 4.1–4.3 lead to the following result.Theorem 4.4. If the function γ satisfies (4.9) and (4.10), then

C∗‖u‖Hs ≤ |||u||| ≤ C∗‖u‖Hs ∀u ∈ Vc(Ω ∪ ΩI),

where C∗ is a positive constants satisfying C−2∗ = max

(γ−1∗ , C(1+4|Ω ∪ ΩI |ε−(n+2s))

)and C∗ = γ∗.

Proof. Lemmas 4.1 and 4.3 imply that

‖u‖2Hs(Ω∪ΩI) ≤ γ−1∗ |||u|||2 +

(1 + 4|Ω ∪ ΩI |ε−(n+2s)

)‖u‖2L2(Ω∪ΩI) ≤ C

−1∗ |||u|||2.

In a similar fashion, Lemmas 4.2 and 4.3 lead to

|||u||| ≤ γ∗|u|2Hs(Ω∪ΩI) ≤ γ∗(|u|2Hs(Ω∪ΩI) + ‖u‖2L2(Ω∪ΩI)

)= C∗‖u‖Hs(Ω∪ΩI).

We then immediately obtain the following equivalence result between constrainedenergy spaces and constrained Sobolev spaces.

Corollary 4.5. If the function γ satisfies (4.9) and (4.10), we then have theequivalence of the constrained spaces Hs

c (Ω ∪ ΩI) and Vc(Ω ∪ ΩI).This theorem and corollary explain that, if the function γ satisfies (4.9) and (4.10),

then V (Ω ∪ ΩI) and its constrained subspace Vc(Ω ∪ ΩI) are compactly embedded inL2(Ω ∪ ΩI) and L2

c(Ω ∪ ΩI), respectively.We note that the equivalence of spaces holds with no restrictions on the exponent

s ∈ (0, 1) because of our consideration of volume constraints in lieu of constraints onthe boundary of the domain (or some other lower dimensional manifold). This is animportant point, particularly for s ≤ 1/2. Indeed, for s ≤ 1/2, there is no well-definedtrace space in the standard manner for functions in the Sobolev space Hs(Ω) which iswhy conventional boundary-value problems have not been discussed in the literaturefor such cases. The volume-constrained problem (1.2b), however, is well-posed for anys ∈ (0, 1), as is demonstrated in Section 4.4.

4.3.2. Case 2. We now demonstrate that, in this case, the constrained spaceVc(Ω ∪ ΩI) = L2

c(Ω ∪ ΩI). We choose to work with the more stringent conditionsgiven in (4.12) rather than other more general assumptions. This allows us to applywell-known results about integral operators. See [2, 4] for the case for which γ isradial and only L1(Ω ∪ ΩI) integrable.

We state the analogue of Lemma 4.2 that can be established through direct cal-culation; see, e.g., [34, 51] for details.

Lemma 4.6. If the function γ satisfies (4.9) and (4.12), then

|||u||| ≤ C2‖u‖L2(Ω∪ΩI) ∀u ∈ Vc(Ω ∪ ΩI)

for some positive constant C2.

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Next, we present a second nonlocal Poincare inequality that relies, in contrast tothe hypotheses of Lemma 4.3, on the compactness of a Hilbert-Schmidt kernel; see,e.g., [8, Chap. 12].

Lemma 4.7. [Nonlocal Poincare inequality II] If the function γ satisfies(4.9) and (4.12), then

C1‖u‖L2(Ω∪ΩI) ≤ |||u||| ∀u ∈ Vc(Ω ∪ ΩI)

for some positive constant C1.Proof. Because

1

2Lu(x) =

∫Ω∪ΩI

u(y) γ(x,y) dy − u(x)

∫Ω∪ΩI

γ(x,y) dy,

we see that the hypothesis on the function γ imply that the nonlocal diffusion operator−L = D

(ΘD∗) is a self-adjoint operator on L2

c(Ω ∪ ΩI). As established at the end ofSection 3.2, −L is a non-negative operator. Moreover, by the properties of Hilbert-Schmidt integral operators, we find that −L is also a compact perturbation of a scalarmultiple of the identity operator and is, in fact, uniformly bounded both above andbelow by positive constant multiples of the identity operator. Furthermore, the kernelof −L in L2

c(Ω ∪ ΩI) contains only the zero element. Therefore,

λ1 := infu∈L2(Ω∪ΩI)

|||u|||2

||u||2L2(Ω∪ΩI)

> 0,

i.e., the smallest eigenvalue of −L is strictly positive, and therefore we have√λ1 ‖u‖L2(Ω∪ΩI) ≤ |||u||| ∀u ∈ L2

c(Ω ∪ ΩI).

