a convergent adaptive finite element algorithm for

A CONVERGENT ADAPTIVE FINITE ELEMENT ALGORITHMFOR NONLOCAL DIFFUSION AND PERIDYNAMIC MODELS∗

QIANG DU† , LI TIAN‡ , AND XUYING ZHAO§

Abstract. In this paper, we propose an adaptive finite element algorithm for the numericalsolution of a class of nonlocal models which correspond to nonlocal diffusion equations and linearscalar peridynamic models with certain non-integrable kernel functions. The convergence of theadaptive finite element algorithm is rigorously derived with the help of several basic ingredients,such as the upper bound of the estimator, the estimator reduction and the orthogonality property.We also consider how the results are affected by the horizon parameter δ which characterizes therange of nonlocality. Numerical experiments are performed to verify our theoretical findings.

AMS subject classifications. 82C21, 65R20, 65M60, 46N20, 45A05

Key words. Nonlocal diffusion, Peridynamics, Adaptive finite element method, A posteriorierror estimation, Convergence, fractional Sobolev space

1. Introduction. In this work, we consider numerical approximations of somenonlocal diffusion models which arise in many fields such as image analysis [10, 25,28], nonlocal diffusion [8, 19], and continuum mechanics [37]. These models offernew alternatives to traditional PDE based models. For instance, the peridynamic(PD) theory proposed in [37] is an integral-type nonlocal continuum theory whichincorporates the nonlocal nature of material interactions. It also connects continuummechanics and molecular dynamics within a single framework [39]. Meanwhile, therehas been much development in the mathematical theory of nonlocal models, see forinstance, an extensive treatment on nonlocal diffusion problems in [4]. In [18, 26], anonlocal vector calculus was developed to provide a more general variational setting fornonlocal models. More theoretical studies of related volume-constraint problems canbe found in [19, 30, 20]. Within the context of PD based nonlocal models, there havebeen a variety of numerical methods implemented for their approximations includingfinite difference, finite element, quadrature and particle-based methods [3, 9, 15, 27,29, 35, 38, 42]. Given the ability of nonlocal PD models to simulate cracks or fractures,adaptive method is a natural ways to reduce the computational cost. Indeed, adaptiverefinement for nonlocal PD type models has been studied in [9] with meshless methods.Utilizing the nice variational structures of the volume-constrained problems associatedwith the linear nonlocal diffusion or PD operators and the strong connection to thevariational PDE problems associated with elliptic operators, it is also natural to studyfinite element and adaptive finite element approximations of nonlocal models [15,19, 21, 42]. Recent works include numerical results on mesh refinement presentedin [9, 15] and a priori error estimates and condition number estimates of nonlocal

∗ This work is supported in part by the U.S. Department of Energy grant DE-SC0005346, theU.S. NSF grant DMS-1016073, the National Nature Science Foundation of China grant 11201462,and China Postdoctoral Science Foundation grant 20110490279.†Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

[email protected]. Part of the work was completed while visiting Beijing Computational ScienceResearch Center.‡Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

[email protected]§Beijing Computational Science Research Center, Beijing 100084, China; Present address:

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, [email protected]

1

2 Q. DU, L. TIAN, X. ZHAO

stiffness matrices presented in [18, 42]. In [21], a residual-based a posteriori errorestimator for the finite element approximation of scalar nonlocal diffusion models andsystems of PD equations has been established based on the Babuska a posteriori errorestimation framework [5]. The equivalence between the estimator and the exact errorwas demonstrated both theoretically and numerically.

An interesting issue that remains to be studied is the convergence of adaptivemethods for nonlocal diffusion or nonlocal PD models using such error estimators,i.e. whether the actual error diminishes with adaptive refinement. Furthermore,motivated by both theoretical and practical considerations, it is also useful to investi-gate how fast the error reduction can be achieved by adaptive refinement for nonlocalproblems, especially in comparison to classical uniform refinement. To carry out thesestudies, it is natural to revisit how the related issues are resolved in the context ofPDEs. Systematic development of a posteriori error estimators for finite element ap-proximations began in the late 1970s [6] and have had much development since then,see the discussions and references in [2, 40]. Some of these approaches have beenextended, for instance in [11, 12, 34] to provide a posteriori error analysis for integralequations that are of a different nature from ones considered here. Meanwhile, for el-liptic PDEs, convergence and optimality of adaptive finite element methods (AFEMs)have been introduced and analyzed in [7, 13, 17, 32, 33, 36] and references therein.In this paper, we extend the framework to establish the convergence of an adaptivefinite element algorithm for a class of linear nonlocal diffusion or PD models. To ourknowledge, this result is the first of its kind in the literature on this particular topic.

The nonlocal operator associated with the class of models to be studied here hasthe following form: for u defined on a given domain Ω,

Lu(x) = −2

∫Bδ(x)

K(x,x′)(u(x′)− u(x)) dx′ (1.1)

where Bδ(x) = x′ : |x′ − x| < δ denotes the δ−neighborhood around x whichcharacterizes the nonlocal range of interactions. The parameter δ > 0 is the horizonparameter, following the convention given in [37]. K = K(x,x′) is called a kernelfunction and in the context of PD models, it is often related to the so-called micro-modulus function that characterizes the nonlocal interaction between material points.

While appearing like an integral operator, the operator L is in fact closer, in termsof its mathematical properties, to a differential operator for small δ [18]. Formally,one may even use the above operator to represent a local partial differential operatorby invoking highly singular kernels, such as the standard Laplacian operator withK(x,x′) being one half of the negative Laplacian of the Dirac delta function of thevariable x′ − x (in the distribution sense, see [18]). On the other hand, with positivekernel functions K = k(x′ − x) that are integrable in x′ − x, the correspondingnonlocal operators and their inverses become bounded operators in the space of squareintegrable functions (L2 space). Development of a posteriori error estimator in thiscase has been given in [21].

Here, we consider a new class of kernels so that the nonlocal operator L on onehand is unbounded in L2, much like elliptic partial differential operators, and on theother hand, still yields a well-defined element-wise residual type a posteriori error es-timator. The inverse of the operator L in this case can be shown to be bounded fromL2 to a smaller subspace of L2, consisting of functions having higher regularity. Thelatter mimics regularity properties of traditional elliptic differential equations [30, 42].The residual error for the nonlocal problem is defined using quantities involving con-tributions from a local patch associated with the neighborhood of interaction which

A convergent AFEM for nonlocal diffusion models 3

replace the contributions due to local flux jump across element boundaries. To de-velop the corresponding theoretical framework, we need to re-derive some essentialproperties of a posteriori error estimators. In fact, the derivation of an upper boundof the estimators becomes more challenging. Moreover, to get the convergence of theAFEM, we need to establish the estimator reduction and orthogonality properties.In parallel with the well-established FEM theory for standard local problems suchas second order elliptic PDEs, we must develop similar theory and tools devoted tononlocal models. For example, in order to prove the upper bound of the defined es-timator, we need to establish an approximation property of some quasi-interpolationoperator in fractional Sobolev spaces. In order to prove the estimator reduction, weneed to prove an inverse estimate between norms associated with the different nonlo-cal spaces. This is done using a special but natural strategy where, for any element,the nonlocal interaction neighborhood is split into the surrounding element-patch andthe rest of the neighborhood. This allows us to estimate the terms defined on thetwo parts separately. We also provide techniques for obtaining estimates with precisedependence on both the mesh size and the nonlocal interaction range. Our estimatesare valid for both triangular/tetrahedral meshes and quadrilateral/hexahedral meshesusing well-established refinement strategies. Moreover, they lead us to a rigorous con-vergence theory of the AFEM. The convergence and effectiveness of the AFEM arefurther illustrated through numerical experiments.

The rest of this paper is organized as follows. In Section 2, we present somebackground material, including the notation and the problem under consideration, aswell as the adaptive finite element method. In Section 3, we analyze the reliability ofthe residual-based a posteriori error estimator and establish approximation propertiesof some quasi-interpolation operator for fractional Sobolev spaces. In Section 4, weprove the estimator reduction, the actual total error reduction and the convergenceof the AFEM for nonlocal models under consideration. We present several numer-ical experiments to verify our theoretical findings in Section 5, together with someconclusions in Section 6.