Thus, the conclusion of this lemma holds with C1 =√λ1.

The following result is an immediate consequence of Lemmas 4.6 and 4.7.Corollary 4.8. If the function γ satisfies (4.9) and (4.12), then Vc(Ω ∪ ΩI) is

equivalent to L2c(Ω ∪ ΩI).

4.4. Well-posedness of nonlocal volume-constrained problems. Sections4.3.1 and 4.3.2 established that the nonlocal constrained energy space Vc(Ω ∪ ΩI)is equivalent to Hs

c (Ω ∪ ΩI) and L2c(Ω ∪ ΩI) for Case 1 and Case 2, respectively.

The following result demonstrates that the minimization problem (4.1) has a uniqueminimizer if the constraint functional Ec(u; 0) satisfies the general requirement (4.14).

Theorem 4.9. The nonlocal variational problem of minimizing E(u; b, 0, 0) overVc(Ω ∪ ΩI) has a unique solution u for any b ∈ V ∗c (Ω ∪ ΩI). Moreover, the Euler-Lagrange equation is given by (4.7) for Ec(u; g) = Edc (u; 0) and (4.8) for Ec(u; g) =Enc (u; 0). Furthermore, there exists a constant C > 0, independent of b, such that

|||u||| ≤ C‖b‖V ∗c (Ω∪ΩI). (4.20)

Proof. The theorem is established via a direct application of the Lax-Milgramtheorem; see, e.g., [8, Section 3.6].

The upper bounds in (4.10) and (4.12b) are not needed to prove Theorem 4.9,i.e., to show well posedness with respect to space Vc(Ω ∪ ΩI) and the nonlocal energynorm ||| · |||. However, those bounds are needed to show the equivalence of that norm

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to standard Sobolev norms and so establish well posedness with respect to standardSobolev spaces.

In Section 4.3, we established the equivalence of the nonlocal constrained energyspace Vc(Ω ∪ ΩI) to Hs

c (Ω ∪ ΩI) or L2c(Ω ∪ ΩI) for Case 1 and Case 2, respectively.

Thus, for problems for which the function γ satisfies the assumptions of Case 1 orCase 2, the estimate (4.20) implies that

‖u‖Hs(Ω∪ΩI) ≤ C‖b‖H−s(Ω∪ΩI) for 0 < s < 1 or (4.21a)

‖u‖L2(Ω∪ΩI) ≤ C‖b‖L2(Ω∪ΩI) (4.21b)

hold, respectively. These should be contrasted with the analogous result for, e.g.,(1.3b) with homogeneous Dirichlet boundary conditions for which we have

‖u‖H1(Ω) ≤ C‖b‖H−1(Ω). (4.22)

The inequalities (4.21a) and (4.21b) show that a gain of regularity of 2s and 0,respectively, occurs for Case 1 or Case 2, respectively. In contrast, the inequality(4.22) results in a gain of regularity of 2. In other words, solutions of the nonlo-cal volume-constrained problems have at most 2s more derivatives than the data bwhereas solutions of the boundary-value problem (1.3b) can have two more derivatives.These regularity conditions are analogous to those established in [51] for restrictedclasses of one- and two-dimensional peridynamic models with constraints suggestiveof nonlocal volume-constrained conditions.

5. Additional comments about nonlocal volume-constrained problems.In this section, we briefly discuss the well-posedness of nonlocal evolution problems,vanishing nonlocality, and nonlocal advection-diffusion problems.

5.1. Well-posedness for nonlocal evolution equations. Using the resultsabout nonlocal operators and variational problems established in this paper, we mayuse standard techniques to establish well-posedness for nonlocal evolution equationssuch as the diffusion equation (1.2a) and the nonlocal wave equation (B.2). Asan illustration, we consider the special case for which the constrained energy spaceVc(Ω ∪ ΩI) associated with the functional (4.2) is established to be a Hilbert spacewith its dual space V ∗c (Ω ∪ ΩI) and that the operator D

(Θ · D∗) is bounded and

coercive in Vc(Ω ∪ ΩI). For that case, we have the following result.Theorem 5.1. Assume that b ∈ L2(0, T ;V ∗c (Ω)) and u0 ∈ Vc(Ω); then, the initial

volume-constrained problem (1.2a) has a unique solution u ∈ C(0, T ;Vc(Ω ∪ ΩI)) ∩H1(0, T ;V ∗c (Ω ∪ ΩI)). Moreover, assume instead that b ∈ L2(0, T ;V ∗c (Ω)) with u0 ∈Vc(Ω) and u1 ∈ L2(Ω); then, the initial volume constrained problem (B.2) has a uniquesolution u ∈ L2(0, T ;Vc(Ω ∪ ΩI)) ∩ L2(0, T ;L2(Ω ∪ ΩI)) ∩H1(0, T ;V ∗c (Ω ∪ ΩI)).

These results are consequences of standard semi-group theory or Galerkin typearguments. We refer to [51] for more detailed proofs of these results in a special casefor which the techniques used are directly generalizable to the problems consideredhere.

5.2. Vanishing nonlocality. The local limit of the operator L = −D(Θ · D∗)was examined in [27]. It was demonstrated that the free-space operator, i.e., L withΩ = Rn, converges to −∇ · (C · ∇) as ε → 0 under suitable conditions on the kernelfunction. More recently, in [4, 46], for kernel functions of radial type γ(x,y) =γ(|x − y|) with γ an element of L1(Ω ∪ ΩI) satisfying (4.9), the local limit of the

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nonlocal diffusion equation has been studied. Further, in [46], finite element solutionsfor nonlocal diffusion and the peridynamic model are also investigated.

As an example of a local limit, let Ω denote a bounded domain independent of εand C := limε→0 Cε, where

(Cε)ij =

∫Bε(0)

γ(|z|) zizj dz for i, j = 1, 2, . . . , n.

Then, recalling that γ satisfies (4.9) so that γ(|z|) = 0 for |z| > ε, a particularconsequence of the results in [46] is

limε→0

∫Ω∪ΩI

∫Ω∪ΩI

D∗v ·(Θ · D∗u

)dy dx

= limε→0

∫Ω∪ΩI

∫Ω∪ΩI

(u(y)− u(x)

)(v(y)− v(x)

)γ(|x− y|) dy dx

=

∫Ω

∇v ·(C · ∇u

)dx

(5.1)

for any u, v ∈ H1(Ω ∪ ΩI) with support in Ω. A more general form of the abovelimit was given in [46] for piecewise smooth functions with respect to a triangulationof the domain that was used to examine the limiting properties of finite elementapproximations and the corresponding error estimators.

By setting u = v, we see from (5.1) that

limε→0

∫Ω∪ΩI

∫Ω∪ΩI

D∗u ·(Θ · D∗u

)dy dx =

∫Ω

∇u ·(C · ∇u

)dx.

This establishes the relationship between the nonlocal norms and the standard localSobolev space norms in the local limit. Moreover, it has been also shown in [51] that,albeit for special nonlocal boundary conditions, the solutions of nonlocal diffusionequation converge, in such a limit, to the solution of the local diffusion equation insuitable spaces. This result relies on a key estimate showing that when Θ is positivedefinite, the smallest eigenvalue of the nonlocal diffusion operator D(Θ · D∗) remainslarger than a positive constant uniformly in ε as ε→ 0. In the context of peridynamicmodels, this is equivalent to the assumption that the materials have well-definedelastic moduli [51]. When such a property holds for more general volume-constrainedproblems considered here, we may also see that the solutions uε of (4.4) converge, atleast weakly in L2(Ω ∪ ΩI), to the unique weak solution u of (2.8) in the local limit.By passing to the limit in the respective weak forms, we recover stronger convergenceresults; in particular, for b bounded in L2(Ω), we obtain

limε→0

∫Ω∪ΩI

∫Ω∪ΩI

D∗uε ·(Θ · D∗uε

)dy dx =

∫Ω

∇u ·(C · ∇u

)dx .