2. Problems and Algorithms. We use standard notations from Lebesgue andSobolev space theories. For a measurable set G ⊂ Rd, we G to denote the closure of Gand |G| to denote the d-dimensional measure of G. If G ⊂ Rd is bounded, but not anopen domain, interior(G) denotes the biggest open subset in G, i.e., interior(G) = Gand interior(G) ⊂ G. We also let (·, ·)G and ‖·‖0,G denote the standard inner productand the corresponding norm in L2(G). More generally, let ‖ · ‖s,G denote the normin the standard (possibly fractional) Sobolev space Hs(G) for s ∈ R. Moreover, letCn(G) denote the space of n-times continuously differentiable functions. Let Pn(G)denote the space of all polynomials of degree no more than n and Qn(G) denotethe space of polynomials of degree no more than n in each variable on the domainG. We note in particular that if G ∈ R is a one dimensional line segment, thenPn(G) = Qn(G). For any point x = (x1, x2, · · · , xd)T ∈ Rd, |x| = (x2

1+x22+· · ·+x2

d)1/2

denotes the distance from x to the origin. Let dist(G1, G2) = infx1∈G1,x2∈G2|x1 −

x2| = infx1∈G1,x2∈G2|x1x2| denote the distance between two sets G1 and G2. For a

compactly supported function f , denote supp(f) the support of f .

We set Ω = interior(Ωs ∪ ΩI) ∈ Rd (d = 1, 2, 3) where Ωs is called the solu-tion domain which is a bounded, open domain with a piecewise flat boundary ∂Ωsand ΩI is called the constraint domain which is an open domain surrounding Ωs,with a width that is no smaller than δ. Then, Ωs ∩ ΩI = ∅, Ωs ∩ ΩI = ∂Ωs anddist(∂Ωs, ∂ΩI\∂Ωs) ≥ δ. Let ∂Ωs = Γ =

∑Ii=1 Γi, where Γi is an endpoint or ver-


tex (d = 1), a straight line segment (d = 2), or a flat face (d = 3). We assume(Γi ∩ Γj) ⊂ (∂Γi ∩ ∂Γj) for d ≥ 2.

2.1. Nonlocal volume-constrained value problem. Let δ denote the horizonparameter following the convention given in [37]. Let β = β(x,x′) : Ω× Ω→ R be ascalar two-point function which is symmetric, that is, β(x,x′) = β(x′,x), and satisfies:

1. Non-negativity: β(x,x′) ≥ 0, ∀ x,x′ ∈ Ω, with |x′ − x| ≤ δ.2. Compact support: β(x,x′) = 0, ∀ x,x′ ∈ Ω, with |x′ − x| > δ.3. Bounded growth property: there exists a positive constant Cβ ,

β(x,x′) ≤ Cβ |x′ − x|−d−2s, ∀ x,x′ ∈ Ω, with |x′ − x| ≤ δ. (2.1)

4. Non-degeneracy: there exists a positive constant cβ such that

β(x,x′) ≥ cβ |x′ − x|−d−2s, ∀ x,x′ ∈ Ω, with |x′ − x| ≤ δ/2. (2.2)

In the above assumptions, δ/2 may be replaced by θδ with a given parameter 0 < θ ≤ 1which is used mainly to ensure the non-degeneracy of β(x,x′). In this paper, weconsider the case s ∈ (0, 1/2) with the main reason documented later in Theorem 3.7.Other cases will be subjects of future studies.

The nonlocal operators in (1.1) can be recast in terms of the nonlocal diver-gence operator and its dual [18]. We recall the definitions here so as to simplify ourpresentation of the nonlocal diffusion model and the associated variational problems.

The nonlocal point divergence operator D is defined as, see [18, 26],

D(v)(x) : = −∫

Ω

(v(x,x′) + v(x′,x)) ·α(x,x′) dx′ , ∀ x ∈ Ω, (2.3)

for any two-point function v : Ω × Ω → Rd, where α(x,x′) : Ω × Ω → Rd is a givenskew-symmetric two-point mapping such that α(x′,x) +α(x,x′) = 0. For simplicity,we take α(x,x′) = (x− x′)/|x− x′| as a unit vector here so that α · α = 1. Theadjoint operator D∗ of D is given by D∗(u) : = (u(x′)−u(x))α(x,x′) for any x′, x ∈ Ωand any one-point function u : Ω→ R.

The nonlocal model we study is given by the following nonlocal diffusion volume-constrained value problem[19]: find u : Ω→ R such that

Lu(x) = D(βD∗(u(x))) = f(x), x ∈ Ωs ,

u(x) = 0, x ∈ ΩI .(2.4)

where f ∈ L2(Ωs) and L, the nonlocal diffusion operator, is, for any v : Ω→ R,

Lv(x) = D(βD∗(v(x))) = −2

∫Ω

βα ·α(v(x′)− v(x)) dx′

= −2

∫Bδ(x)

β(v(x′)− v(x)) dx′ , ∀x ∈ Ωs .(2.5)

The equation (2.4) appears formally very much like a second order elliptic differentialequation when D∗ is identified with the usual (local) gradient operator. The conven-tional boundary condition for elliptic operators is replaced by a volume constraint.

For the nonlocal model (2.4), properties of it solution depend crucially on thekernel β. We refer to [19, 23, 39, 30, 42] and the references cited therein for more


discussions on the related problems including the well-posedness studies and connec-tions between (2.4) and its local differential equation limit. Other volume constrainedvalue problems associated with the nonlocal operator (2.5) have also been studiedin [19]. For a large class of kernel functions, nonlocal diffusion operators L can belinked with fractional differential operators, so that the solution space of (2.4) canbe identified with a subspace of some fractional Sobolev space Hs(Ω). We thus firstintroduce some notation about function spaces and norms.

For a positive real number t = k + τ where τ ∈ (0, 1) and k being an integer, wedefine a semi-norm on Ht(Ω) as

|u|t,Ω :=

∑|r|=k

∫Ω

∫Ω

|∂ru(x)− ∂ru(x′)|2

|x− x′|d+2τdx′ dx

1/2

(2.6)

and the norm as ‖u‖t,Ω :=(‖u‖2k,Ω + |u|2t,Ω

)1/2

. Ht(Ω) is the set of all functions u

such that ‖u‖t,Ω < +∞. For s as specified in assumptions (2.1)-(2.2), we denote thesolution space of model problem (2.4) as

V := u ∈ Hs(Ω) and u = 0 a. e. on ΩI.

By the nonlocal Green’s identity [18], we have that for any u, v ∈ Hs(Ω),

−∫

Ωs

v(x)D(βD∗(u(x)))dx +

∫Ω

∫Ω

D∗(u(x))βD∗(v(x)) dx′ dx

=

∫ΩI

v(x)D(βD∗(u(x)))dx.

(2.7)

For any u, v ∈ V , we define the bilinear form as

B(u, v) =

∫Ω

∫Ω

D∗(u(x))βD∗(v(x)) dx′ dx

=

∫Ω

∫Bδ(x)∩Ω

D∗(u(x))βD∗(v(x)) dx′ dx.(2.8)

With the nonlocal Green’s identity (2.7) and the homogeneous volume constraintcondition, we get the weak form of (2.4): find u ∈ V , such that for any v ∈ V ,

B(u, v) = (f, v) (2.9)

where (f, v) =∫

Ωfvdx =

∫Ωsfvdx.

Given Ω′ ⊂ Ω, we define 9 · 9Ω′ for any v ∈ Hs(Ω) as

9v9Ω′ :=

(∫Ω′

∫Ω

β|D∗(v(x))|2 dx′ dx) 1

2

. (2.10)

We use 9v9 to denote 9v9Ω for simplicity. The following lemma shows the energynorm 9·9 is equivalent to the semi-norm of fractional Sobolev space under the assump-tions on the kernel functions given earlier. Moreover, we notice that the equivalenceholds without using the volume constraint condition.