One may draw analogies of the above results with other existing studies on thecharacterization of Sobolev spaces and their norms; for instance, using the character-ization established in [17], it was shown in [7] that

limm→∞

∫Ω∪ΩI

∫Ω∪ΩI

(u(x)− u(y)

)2|y − x|2

ρm(|y − x|) dy dx ∝ |u|H1(Ω)

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for a sequence of radial mollifiers ρm. In particular, in [7], it is demonstrated

that the norm induced by∫

Ω∪ΩI

∫Ω∪ΩI

(u(x)−u(y))2

|y−x|2 ρm(|y − x|) dy dx is equivalent to

|u|Hs0 (Ω) for 1/2 < s < 1. We note that our results cover wider classes of Sobolev

spaces and kernel functions.

5.3. Nonlocal advection-diffusion problems. Nonlocal advection-diffusionproblems can be defined from (3.12) by having ν account for an advective flux inaddition to a diffusive flux. To this end, we now set, for a given symmetric vectorfunction µ(x,y),7

ν(x,y)= Θ(x,y) · D∗(u)(x,y) +u(x) + u(y)

2µ(x,y)

= −(u(y)− u(x)

)Θ(x,y) ·α(x,y) +

u(x) + u(y)

2µ(x,y).

(5.2)

The term 12

(u(x)+u(y)

)µ is used to model a nonlocal convective flux (see [28]) and is

analogous to the use of u(x)v(x) as a model for the conventional local advective flux.Substitution of (5.2) into (3.13) results in the nonlocal advection-diffusion equation

ut +D(Θ · D∗u

)+D(µu) = b ∀x ∈ Ω, t > 0. (5.3)

We have that, using the symmetry of µ,

D(µu)(x)=

∫Rn

(u(x) + u(y)

)µ(x,y) · α(x,y) dy

= u(x)

∫Rn

(µ(x,y) + µ(y,x)

)· α(x,y) dy

+

∫Rn

(u(y)− u(x)

)µ(x,y) · α(x,y) dy

so that

D(µu) = uD(µ)−∫Rn

µ · D∗(u) dy

which is the nonlocal analog of the local product formula ∇ · (uv) = u∇ · v + v · ∇u.Selecting the special kernel α(x,y) = −∇yδ(y − x) results in

D(µu)(x) = ∇ ·(u(x)µ(x,x)

)i.e., in conventional advection so that the nonlocal product rule is equivalent to theconventional product rule in a distributional sense.

The one-dimensional case of a nonlinear advective conservation law is the subjectof [28] and nonlocal convection-diffusion is investigated in [35].

6. Finite-dimensional approximations. Given the variational formulation(4.1) of the nonlocal volume-constrained problem (1.2b), one may naturally considerits finite-dimensional approximations within that variational framework. Here, forboth Case 1 and Case 2, we establish a priori error and condition number esti-mates for finite-dimensional approximations of the nonlocal volume-constrained prob-lem (1.2b). These results are analogous to those established in [51] for a restricted

7We again suppress explicit reference to dependences on t.

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class of one- and two-dimensional volume-constrained problems associated with linearperidynamic models.

Let V Nc ⊂ Vc(Ω ∪ ΩI) denote a sequence of finite-dimensional subspaces andassume that, as N → ∞, V Nc is dense in Vc(Ω ∪ ΩI), i.e., for any v ∈ Vc(Ω ∪ ΩI),there exists a sequence vN ∈ V Nc such that

|||v − vN ||| → 0 as N →∞. (6.1)

Throughout the remainder of this section, we let u ∈ Vc(Ω ∪ ΩI) denote the solution ofthe variational problem (4.1) posed on Vc(Ω ∪ ΩI), or equivalently, of (4.4). We seekthe Ritz-Galerkin approximation uN ∈ V Nc of u determined by posing the variationalproblem (4.1) on V Nc ⊂ Vc(Ω ∪ ΩI); this approximation falls within the class of“internal” (see [9, p. 86]) or “conforming” approximations.

For the sake of brevity, throughout this section we again confine the discussionto the homogeneous nonlocal “Dirichlet” and “Neumann” problems (4.7) and (4.8),respectively.

6.1. Convergence and error estimates. We first state an abstract conver-gence result that gives the best approximation property of the finite-dimensionalRitz-Galerkin solution.

Theorem 6.1. If the function γ satisfies (4.9) and either (4.10) or (4.12), then,for any b ∈ V ∗c (Ω), we have

|||u− uN ||| ≤ minvN∈V N

c

|||u− vN ||| → 0 as N →∞. (6.2)

Proof. Standard variational argument shows that the Ritz-Galerkin approxima-tion uN is the best approximation to u in V Nc with respect to the energy norm. This,together with (6.1), gives the result of the theorem.

We now consider a concrete example of finite dimensional approximations, namelyfinite element approximations for the case that both Ω ∪ ΩI and Ω are polyhedraldomains. For a given triangulation of Ω ∪ ΩI that simultaneously triangulates Ω,we let V Nc consist of those functions in Vc(Ω ∪ ΩI) that are piecewise polynomials ofdegree no more than m defined with respect to the triangulation. We assume thatthe triangulation is shape-regular and quasiuniform [18] as h → 0, i.e., as N , thedimension of the approximation space V Nc , goes to ∞; here h denotes the diameterof the largest element in the triangulation. Note that generally N is of order h−n forsmall h. If the exact solution u is sufficiently smooth, we have the following result.

Theorem 6.2. Let m be a non-negative integer and 0 < s < 1.Case 1. Suppose that u ∈ Vc(Ω ∪ ΩI) ∩ Hm+t(Ω ∪ ΩI), where 0 ≤ r ≤ s ands ≤ t ≤ 1. Then, there exists a constant C such that, for sufficiently small h,

‖u− un‖Hr(Ω) ≤ Chm+t−r‖u‖Hm+t(Ω∪ΩI). (6.3a)

Case 2. Suppose that u ∈ Vc(Ω ∪ ΩI)∩Hm+t(Ω ∪ ΩI) where 0 ≤ t ≤ 1. Then, thereexists a constant C such that for sufficiently small h,

‖u− un‖L2(Ω∪ΩI) ≤ Chm+s‖u‖Hm+s(Ω∪ΩI). (6.3b)

Proof. The proof follows similar derivations as that given in [51] for a linearperidynamic model. We thus only outline the main ingredients. By Theorem 6.1

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and the norm equivalence established in earlier sections, the error estimate (6.3a) forthe case r = s and the estimate (6.3b) follow from standard approximation prop-erties in Sobolev spaces [18] of integer order. The estimate for r = 0 can then beobtained via a standard duality argument as that for the conforming (internal) finiteelement approximations of second-order elliptic equations [9, 18]. One may then useinterpolation theory for the more general case of (6.3a) for r ∈ (0, s).

In particular, if m = 1, then second-order convergence with respect to theL2(Ω ∪ ΩI) norm can be expected for linear elements by setting r = 0 and t = 1for Case 1 and t = 1 for Case 2.

It is important to note that for Case 1 with s < 1/2 and for Case 2, discontin-uous (across element boundaries) finite element spaces are conforming. This shouldbe contrasted with discontinuous Galerkin methods for second-order elliptic partialdifferential equations that are nonconforming and thus require special handling, e.g.,penalty terms, at element boundaries. For nonlocal volume-constrained problems, nosuch special handling is needed if s < 1/2.

6.2. Condition number estimates. For finite element approximations of thenonlocal operator L using basis functions φiNi=1, consider the nonlocal N ×N stiff-ness matrix K, where the entries Kij are defined by

Kij = −(L(φj), φi) for i, j = 1, . . . , N.

The condition number of the stiffness matrix is an indicator of the sensitivity of thediscrete solution with respect to the data and the performance of iterative solverssuch as the conjugate-gradient method. Our condition number estimates allow thedevelopment of preconditioners for nonlocal problems and extend the existing resultsin [3] to the case when the inverse of −L smooths the data.

The particular choice of a basis can affect the dependence of condition numberson the grid size. Consider the case in which a conventional nodal finite element basisφini=1 is used [18] so that under shape-regular and quasi-uniform mesh assumptions,there exist positive constants c1 and c2 such that, for h small,

c1hn

N∑i=1

u2i ≤

∥∥ N∑i=1

uNφi∥∥2

L2(Ω∪ΩI)≤ c2hn

n∑i=1

u2i

holds for any uh =∑Ni=1 uiφi ∈ V Nc . We then have the following condition number

estimates.Theorem 6.3. For the nonlocal stiffness matrix K, we have, for h small,

a) Case 1. if γ satisfies (4.9) and (4.10), then for some generic constant c > 0,

cond(K) ≤ ch−2s. (6.4a)

b) Case 2. if γ satisfies (4.9) and (4.12), then for some generic constant c > 0,

cond(K) ≤ c. (6.4b)

Proof. The proof again follows the same line of derivations as that given in[51] for a linear nonlocal peridynamic model. The main ingredients are the normequivalences as established in earlier sections and, for any finite element functionuh =