Lemma 2.1. For any v ∈ Hs(Ω) with s being as in (2.1) and (2.2), it holds that

Cδ|v|s,Ω ≤ 9v9 ≤ Cβ |v|s,Ω, (2.11)


where Cδ is a generic positive constant depending on δ, Ω, and the kernel function.Proof : It is easy to see from (2.1) that 9v9 ≤ Cβ |v|s,Ω. By noticing 9v9 = 9v−c9and |v|s,Ω = |v− c|s,Ω for any constant c together with (2.2), one can rewrite Lemma7.1 in [19] as

|v|2s,Ω ≤ c−1β 9 v 9 +4|Ω|

(δ

2

)−(d+2s)

‖v − v‖20,Ω

where v =∫

Ωvdx/|Ω|. Utilizing the Poincare inequality ‖v− v‖20,Ω ≤ C(δ)9 v9 given

in [19], the above inequality leads to the conclusion. The coercivity and continuity of the bilinear form B(·, ·) can also be proved [19],

that is, there exist generic positive constants C1 and C2 such that for u, v ∈ V ,

B(u, u) ≥ C1‖u‖2s,Ω, B(u, v) ≤ C2‖u‖s,Ω‖v‖s,Ω (2.12)

which then lead to existence and uniqueness of the solution of (2.9) [14, 19].

2.2. Discretization of domain. Given any nonnegative integer k, let Tk be apartition with line segments (d = 1), triangles or quadrilaterals (d = 2), tetrahedraor hexahedra (d = 3) of the domain Ωs. For simplicity, we assume that Tk+1 is arefinement of the coarser mesh Tk and we require that each mesh is shape regular inthe sense of [14, 31].

For triangular or tetrahedral meshes, we further require that each mesh is con-forming (without hanging nodes). As for quadrilateral or hexahedral meshes, weaccept meshes with hanging nodes but require that each mesh Tk is a graded mesh inthe following sense: for any element T ∈ Tk, we define the refinement level L(T ) tobe the number of refinement steps (e.g., bisections in 1D) that are needed to generatethe element T from an element T0 of the initial mesh T0, then there exists a genericconstant M > 0 such that |L(T1)− L(T2)| ≤M for any pair of neighboring elementsT1, T2 ∈ Tk with T1 ∩ T2 6= ∅. We take M = 1 which is usually the choice takenin most practical implementations and the corresponding meshes are the so-called1-irregular meshes [1, 22, 41]. We use Ek = ∪T∈Tk,E⊂∂TE to stand for the set of allthe endpoints(d = 1)/edges(d = 2)/faces(d = 3) of each element in the mesh Tk.

The meshes considered above all share the local quasi-uniform property, i.e. thereexist generic constants ca and Ca, such that

ca ≤ |T1|/|T2| ≤ Ca, ∀ T1, T2 ∈ Tk, with T1 ∩ T2 6= ∅, (2.13)

uniformly holds for any mesh Tk. Let hT be the diameter of T (largest distancebetween points in T ), and hk = maxT∈Tk hT .

For our discussion later, we let xjJj=1 be the set of all the nodal points of the

mesh Tk related to the degrees of freedom, and φjJj=1 be the set of the corresponding

nodal basis functions. Let xjJj=1 be the set of all the interior nodal points and

φjJj=1 be the set of the corresponding interior nodal basis functions. For any xj ,define sj = supp(φj), which is referred to as the star surrounding xj . A suitablemodification of the star region will be introduced later for our nonlocal problem.

2.3. Finite Element Approximation. We consider the continuous finite ele-ment space Vk, which is a subspace of V , over simplicial (line segment, triangle ortetrahedron) meshes, or quadrilateral/hexahedral meshes Tk as follows: if Tk is asimplicial mesh,

Vk = v ∈ C0(Ω), v|ΩI ≡ 0, v|T ∈ Pn(T ), ∀ T ∈ Tk,


where Pn(T ) is the set of all polynomials of degree no larger than n over T ; if Tk is aquadrilateral/hexahedral mesh,

Vk = v ∈ C0(Ω), v|ΩI ≡ 0, v|T FT ∈ Qn(T ), ∀ T ∈ Tk,

where FT : T → T is the bilinear/trilinear mapping between the reference elementT := [−1, 1]d, (d = 2, 3) and the original element T , and Qn(T ) is the set of allpolynomials of degree no larger than n in each variable over T .

The discrete weak form of (2.4) over Tk reads: find uk ∈ Vk such that

B(uk, vk) = (f, vk), ∀ vk ∈ Vk. (2.14)

Utilizing the classical approximation theory, we can get [19]:Theorem 2.2. Let u ∈ V and uk ∈ Vk be the solutions of (2.9) and (2.14)

respectively. If u ∈ Hµ(Ω) with µ > s, then there exits a constant C such that

9u− uk9 ≤ C hµ−sk ‖u‖µ,Ω, (2.15)

and if µ = s, then

9u− uk9→ 0, as hk → 0. (2.16)

2.4. Adaptive Finite Element Loop. The typical structure of an adaptivealgorithm is made up of four modules: Solve =⇒ Estimate =⇒Mark =⇒ Refine.The module Solve solves the discretized problem on the given mesh numerically, usu-ally by iterative methods for large-scale problems. The module Estimate evaluatesthe a posteriori error estimator. In this paper, we consider the following residual-basederror estimator: for any vk ∈ Vk, we define the residual term

R(vk) = f − Lvk, (2.17)

given T ∈ Tk, we then define the local a posteriori estimator over T as

η2k(vk, T ) := h2s

T ‖f − Lvk‖20,T , (2.18)

where hT := |T |1/d. For any subset T ′k ⊂ Tk, we define η2k(vk, T ′k) :=

∑T∈T ′

kη2k(vk, T );

for T ′k = Tk and vk = uk being the numerical approximation thereon, we use ηk todenote ηk(uk, Tk) for short.

With the a posteriori error estimator, one can identify some elements on whichthe estimator is relatively large, from the current mesh using the module Mark, andthe set composed of the selected elements is called the marking set. Here, we considerthe so-called Dorfler’s marking strategy [17] and the details are given in Algorithm2.3. The module Refine is to refine the current mesh locally based on a marking set.As for the triangular/tetrahedral meshes, one can choose the widely used bisectionlocal refinements such as the newest vertex bisection method and the longest edgebisection method [40], which can guarantee that the resulting meshes are alwaysshape regular and without any hanging nodes. As for 1D meshes with line segments,higher dimensional quadrilateral and hexahedral meshes, it is easy to split an elementinto 2d smaller parts by connecting the midlines to realize local refinement. One only


needs to guarantee the 1-irregular property during the refining process [1, 22, 41]. Itis worth noting that all the refined meshes are nested.

We now present the following standard adaptive finite element algorithm:

Algorithm 2.3. Choose the Dorfler marking parameter (cf. [17]) θ ∈ (0, 1], aninitial mesh T0, and set k = 0.

1. Solve the problem (2.14) to get the discrete solution uk over the mesh Tk;2. Evaluate the a posteriori error estimator ηk(uk, T ) for each element T ∈ Tk;3. Mark a set Mk of Tk with a minimal cardinality such that

η2k(uk,Mk) ≥ θ η2

k; (2.19)

4. Refine elements in Mk by a shape regular and nested local refinement proce-dure described above to get Tk+1;

5. Set k := k + 1 and go to step (1).

In later sections, a residual-based a posteriori error estimator will be derived in(2.18). The convergence of the corresponding adaptive finite element algorithm willalso be established.

3. A Posteriori Error Analysis. For the residual-based a posteriori error es-timator ηk in (2.18), we now aim to show that it can serve to approximate the exacterror. More specifically, the reliability of the estimator is first proved, which means theestimator bounds the exact error on each level. To establish the reliability result, weneed to develop a local approximation property of some quasi-interpolation operatorfor fractional Sobolev spaces Ht(Ω). We note that this property may of independentinterest whenever approximation properties of interpolation operator are needed fornonlocal problems. As the interpolation operator needs to be properly modified toaccount for the nonlocal constrained values associated with the function space, wefirst present the construction of auxiliary element and modified star, then we recallthe Clement interpolation operator [16] in brief and present a slight modification.