∑Ni=1 uiφ ∈ V Nc (Ω ∪ ΩI), an inverse inequality of the type

‖uh‖2Hs(Ω∪ΩI) ≤ ch−2s‖uh‖2L2(Ω∪ΩI)

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for conventional Sobolev space norms [18].These results are again consistent with the ones given in [51] for special boundary

conditions corresponding to special peridynamic nonlocal models.We again observe that if s ∈ (0, 1/2), the error and condition number estimates

also hold for discontinuous Galerkin approximations, because in this case all piecewisepolynomial spaces with respect to the triangulation, whether globally continuous ornot, are conforming for the internal discretization of the nonlocal problem; see [23, 51].Moreover, note that for Case 1 with s ∈ (0, 1), the condition number increases at aslower rate as h decreases than that for elliptic partial differential equations for whichthe condition number increases with h−2. In contrast, for Case 2, the conditionnumber is bounded independently of h.

Interestingly, related to the discussion given in Appendix B.3, the finite-dimensionalstiffness matrix may also be related to a graph Laplacian matrix. The inequality (6.4b)implies that the condition number is uniformly bounded with respect to the systemsize. Such results help explain why the development of preconditioners and solversremains a challenge.

Appendix A. Relations between the operator L and fractional Laplacianand fractional derivative operators.

In this section, we discuss two approaches for the modeling of anomalous su-perdiffusion that replace ∇ · ∇u with the fractional Laplacian or a fractional deriva-tive operator. In Section 3.1, we explained how, compared to problems involvingfractional Laplacian or fractional derivative operators, a nonlocal volume-constrainedproblem leads to the expedient modeling of a broader range of anomalous diffusionsover general domains in Rn. In particular, here we demonstrate that both the frac-tional Laplacian and fractional derivative operators are special cases of the integraloperator L. Moreover, the notion of volume constraints and the nonlocal vector cal-culus has enabled us to discuss well posedness over bounded domains in Rn for a moregeneral class of diffusion problems.

A.1. The fractional Laplacian as a special case of the operator L. Thefractional Laplacian is the pseudo-differential operator with Fourier symbol F satis-fying [5]

F((−∆)su

)(ξ) = |ξ|2su(ξ), 0 < s ≤ 1,

where u denotes the Fourier transform of u. Suppose that u ∈ L2(Rn) and that∫Rn

∫Rn(u(x) − u(y))2|y − x|−(n+2s)dy dx < ∞; the vector space of such functions

defines, for 0 < s < 1, the fractional Sobolev space Hs(Rn) defined by (4.15). TheFourier transform can be used to show that an equivalent characterization of thefractional Laplacian is given by [5]

(−∆)su = Cn,s

∫Rn

u(x)− u(y)

|y − x|n+2sdy, 0 < s < 1,

for some normalizing constant Cn,s. When Ω = Rn and γ(x,y) = Cn,s|y−x|−(n+2s)/2,then

−DD∗ = L = −(−∆

)s, 0 < s < 1, (A.1)

thus establishing that, when Ω = Rn, the fractional Laplacian is the special case ofthe operator L defined in (1.1) for the choice of γ(x,y) proportional to 1/|y−x|n+2s.

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A standard definition for the fractional Laplacian on a bounded domain Ω is∫Ω

u(x)− u(y)

|y − x|n+2sdy, 0 < s < 1.

However, in order for the constrained minimization problem

minu∈Hs(Ω)

1

2

∫Ω

∫Ω

(u(x)− u(y)

)2|y − x|n+2s

dy dx−∫

Ω

ub dx subject to u = 0 on ∂Ω (A.2)

to be well-posed, the condition 1/2 < s < 1 is required. On the other hand, as we havedemonstrated, replacing the above boundary constraint with the volume constraint

u = 0 on ΩI ,

where ΩI has nonzero volume, results in a well-posed minimization problem for 0 <s < 1. This in turn demonstrates that the volume-constrained problem (1.2b) is a well-posed reformulation of constrained value problems involving the fractional Laplacianoperator on bounded domains for 0 < s < 1.