3.1. Construction of auxiliary element and modified star. Different fromthe standard Clement interpolation operator, for a nodal point xj located on theboundary ∂Ωs, we need to modify the original star sj in order to use the volumeconstraint condition. The key herein is to construct an auxiliary element which shouldbe shape regular and in the constraint domain ΩI .

For the 1D case, this can be done by drawing a line segment T ′ with the samelength as the element containing xj as one of its endpoints. However, the construc-tion in higher dimensional case is more involved. We now present some possibleconstructions (the choices are not unique) of such an auxiliary element for triangu-lar/tetrahedral meshes and quadrilateral/hexahedral meshes respectively. The auxil-iary element is then added to the original star as the modified star. See Figure 3.1and Figure 3.2.

The details are descried in the following algorithms, for which we assume that Ωsis a bounded, polygonal/polyhedral domain with the boundary ∂Ωs =

∑ni=1 Γi, Γi

are flat and (Γi ∩ Γj) ⊂ (∂Γi ∩ ∂Γj), which is decomposed into the initial mesh T0.We begin with the construction for two dimensional triangular meshes.

Algorithm 3.1 (Auxiliary element - 2D triangular mesh). Construct a triangleTi with Γi as its base edge outside Ωs for each Γi and the two dimensional measureof Ti ∩ Tj for i 6= j is zero. For the current mesh Tk, construct the auxiliary trianglefor any nodal point on the boundary as follows:


iT

i

1i

1i

T'

iT

T'

2S

Initial mesh k-th level mesh

3S

3x

1S

1x

2x

Fig. 3.1. The modified star S1, S2 and S3 for triangular meshes.

i

1i

1i

C

A

Initial mesh k-th level mesh

i

1i

1i

C

A

1K

T'1x

1S

1K

2K

2K

2x

2S

B B

D

E

F

G

Fig. 3.2. The modified star S1 and S2 for quadrilateral meshes.

1. for any xj on ∂Ωs, xj must be on a boundary edge el ∈ Ek and el ⊂ Γi,construct a triangle T ′ with el as its base edge such that the angle betweeneach edge in ∂T ′\el and el is π/4. Denote O as the vertex opposite to el inT ′. See Figure 3.3;

2. if T ′ does not completely lie in Ti (see the pictures (b) and (c) in Figure3.3), then we carve out a smaller triangle inside Ti and T ′ as follows: denoteel = B1B2, Γi = A1A2 and Ti = QA1A2. T ′ must be cut by some side(s) ofTi. If T ′ is cut by the line A1Q, find its projection P on the edge el. Sinceevery base angle is π/4, P stays in the interior of the edge el. There mustbe only one point between B1 and B2 in the line segment A1P . Without lossof generality, still let B1 be the point. Find the line segment OB2 which isopposite to B1 and find the intersection point O1 between the line A1Q andthe line OB2. It is easy to see the triangle O1B1B2 does not intersect theline A1Q, and the angle ∠O1B1B2 is smaller than the angle ∠OB1B2, butnot smaller than ∠QA1A2. Update O = O1 and T ′ = O1B1B2;

3. if the updated T ′ remains to be cut by the line A2Q, then repeat step 2 tocarve out a smaller triangle O2B1B2 and update T ′ = OB1B2 with O = O2.

i

1A 2AiT

1B 2B

O

Q 1Oi

1A 2A

iT

1B 2B

OQ

1O

P P2O

1P

i

1A 2A

iT

1B 2B

O

Q

(a)

jX

le

(b) (c)

'T

Fig. 3.3. Construction of auxiliary element (triangle meshes).


The construction in 3D follows similar steps but is slightly more involved.Algorithm 3.2 (Auxiliary element - 3D tetrahedral mesh.). Construct a pyra-

mid Ti with Γi as its base face outside Ωs for each Γi and the three dimensionalmeasure of Ti ∩ Tj for i 6= j is zero. For the current mesh Tk, construct the auxiliarytetrahedron for any nodal point on the boundary as follows:

1. for any xj on ∂Ωs, xj must be on a boundary triangle el ∈ Ek and el ⊂ Γi,construct a tetrahedron T ′ with el as its base face such that all the dihedralangles between each face in ∂T ′\el and el are π/4. Denote O as the vertexopposite to el in T ′;

2. if T ′ does not lie completely in Ti (see Figure 3.4), then we carve out asmaller tetrahedron inside Ti and T ′ by a recursive process: denote the triangleel = B1B2B3, Γi = A1A2 · · ·Ai and Ti = QA1A2 · · ·Ai. T ′ must intersectsome face(s) of Ti. If T ′ is cut by the face QA1A2, find the projection pointP ∈ el of O onto el. Since every base angle is π/4, P must be in the interior ofel. Find a point D1 on the line A1A2 such that PD1⊥A1A2. The line segmentPD1 must be cut by one edge of the triangle el. Without loss of generality,let B1B2 be such an edge. Then find the edge of T ′ (denoted by OB3) whichis opposite to the edge B1B2. Find the intersection point O1 between OB3

and the face QA1A2. It is easy to see that the dihedral angle formed by facesO1B1B2 and el is smaller than the one formed by faces OB1B2 and el, but notsmaller than that formed by faces QA1A2 and Γi. The tetrahedron O1B1B2B3

does not intersect the face QA1A2. Update O = O1 and T ′ = O1B1B2B3;3. if the updated T ′ is cut by another face of Ti, then repeat the step 2 to carve

out a smaller tetrahedron until T ′ has no intersection with any face QAjAj+1,j = 1, 2, · · · , i with j + 1 mod i.

s 1A

2A3A

4A

1OQ

3A

2B

Q

1B1D

P3B

1A4A

2A

O

s

2B

1B

3Bi

le

jX

Fig. 3.4. Construction of auxiliary element (tetrahedral meshes)

For quadrilateral/hexahedral meshes in d dimensional space (d = 2 or 3), we takethe following procedure instead.

Algorithm 3.3 (Auxiliary element - quadrilateral/hexahedral mesh).1. For the initial mesh T0, construct a quadrilateral/hexahedral Km for each

initial boundary edge (d = 2) or face (d = 3) Em ∈ E0 outside Ωs such thatKm ∩ Ωs = Em and the d dimensional measure of Km ∩ Kn for m 6= n iszero. For example, the element K1 corresponds to the initial edge AB andK2 to BC, see Figure 3.2.

2. For d = 2, let rm be the length of Em. For d = 3, let the quadrilateralEm = A1A2A3A4 and let rm = maxdist(A1A2, A3A4), dist(A4A1, A2A3).We can further require that for each Km, the length of each of its two (d = 2)or four (d = 3) edges having only one endpoint in Em is not larger than rm


(otherwise, one can carve out a smaller quadrilateral/hexahedral inside Km

to satisfy the above condition). See Figure 3.2 for an illustration of the twoedges |AD| ≤ |AB| and |BE| ≤ |AB| in the element K1.

3. For the current mesh Tk, construct the auxiliary quadrilateral/hexahedral cor-responding to Tk by: for any xj on the boundary ∂Ωs, xj must be on anboundary edge/face el ∈ Ek with el ⊂ ∂Ωs, then we construct a quadrilat-eral or hexahedral T ′ with el as one edge/face such that T ′ is geometricallyproportional to Km (notice that el is obtained by bisecting/quadrisecting Emrepeatedly through the local refinement for quadrilateral/hexahedral meshes),see Figure 3.2.