As an added bonus, note that (A.1) shows that by introducing the nonlocal op-erator D and its adjoint operator D∗, we are able to provide, for 0 < s < 1, thedecomposition (−∆)s = DD∗ of the fractional Laplacian operator analogous to thedecomposition −∆ = −∇ · ∇ = ∇ · (∇·)∗ for the Laplacian operator. Having sucha decomposition available has useful consequences. First, in addition to having thatthe factional Laplacian operator is a special case of the operator L = −DD∗, we candefine more general operators −D(Θ ·D∗), where Θ(x,y) denotes a second-order ten-sor function, that have similar properties to the fractional Laplacian. Furthermore,for 0 < s < 1, we are able to define the weak formulation (4.4) for problems involvingthe fractional Laplacian; this itself is useful for the analysis of such problems and fordeveloping finite element discretizations, as we have demonstrated in this paper.

A.2. Fractional derivative operators as a special case of the operatorL. In [42], the free-space fractional dispersion problem

ut(x, t) = c∇2sMu(x, t) x ∈ Rn, t > 0

u(x, 0) = u0(x) x ∈ Rn(A.3)

is proposed for 0 < s ≤ 1, where the Fourier symbol of ∇2sM is given by

F(∇2sMu(x)

):= u(ξ)

∫‖θ‖=1

(iξ · θ

)2sM(dθ),

M(dθ) denotes an arbitrary probability measure on the unit sphere, and u denotesthe Fourier transform of u. The authors of [42] describe M as a mixing measurebecause the integral involves directional derivatives over all radial directions on theunit sphere; the authors also explain that the solutions to (A.3) yield every possiblemultivariable Levy motion of index 2s, s 6= 1. The operator ∇2s

M is a generalizationof the fractional Laplacian; the latter operator is recovered when the measure M(dθ)is the uniform measure over the unit sphere. In [41], a fractional divergence operatoris introduced that enables the consideration of a fractional flux.

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In the special case when M(dθ) corresponds to a symmetric measure ω(dθ), i.e.,ω(dθ) = ω(−dθ), then the Fourier symbol of ∇2s

ω is given by [42, Eq. 8]

F(∇2sω u(x)

):= u(ξ) cos(πs)

∫‖θ‖=1

|ξ · θ|2s ω(dθ).

The inverse Fourier transform may then be used to determine a kernel γ so that whenΩ ≡ Rn,

L = Cn,s,ω∇2sω , 0 < s ≤ 1,

for a constant Cn,s,ω. In words, the fractional derivative operator∇2sω and the nonlocal

operator L are equivalent on all of space, i.e., for Ω = Rn.As was the case for fractional Laplacian operator equations, for 1/2 < s < 1, a

limitation of fractional derivative-based approaches for modeling anomalous diffusionoccurs on general bounded domains in Rn whenever the field u is constrained, e.g., byboundary conditions. This limitation is apparent when developing numerical methodsfor fractional partial differential equations; see [45] for recent work including manycitations to the literature. In [30], an impressive attempt is made to improve numer-ical methods for fractional partial differential equations; an equivalent reformulationof the fractional dispersion equation (A.3) on bounded domains in Rn, including asystematic numerical method, is considered. A fractional derivative function space isdefined and demonstrated to be equivalent to the fractional Sobolev space Hs(Ω) fors > 0, excluding integer multiples of 1/2. However, instead of applying the volumeconstraint operator V of (1.2b), homogeneous Dirichlet boundary conditions are usedand thus the well-posedness over Hs

0(Ω) for only 1/2 < s < 1 can be considered in [30].Hence, a restricted notion of steady-state diffusion is addressed; see [30, Theorem 6.1].In contrast, the volume-constrained problem (1.2b) is a well-posed reformulation offractional derivative operator equations on bounded domains for 0 < s < 1. Fur-thermore, the development of finite element methods follows naturally from the weakformulation (4.4) of such volume-constrained problems.

Appendix B. Other applications of the nonlocal operator L.We briefly describe four applications in which the operator L defined in (1.1)

(or a generalization to vector fields) arise. Applications to nonlocal diffusion andadvection-diffusion problems are discussed in Sections 3.3 and 5.3, respectively, andto reformulations of fractional Laplacian and fractional derivative operator equationsare discussed in Sections (A.1) and (A.2), respectively.

B.1. The peridynamic continuum model for mechanics. In [48], the lin-earized peridynamic balance of linear momentum is derived as

utt(x, t) = Λu(x, t) + b(x, t), x ∈ Rn, t > 0, (B.1)

where u : Ω× (0, T ]→ Rn and

Λu(x, t) :=

∫Rn

(y − x

)⊗(y − x

)σ(|y − x|)

(u(y, t)− u(x, t)

)dy.