Remark 3.4. The construction of the triangle/pyramid Ti with Γi as its baseedge/face for triangular/tetrahedral meshes and the construction of the quadrilat-eral/hexahedral Km for quadrilateral/hexahedral meshes are independent of δ but maydepend on ∂Ωs and the initial mesh T0. Therefore, the construction of the auxiliaryelement may depend on Ωs. However, if we further require that Ωs is convex, thenthe construction of the auxiliary element can be simplified, independently of Ωs, bydirectly drawing an auxiliary element T ′, without first constructing Ti or Km, as fol-lows: draw an isosceles right triangle T ′ with el as its long base edge for each boundaryedge el ∈ Ek for the k−th level triangular mesh Tk; draw a tetrahedron T ′ with el asits base face and the three dihedral angles between each face in ∂T ′\el and el equal toπ/4 for each boundary triangle el ∈ Ek for the k−th level tetrahedral mesh Tk; draw asquare T ′ with el as one edge/face for each boundary edge el ∈ Ek for the k−th levelquadrilateral mesh Tk; and draw a straight quadrangular prism with the length of thefour edges which are perpendicular to the base face Em equal to the maximum lengthof the four edges of the base quadrilateral Em for each boundary quadrilateral el ∈ Ekfor the k−th level hexahedral mesh Tk. We also note that the above construction ofauxiliary element and those stated in Algorithms 3.1-3.3 are independent of δ and theymaintain the shape-regularity.

We define the modified star as Sj = interior(sj ∪ T ′) for any boundary nodalpoint xj . For each interior nodal point xj , we define the modified star as the originalstar, that is, Sj = sj . See Figures 3.1 and 3.2. We use Ω0 = interior(∪T ′) to denotethe domain composed of all the elements T ′. For any T ∈ Tk, we let ΩT be a domainconsisting neighboring elements of T in Tk, i.e.

ΩT = interior(∪sj : xj ∈ T , j = 1, 2, · · · , J)= interior(∪T ′ : T ′ ∩ T 6= ∅, T ′ ∈ Tk).

(3.1)

We also define

ωT = interior(∪Sj : xj ∈ T , j = 1, 2, · · · , J). (3.2)

3.2. Interpolation estimates. For any sj , let sj be the corresponding referenceconfiguration and Fj be a C0-diffeomorphism from sj to sj such that Fj |T ′ is affine

(or bilinear/trilinear if Tk is a quadrilateral/hexahedral mesh) for any T ′ ⊂ sj . Define

Rj : L1(sj)→ P1(sj) as the L2 projection operator on sj ,

Rj v ∈ P1(sj) :

∫sj

(Rj v)wdx =

∫sj

vwdx, ∀w ∈ P1(sj). (3.3)

For any v ∈ L1(sj), denote by Rjv = Rj v F−1j . For the mesh Tk defined on Ωs and

any function v ∈ L1(Ωs) with v = 0 a.e. in ΩI , the Clement interpolation operator


Πk is defined by

Πkv =

J∑j=1

Rjv(xj)φj =

J∑j=1

Rj v(F−1j (xj))φj . (3.4)

For any function v ∈ L1(Ωs) without the zero volume constraint condition on ΩI , theClement interpolation operator Πk is defined by

Πkv =

J∑j=1

Rjv(xj)φj =

J∑j=1

Rj v(F−1j (xj))φj . (3.5)

We have the following local approximation property on fractional Sobolev spaces.Lemma 3.5. Let Πk be the Clement interpolation operator defined in (3.4), and

0 < t < 1, there exists a constant C independent of hT such that

‖v −Πkv‖0,T ≤ ChtT |v|t,ΩT , ∀ v ∈ Ht(Ωs), and ∀ T ∈ Tk, (3.6)

and for any v ∈ Ht(interior(Ωs ∪ ΩI)

)with v = 0 a.e. in Ω0,

‖v −Πkv‖0,T ≤ ChtT |v|t,ωT , ∀ T ∈ Tk . (3.7)

Proof : We only prove (3.7) since (3.6) can be similarly proved. Our proof is dividedinto three steps. First, from the fact

∑xj∈T φj ≡ 1, it is easy to see that

‖v −Πkv‖0,T = ‖∑xj∈T

(v −Rjv(xj))φj‖0,T ≤ C∑xj∈T

‖v −Rjv(xj)‖0,T .

If xj is on the boundary Ωs, then Rjv(xj) = 0. If xj is an interior nodal point, it iseasy to see that for any constant function w = cj ∈ R on sj , Rjw(xj) = cj . Therefore,

‖v −Πkv‖0,T ≤ C( ∑xj∈T∩Ωs

‖v −Rjv(xj)‖0,sj +∑

xj∈T∩∂Ωs

‖v‖0,sj)

≤ C( ∑xj∈T∩Ωs

‖v − cj‖0,Sj +∑

xj∈T∩∂Ωs

‖v‖0,Sj)

≤ C( ∑xj∈T∩Ωs

‖v − cj‖t,Sj +∑

xj∈T∩∂Ωs

‖v‖t,Sj).

(3.8)

Secondly, we make claim of the following Poincare type inequality: there exists aconstant C such that

‖|v − ¯v‖t,Sj ≤ C|v|t,Sj , ¯v =1

|Sj |

∫Sj

vdx (3.9)

for interior nodal point xj , and

‖v‖t,Sj ≤ C|v|t,Sj , (3.10)

for boundary nodal point xj . These results follow from standard argument using the

reflexivity of Ht(Sj) and its compactness in L2(Sj). We omit the details.


Thirdly, by scaling arguments and using (3.8),-(3.9)-(3.10), one can show that

‖v −Πkv‖0,T ≤ Chd/2T

∑xj∈T

‖v − Rj v(xj)φj‖0,Sj

≤ Chd/2T

∑xj∈T

|v|t,Sj ≤ ChtT

∑xj∈T

|v|t,Sj ≤ ChtT |v|t,ωT .

As there are only finite number of different reference configurations Sj , the aboveconstant C is independent of hT but may depend on the mesh shape regularity.

Remark 3.6. Notice that the constant C in (3.6) is independent of Ωs. Onthe other hand, the constant C in (3.7) may depend on Ωs in general, but it can beindependent of Ωs for convex Ωs using constructions outlined in Remark 3.4.

According to Algorithms 3.1-3.3, it is easy to see that if the mesh Tk satisfies thecondition that hk ≤ δ, then each auxiliary element T ′ ⊂ ΩI .

3.3. Reliability of a posteriori error estimator. We now show the reliabilityof the a posteriori error estimator in the next theorem.

Theorem 3.7 (Upper bound). Assume that the mesh Tk satisfies the conditionhk ≤ δ. Let u ∈ V and uk ∈ Vk be the solutions of problems (2.9) and (2.14) over Tk,respectively. There exists a constant C3 > 0 which depends on ca and Ca in (2.13),cβ in (2.2), s and δ, such that

9u− uk92 ≤ C3η2k . (3.11)

Moreover, if the mesh Tk satisfies the condition hk ≤ δ/6, then the above constant C3

is independent of the horizon parameter δ.Proof : Since 0 < s < 1/2, it is easy to see that

‖Luk‖20,Ωs =∑T∈Tk

‖Luk‖20,T <∞, (3.12)

which means that ‖R(uk)‖20,Ωs =∑T∈Tk ‖f − Luk‖

20,T is well-defined, and

B(uk, v) = (Luk, v), ∀ v ∈ V. (3.13)

Set ek = u − uk. Using orthogonality, the equation (3.13), the Cauchy-Schwarzinequality and (3.7), we have for any v ∈ V ,

B(ek, v) = B(ek, v −Πkv) =∑T∈Tk

∫T

(f − Luk)(v −Πkv) dx

≤∑T∈Tk

‖R(uk)‖0,T ‖v −Πkv‖0,T ≤ C∑T∈Tk

hsT ‖R(uk)‖0,T |v|s,ωT .

Since the mesh Tk satisfies the condition hk ≤ δ, one can see from the constructionof the modified star Sj that ωT ⊂ Ω. Let v = ek in the above formula, by using theCauchy-Schwarz inequality and the finite overlapping of ωT together with the normequivalence (2.11), we have

9ek92 ≤ C∑T

hsT |ek|s,ωT ‖R(uk)‖0,T


≤ Cηk(∑T∈Tk

|ek|2s,ωT )1/2 ≤ Cηk|ek|s,Ω ≤ Cηk 9 ek9,

which leads to the final result and the constant C depends on ca and Ca in (2.13), s,δ and the kernel function.

Remark 3.8. Since (2.11) and (3.7) are used in the above derivation, the con-stant C3 in (3.11) may depend on Ωs in general.

Remark 3.9 (Patch Comparison Technique). For the mesh-size hk of Tk, ifδ ≥ 6hk, then ωT ⊂ Bδ/2(x) holds for any x ∈ ωT (see Figure 3.5), which, together

with the lower bound (2.2), lead to |ek|2s,ωT ≤ c−1β 9 ek92

ωT . Therefore,

9ek92 ≤ C( ∑T∈Tk

|ek|2s,ωT)1/2

ηk ≤ C( ∑T∈Tk

9ek 92ωT

)1/2ηk ≤ C3 9 ek 9 ηk

where the finite overlapping of ωT is used and C3 may depend on ca, Ca in (2.13),s and cβ in (2.2) but is independent of δ. Moreover, if Ωs is convex, then C3 isindependent of Ωs. We name this above derivation a patch comparison technique.

Remark 3.10. For the case of bounded nonlocal operators with integrable kernelfunctions, one can similarly use the patch comparison technique in the last remarkto show the reliability result (3.11) with a constant C ′3 [21]. Since the interpolationoperator Πk is not needed in that case, the term |v|s,ωT is replaced by ‖v‖0,T . There-fore, this implies also that in the reliability result given in [21], the constant C ′3 isindependent of δ under the condition hk ≤ δ/2 (or say, T ⊂ Bδ/2(x) for any x ∈ T ).In addition, we note that if s ≥ 1/2, then ‖Luk‖0,Ωs is undefined so that the errorestimator (2.18) is no longer valid for this case, which is why we require s ∈ (0, 1/2)in this work. The case with s ∈ [1/2, 1) will be studies in future works.

T

A

B

2

x'

x

Fig. 3.5. Patch comparison technique. For any θ ∈ (0, 1], let δ/2 be replaced by θδ in (2.2),then ωT ⊂ Bθδ(x) for any x ∈ ωT under the condition θδ ≥ 3hk.

4. Convergence of Adaptive Finite Element Method. In this section wepresent the main results of this paper, i.e. the convergence of the adaptive finiteelement algorithm for nonlocal diffusion problems. This result is established withthe help of several ingredients: the upper bound of the estimator established in theprevious section, the estimator reduction and the orthogonality property, where thelatter two are to be discussed next.


4.1. Inverse estimates. Let us present two technical lemmas which mimic theclassical, local, inverse estimates. The first one is essentially related to the fractionalorder Sobolev space norms.

Lemma 4.1. For any two real numbers 0 < t1 < t2 < 1, v ∈ Vk and T ∈ Tk,there exists a constant C depending only on t1, t2, the degree n of shape functions ofthe finite element space and the shape regularity such that∫

T

∫ωT

|v(x′)− v(x)|2

|x′ − x|d+2t2dx′ dx ≤ Ch2(t1−t2)

T

∫T

∫ωT

|v(x′)− v(x)|2

|x′ − x|d+2t1dx′ dx. (4.1)

Proof : Let xjJ0j=1 be the set of nodes of the mesh Tk (the set of vertices of allelements in the mesh Tk). For any xj , let sj be the original star surrounding xj .Since Tk is shape regular, the number of elements in sj is bounded by a constant.Consequently, the macro-elements sj can only assume a finite number of differentconfigurations. We define the following set as

Wj := v|sj − v(xj) : v ∈ Vk, (4.2)

where v|sj means the restriction of v on sj . Notice Wj is a space and any w ∈ Wj

contains at least one zero point. For any T ∈ Tk, denote all the vertexes of T asxj1 ,xj2 , · · · ,xjm . The corresponding stars and function spaces are sj1 , sj2 , · · · , sjmand Wj1 ,Wj2 , · · · ,Wjm respectively. For any w ∈ Wji with i = 1, 2, · · · ,m and anyt ∈ (0, 1), we define ‖ · ‖∗t,sji on sji to be

‖w‖∗t,sji :=

(∫T

∫sji

|w(x′)− w(x)|2

|x′ − x|d+2tdx′ dx

)1/2

. (4.3)

It is easy to check that ‖ · ‖∗t,sji is a norm on the function space Wji . For any sji ,

let sji be the corresponding reference configuration and Fji be a C0-diffeomorphism

from sji to sji such that the restriction of Fji on T ′ is affine (bilinear/trilinear if Tkis a quadrilateral/hexahedral mesh) for any T ′ ⊂ sji . Denote by Λ = sji the set ofreference configurations. Obviously, the number of possible reference configurationsin Λ is finite. By a scaling argument and the equivalence of any two norms on thefinite dimensional function space Wji defined on the reference macro-elements sji , onecan obtain

‖w‖∗t2,sji ≤ Chd/2−t2T ‖w‖∗t2,sji ≤ Ch

d/2−t2T ‖w‖∗t1,sji ≤ Ch

t1−t2T ‖w‖∗t1,sji .

Therefore,∫T

∫ωT

|v(x′)− v(x)|2

|x′ − x|d+2t2dx′ dx ≤

m∑i=1

‖w‖∗ 2t2,sji

≤ Ch2(t1−t2)T

m∑i=1

‖w‖∗ 2t1,sji

≤ Ch2(t1−t2)T

∫T

∫ωT

|v(x′)− v(x)|2

|x′ − x|d+2t1dx′ dx,

hence the conclusion follows. We next give a result which resembles the inverse inequality in standard finite

element theory, but is adapted to norms associated with nonlocal operators. To obtainthis element-wise result on each element T , we need to deal with the complication


due to the nonlocal δ-neighborhood in the estimation. To this end, we develop a newlocal analysis strategy which splits the nonlocal δ-neighborhood into two parts: anelement patch ωT and Bδ(x)\ωT , and estimate each part respectively.

Lemma 4.2. If hk ≤ δ/4 with hk being the mesh size of Tk, then there existsa constant C4 > 0 dependent on ca and Ca in (2.13), Cβ in (2.1) and cβ in (2.2),s in (2.5), the order n of shape functions of the finite element space, and the shaperegularity of Tk while independent of δ in (2.5) and Ωs, such that

‖Lvk‖0,T ≤ C4h−sT 9 vh9ωT , ∀ vk ∈ Vk, ∀T ∈ Tk (4.4)

where ωT is defined in (3.2).Proof : We use the patch comparison technique stated in Remark 3.9 here. In fact,it not only can demonstrate the independence between the constant C4 in the abovelemma and the horizon parameter δ but also simplifies the proof.

Since hk ≤ δ/4, it holds that ωT ⊂ Bδ/2(x) for any x ∈ T using the patchcomparison technique in Remark 3.9. It follows from Young’s inequality that

h2sT ‖Lvk‖20,T ≤ h2s

T

∫T

(∫Bδ(x)

β∣∣vk(x′)− vk(x)

∣∣ dx′)2

dx

≤ 8h2sT

∫T

(∫Bδ(x)\ωT


∣∣ dx′)2

dx

+

∫T

(∫ωT


∣∣ dx′)2

dx

)= 8h2s

T (S1 + S2) .

(4.5)

Using the Cauchy-Schwarz inequality and the upper bound of the kernel (2.1), wefirst approximate S1 by

S1 ≤∫T

(∫Bδ(x)\ωT

β dx′∫Bδ(x)\ωT


∣∣2 dx′) dx

≤ C∫T

(∫Bδ(x)\ωT

dx′

|x′ − x|d+2s

∫Bδ(x)\ωT

D∗(vk(x))βD∗(vk(x)) dx′

)dx.