The operators L and Λ coincide when n = 1 and γ(x, y) = (y− x)2/(2σ(|y− x|)

). In

[51], results are provided about the well-posedness of both (B.1) and the associatedequilibrium equation Λu + b = 0. In [29], analyses are provided for model one- and

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two-dimensional volume-constrained problems on bounded domains that are evocativeof boundary-value problems with Dirichlet and Neumann boundary conditions. Thetheory developed in [29, 51] relies on the analytic properties of σ, showing how the ker-nel γ determines the regularity (or lack thereof) of the solution of volume-constrainedproblems involving the operator L.

B.2. Nonlocal wave equation. The operator L = −D(Θ · D∗) appears in thenonlocal wave equation

utt +D(Θ · D∗u

)= 0 ∀x ∈ Ω , t > 0

Vu = 0 ∀x ∈ ΩI , t > 0

u(x, 0) = u0(x) ∀x ∈ Ω

ut(x, 0) = u1(x) ∀x ∈ Ω

(B.2)

that can be viewed as a special case of the time-dependent peridynamic model. Theone-dimensional free-space problem for the nonlocal wave equation was studied in[50].

Define the energy

E(t) =1

2

∫Ω

u2t dx +

1

2

∫Ω∪ΩI

∫Ω∪ΩI

(D∗u) · (Θ · D∗u) dy dx.

Applying the nonlocal Green’s first identity (3.10) we obtain

d

dtE(t) =

∫Ω

(utt +D(Θ · D∗u)

)ut dx−

∫ΩI

N (Θ · D∗u))ut dx

so that if the nonlocal wave equation is satisfied, i.e., if the first equation in (B.2)holds, we obtain

d

dtE(t) = −

∫ΩI

N (Θ · D∗u))ut dx.

If the volume constraint V in (B.2) implies that the last integral vanishes, as it doesfor either of the volume constraints (4.7b) and (4.8b), then dE/dt = 0 so that thenonlocal wave equation conserves energy. This is an instance of the peridynamicbalance of energy; see, e.g., [48, Section 4].

B.3. Graph Laplacians. In [40], a precise notion of the limit of a sequence ofdense finite graphs8 is introduced. The limit is a symmetric measurable function W :[0, 1]× [0, 1] 7→ [0, 1] and represents the continuum analog of an adjacency matrix for asimple unweighted graph. When γ = W and Ω = (0, 1), the operator L represents thecontinuum analog of the graph Laplacian for a simple unweighted graph. This allowsconsideration of many properties of a graph associated with its Laplacian matrix tobe independent of the size of the graph or its connectivity. This includes a continuumanalog of diffusion on a graph, where Ω then corresponds to diffusion occurring onthe limit of a sequence of dense finite graphs. See, for instance, [25, Chapter 8]for an introduction to “Dirichlet” and “Neumann” boundary conditions for a graphLaplacian.

8A graph with m vertices is dense if the number of edges normalized by the number of verticesis proportional to m.

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We note that a discrete calculus has precedence in the graph theory and machinelearning literature; see, e.g., [26, Sect. 3] and [10, 37, 36] for some recent work andcitations to the literature. Our nonlocal vector calculus, then, is a generalization ofa discrete vector calculus to a graph with an uncountable number of vertices. Thefascinating results of [40] then suggest that a continuum analogue of a discrete vectorcalculus and its analysis and applications are of interest.

B.4. Symmetric jump processes. The operator L is the infinitesimal gener-ator for a symmetric jump process9 and has been the subject of much recent interest.For instance, Harnack inequalities, heat kernel estimates, and Holder continuity for Lare the subjects of [13, 14]; the Dirichlet fractional Laplacian and Cauchy martingaleproblems for L are studied in [24] and [1], respectively. The stochastic interpretationassociated with volume constraints is that the sample path for a symmetric jumpprocess exhibits discontinuous behavior and so “jumps” to a point in the exterior ofa bounded domain; this exterior region, or volume, constrains the sample path. Forexample, a statistic of interest is the time for a process to exit a domain; see, e.g.,[20, 21, 22] for further discussion. Our results complement these probabilistic analysesand provides a variational approach useful for numerical simulations.

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analysis and approximation of nonlocal … · and fractional derivative models for anomalous di...

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