(4.6)

Denote the boundary of ωT as ∂ωT . For any x ∈ T and x′ ∈ ∂ωT , it is easy to seefrom (2.13) that |x′ − x| ≥ ChT . Therefore, direct calculation gives∫

Bδ(x)\ωT

1

|x′ − x|d+2sdx′ ≤

∫Rd\ωT

1

|x′ − x|d+2sdx′ ≤ Csh−2s

T , (4.7)

where Cs is a constant dependent on s, d and the shape regularity. Thus,

S1 ≤ Ch−2sT

∫T

∫Bδ(x)\ωT

D∗(vk(x))βD∗(vk(x)) dx′ dx. (4.8)

We then estimate the term S2. Applying the upper bound of the kernel (2.1) andCauchy-Schwarz inequality, we get

S2 ≤ C∫T

∣∣∣∣∫ωT

vk(x′)− vk(x)

|x′ − x|d+2sdx′∣∣∣∣2 dx


≤ C∫T

(∫ωT

1

|x′ − x|d−τdx′ ·

∫ωT

|vk(x′)− vk(x)|2

|x′ − x|d+4s+τdx′)dx,

where τ = (1 − 2s)/2. Using (2.13), one can see for any x ∈ T and any x′ ∈ ∂ωT ,|x′ − x| ≤ ChT for some constant C. Thus, some direct calculations lead to∫

ωT

1

|x′ − x|d−τdx′ ≤ ChτT .

Combining the above two inequalities yields

S2 ≤ ChτT∫T

∫ωT

|vk(x′)− vk(x)|2

|x′ − x|d+4s+τdx′ dx.

Notice 4s+τ2 ∈ (0, 1). Based on Lemma 4.1, we arrive at

S2 ≤ Ch−2sT

∫T

∫ωT

|vk(x′)− vk(x)|2

|x′ − x|d+2sdx′ dx.

For any x ∈ T , it holds ωT ⊂ Bδ/2(x) due to hk ≤ δ/4. Therefore, it follows from thelower bound of the kernel (2.2) that

S2 ≤ Ch−2sT

∫T

∫ωT

D∗(vk(x))βD∗(vk(x)) dx′ dx. (4.9)

Combining (4.5), (4.8), and (4.9), we get the conclusion. By summing over the squares of (4.4) over all T ∈ Tk and using the overlapping

property of ωT , we obtain the following corollary.Corollary 4.3. If hk ≤ δ/4, there exists a constant C5 > 0 dependent on ca

and Ca in (2.13), Cβ in (2.1), cβ in (2.2), s in (2.5), the degree n of shape functionsof the finite element space, and the shape regularity of Tk while independent of δ andΩs, such that for any vh ∈ Vh,∑

T∈Th

h2sT ‖Lvh‖20,T ≤ C5 9 vh 92 . (4.10)

4.2. Estimator Reduction. We now present the estimator reduction theorem,Theorem 4.4 (Estimator Reduction). For any k, let Tk+1 be a nested refinement

of Tk based on the marking set Mk. If hk ≤ δ/4, there exists a constant C5 > 0 as inCorollary 4.3, such that for any vk ∈ Vk and vk+1 ∈ Vk+1

η2k+1(vk+1, Tk+1) ≤ (1+σ)η2

k(vk, Tk)−λ(1+σ)η2k(vk,Mk)+C5(1+1/σ)9vk+1−vk92 ,

where λ = 1− 2−2s/d ∈ (0, 1) and σ > 0 can be any positive real number.Proof : The proof is similar to that for Corollary 3.4 in [13]. One only needs to noticethe fact that for a marked element T ∈Mk ⊂ Tk, let Tk+1,T := T ′ ∈ Tk+1 | T ′ ⊂ T.From the local refinement, it is easy to see |T ′| ≤ 1

2 |T |, i.e., hT ′ ≤ hT21/d , which yields

η2k+1(vk, T ) =

∑T ′∈Tk+1,T

η2k+1(vk, T

′) ≤ 2−2s/dη2k(vk, T ) . (4.11)


On the other hand, one can obtain

ηk+1(vk+1, T ) ≤ ηk(vk+1, T ) , ∀ T ∈ Tk+1\Mk . (4.12)

The rest of the proof is nearly the same as that in [13, 36] which is repeated brieflybelow for completeness, but with many details skipped. First, using (4.11) and (4.12)and summing over all T ∈ Tk+1, we get

η2k+1(vk, Tk+1) ≤ η2

k(vk, Tk)− λη2k(vk,Mk) .

Then, by the triangle and Holder inequalities and the above estimate, we have

η2k+1(vk+1, T ) ≤ (1 + σ)η2

k+1(vk, T ) + (1 +1

σ)h2sT ‖L(vk − vk+1)‖20,T

≤ (1 + σ)η2k(vk, Tk)− λ(1 + σ)η2

k(vk,Mk) + C5(1 + 1/σ) 9 vk+1 − vk92

where in the final step we have used the inverse inequality given in (4.10).

4.3. Error Reduction and Convergence of the AFEM. It is clear that withall the refined meshes being nested, the continuous finite element spaces Pn or Qnover the resulting meshes are also nested, which leads to the following orthogonalityproperty based on the Galerkin-orthogonality B(u− uk′ , uk′ − uk) = 0.

Lemma 4.5. Let Tk′ be some refinement of Tk (k′ > k), u be the solution of (2.9),uk′ and uk be the numerical approximation of (2.14) on Tk′ and Tk, respectively. Thefollowing orthogonality holds,

9u− uk92 = 9u− uk′ 92 + 9 uk′ − uk 92 . (4.13)

Now, we can get the main result of this paper, i.e. the reduction of total error inform of 9u− uk 92 +γη2

k with a proper parameter γ.Theorem 4.6 (Error Reduction). Assume that the initial mesh-size satisfy h0 ≤

δ/6. Let θ ∈ (0, 1] and let Tk, Vk, ukk≥0 be the sequence of meshes, finite elementspaces, and discrete solutions produced by the Algorithm 2.3, then there exist constantsγ > 0 and 0 < ρ < 1, dependent on ca and Ca in (2.13), Cβ in (2.1), cβ in (2.2), s in(2.5), the marking parameter 0 < θ ≤ 1, the order n of shape functions of the finiteelement space and the shape regularity of Tk, while independent of δ, such that

9u− uk+1 92 +γη2k+1 ≤ ρ(9u− uk 92 +γη2

k). (4.14)

Proof : The proof follows a similar proof of Theorem 4.1 in [13], with the help ofLemma 4.5, Theorem 4.4 and Theorem 3.7. For completeness, the main steps are asfollows: for any β > 0,

9u− uk+1 92 +βη2k+1 = 9u− uk 92 − 9 uk − uk+1 92 +βη2

k+1

≤ 9u− uk 92 − 9 uk − uk+1 92

+β((1 + σ)η2

k − λ(1 + σ)η2k(uk,Mk) + C5(1 + 1/σ) 9 uk − uk+1 92

)≤ 9u− uk 92 − 9 uk − uk+1 92

+β(1 + σ)η2

k − λθβ(1− χ+ χ)(1 + σ)η2k + C5β(1 + 1/σ) 9 uk − uk+1 92

= 9u− uk 92 − 9 uk − uk+1 92 +β(1 + σ)(1− λθ(1− χ)

)η2k


− β λ θ χ (1 + σ)η2k + C5β(1 + 1/σ) 9 uk − uk+1 92

≤(

1− βλθχ(1 + σ)C−13

)9 u− uk 92 +β(1 + σ)

(1− λθ(1− χ)

)η2k

+(C5β(1 + 1/σ)− 1

)9 uk − uk+192,

where C3 was defined in Theorem 3.7, C5 was defined in Theorem 4.4. Moreover, theparameters β, λ, σ, θ and χ are set to satisfy the following conditions:

• 1− βλθχ(1 + σ)C−13 < 1, which is always true for positive parameters;

• 0 < (1 + σ)(1− λθ(1−χ)

)< 1, i.e., 0 < χ < 1− σ/(λθ(1 + σ)) so that it can

be achieved if σ is small enough; and• C5β(1 + 1/σ)− 1 ≤ 0, which requires 0 < β < 1/(C5(1 + 1/σ)).

Therefore, given the parameters satisfying the above properties, we set

0 < ρ := max

1− βλθχ(1 + σ)C−13 , (1 + σ)

(1− λθ(1− χ)

)< 1,

which yields the final conclusion (4.14). Remark 4.7. Combining the mesh-size conditions in Remark 3.9 and Theorem

4.4 derived by the patch comparison technique, it is worth noting that if h0 ≤ δ/6,then hk ≤ δ/6 for any k which means that γ and ρ are independent of δ. In addition,since C3 in (3.11) is used in the above derivation, the constants γ and ρ may dependon Ωs, but for convex Ωs, the above two constants are independent of Ωs.

Remark 4.8. In order to examine the behavior of the solution and the approxima-tions in the local limit as δ → 0, assuming that the solution has sufficient smoothness[26], there is a need to introduce a scaling factor ζ = δ2s−2 in front of L. We mayrewrite (2.4) as L(ζ)u = f (ζ) where L(ζ) = ζL and f (ζ) = ζf , denote 9 · 9(ζ) and

η(ζ)k the corresponding energy norm and error estimator, respectively. It is easy to see

9 · 9(ζ) =√ζ 9 ·9 and η

(ζ)k (vk, Tk) = ζ ηk(vk, Tk). Notice that the same sequence of

meshes and approximations may be obtained either by solving L(ζ)u = f (ζ) with the

estimator η(ζ)k or solving Lu = f with ηk, then it follows from (4.14) that

ζ−19u− uk+19(ζ) 2 + γζ−2η(ζ) 2k+1 ≤ ρ

(ζ−19u− uk9(ζ) 2 + γζ−2η

(ζ) 2k

),

i.e.,

9u− uk+1 9(ζ) 2 +γζ−1η(ζ) 2k+1 ≤ ρ

(9u− uk 9(ζ) 2 +γζ−1η

(ζ) 2k

).

with the constants ρ, γ independent of δ.

5. Numerical experiments. In this section, we use numerical experiments toverify the convergence result proved for the adaptive finite element algorithm 2.3, fornonlocal diffusion equations (2.4) and demonstrate the effectiveness of the adaptivescheme for solutions which lack sufficient regularity. We focus on 1D examples here.As an illustration, we assume the following kernel function β = δ−3/2|x′ − x|−3/2,which corresponds to s = 1/4 in Lemma 2.11, with the energy space being equivalentto H1/4(Ω). The 1D nonlocal diffusion equation on the domain [0, 1] is given by−

1

δ3/2

∫ x+δ

x−δ

u(x′)− u(x)

|x′ − x|3/2dx′ = f(x), for x ∈ [0, 1],

u(x) = g(x), for x ∈ [−δ, 0] ∪ [1, 1 + δ] .

(5.1)


We use continuous piecewise linear basis function for the finite element approxima-tion for illustration, although other types of finite element spaces like the piecewiseconstant or discontinuous piecewise linear elements can also be used. In the imple-mentation, we follow the adaptive finite element algorithm 2.3, and use the bisectionmethod for the module Refine.

Example 5.1. Consider (5.1) where we set δ = 0.2, and choose an exact solutionu(x) = 60(x−0.5)6, to determine the volume-constraint condition g(x) directly outside(0, 1) and the right side term f(x) according to (5.1). Detailed calculations are omittedhere. We choose the Dorfler marking parameter θ = 0.7.

For this example, the solution is a smooth function in (0, 1) but with relativelylarger solution variations around the endpoints of domain [0, 1] where we expect toobserve more refinement. We initially divide the interval [0, 1] into 20 cells uniformly(degrees of freedom N=21), then carry out numerical approximations with both uni-form and adaptive refinement. The computational results are presented in Figure 5.1which illustrates both the solution on the initial mesh and the solution after six stepsof adaptive refinement. In Figure 5.2, a comparison of convergence rates is shown,as measured by the estimator and the exact error (in energy norm). A reference lineis provided which shows the expected convergence rate (CR). As shown in the twographs in Figure 5.2, the optimal convergence rate N−1.75 is achieved for both theerror in the energy norm and the error estimator not only for adaptive refinement butalso for classical uniform refinement.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact Solution

Numerical Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact Solution

Numerical Solution

Fig. 5.1. Adaptive refinement of Example 5.1: solutions on initial mesh (left) and adaptivelyrefined mesh after 6 levels of refinement.

102

10−2

10−1

Posteriori Error

Energy Norm Error

Reference (CR=1.75)

102

10−3

10−2

10−1

100

Posteriori Error

Energy Norm Error

Reference (CR=1.75)

Fig. 5.2. Convergence rate of uniform (left) and adaptive (right) refinements for Example 5.1:errors (vertical axis) vesus total degrees of freedom (horizontal axis).

Example 5.2. We consider (5.1) with an exact solution given by a discontinuous


function with a discontinuity at 0.5 as follows,

u(x) =

x, for x ∈ [0, 0.5),

x2, for x ∈ [0.5, 1],(5.2)

where the volume-constraint condition g(x) and the right side term f(x) also can bedetermined by (5.1). Again we take δ = 0.2 and a relatively large Dorfler markingparameter θ = 0.98.

Given the exact solution being discontinuous, we expect greater refinement nearthe point of discontinuity. The computational results can be found in Figures 5.3and 5.4, where for Figure 5.4 the axes have the same meaning as those in Figure5.2. We initially divide the computational domain [0, 1] into 10 uniform cells (N=11),then perform uniform and adaptive numerical solutions. It is clear from Figure 5.3that the grids near the discontinuity point 0.5 are heavily refined while the rest arebarely modified, this is consistent with our expectation. For the convergence rate, dueto the lack of regularity of the solution, classical uniform refinement only reaches aconvergence rate of N−0.25, which is much slower than the rate N−1.75 obtained underadaptive refinement even when we use a relatively large Dorfler marking parameterθ = 0.98. Thus the adaptive method is particularly effective for these problems.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact Solution

Numerical Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact Solution

Numerical Solution

Fig. 5.3. Adaptive refinement of Example 5.2: solutions on initial mesh (left) and adaptivelyrefined mesh after 6 levels of refinement (right).

101

102

103

100

Posteriori Error

Energy Norm Error

Reference (CR=0.25)

101.1

101.4

101.7

10−2

10−1

100

101

Posteriori Error

Energy Norm Error

Reference (CR=1.75)


Example 5.3. We now consider (5.1) with f(x) = 0.5δ−3/2(0.5−x)/√|0.5− x|3

for x ∈ (0.5 − δ, 0.5 + δ) and zero otherwise. We choose the homogeneous boundarycondition, i.e. g = 0, and take δ = 0.2 with the marking parameter θ = 0.9.

While the exact solution for Example 5.3 does not have a simple analytic form,we observe that it has a discontinuity at x = 0.5 as shown in Figure 5.5. Therefore,


0 0.2 0.4 0.6 0.8 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Numerical Solution

0 0.2 0.4 0.6 0.8 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Numerical Solution

Fig. 5.5. Adaptive refinement of Example 5.3: solutions on initial mesh (left) and adaptivelyrefined mesh after 15 levels of refinement (right).

102

100.6

100.7

100.8

100.9

Posteriori Error

Reference (CR=0.25)

101.4

101.6

101.8

100

101

Posteriori Error

Reference (CR=1.75)


we expect to see similar convergence result as that in Example 5.2. Since we can notevaluate the exact error, only the a posteriori errors are compared between uniformand adaptive refinements. We initially divide the interval [0, 1] into 20 cells uniformly(degrees of freedom N=21) and perform the refinement. Again, we can observe thatclassical uniform refinement results in a lower convergence rate N−0.25, while adaptiverefinement still gives the convergence rate N−1.75.

6. Conclusion. In this paper, an adaptive finite element method is developedfor nonlocal diffusion models with singular kernels by establishing the residual-baseda posteriori error estimator. By extending similar frameworks for elliptic PDEs tothe nonlocal setting and deriving some new basic estimates in the fractional Sobolevspaces, the convergence of the estimator is shown when certain refining algorithm isused, which gives the theoretical foundation for adaptive methods in their applicationto nonlocal models. We also reveal that the theoretical results are independent ofhorizon parameter δ if the initial mesh size is smaller than δ/6. Such an observationis of interest to the multiscale materials modeling community as δ may vary in sizedepending on the nature of the materials behavior one wishes to capture. Numericalexamples are also performed, which strongly support the theory.

Acknowledgements: The authors are very thankful to the anonymous referees whomade many valuable comments and suggestions which helped us improving the pre-sentation of our work.

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a convergent adaptive finite element algorithm for

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