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International Journal of Computer Mathematics

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A discontinuous least-squares finite-elementmethod for second-order elliptic equations

Xiu Ye & Shangyou Zhang

To cite this article: Xiu Ye & Shangyou Zhang (2019) A discontinuous least-squares finite-elementmethod for second-order elliptic equations, International Journal of Computer Mathematics, 96:3,557-567, DOI: 10.1080/00207160.2018.1445230

To link to this article: https://doi.org/10.1080/00207160.2018.1445230

Accepted author version posted online: 26Feb 2018.Published online: 07 Mar 2018.

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS2019, VOL. 96, NO. 3, 557–567https://doi.org/10.1080/00207160.2018.1445230

A discontinuous least-squares finite-element method forsecond-order elliptic equations

Xiu Ye a and Shangyou Zhang b

aDepartment of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, USA; bDepartment ofMathematical Sciences, University of Delaware, Newark, DE, USA

ABSTRACTIn this paper, a discontinuous least-squares (DLS) finite-element method isintroduced. The novelty of this work is twofold, to develop a DLS formula-tion that works for general polytopal meshes and to provide rigorous erroranalysis for it. This newmethod provides accurate approximations for boththe primal and the flux variables. We obtain optimal-order error estimatesfor both theprimal and the flux variables. Numerical examples are tested forpolynomials up to degree 4 on non-triangular meshes, i.e. on rectangularand hexagonal meshes.

ARTICLE HISTORYReceived 7 August 2017Revised 27 November 2017Accepted 19 January 2018

KEYWORDSDiscontinuous Galerkin;finite-element methods;least-squares finite-elementmethods; second-orderelliptic problems; polygonalmesh

2010MSC SUBJECTCLASSIFICATIONSPrimary: 65N15; 65N30;76D07; Secondary: 35B45;35J50

1. Introduction

We consider the model problem: seeking an unknown function u satisfying

−∇ · (a∇u) + cu = f , in �, (1)

u = 0, on ∂�, (2)

where c ≥ 0 and� is a polytopal domain in Rd with d= 2,3,∇u denotes the gradient of the function

u, and a is a d × d tensor that is uniformly symmetric positive definite in the domain. The partialdifferential equation (1) is a benchmark testing problem for new discretization techniques.

The goal of this work is to develop a discontinuous least-squares finite-element method for thesecond-order elliptic problem (1) and (2), and provide rigorous error analysis for the method. Thisnew DLS finite-element method has two unique features: 1. provide accurate approximation for boththe primal and the flux variables and lead to a positive and definite system, 2. allow the use of generalpolytopal meshes.

The least-squares finite-element method is a discretization technique for solving partial differen-tial equations. The method receives its name by minimizing the residuals in a least-squares fashion.Least-squares finite-element methods have been developed for the second-order elliptic problems

CONTACT Shangyou Zhang szhang@udel.edu Department of Mathematical Sciences, University of Delaware,Newark, DE 19716, USA

© 2018 Informa UK Limited, trading as Taylor & Francis Group

558 X. YE AND S. ZHANG

in [3,6,8,9,13,17] and references therein. The research of finite-element methods with discontinu-ous approximations has received extensive attention in the past two decades. Thousands of papershave been published on discontinuous Galerkin techniques; a few representatives include an inte-rior penalty discontinuous Galerkin method [1], local discontinuous Galerkin method [11] andhybridizable discontinuous Galerkin method [12].

Discontinuous least-squares methods [14,15] have been developed for singularly perturbed reac-tion–diffusion problems. These methods are the first-order methods with nonsymmetric system.Discontinuous least-squares methods have been developed in [4,5,10] for the Stokes equations. Opti-mal or near-optimal convergence rates of the methods are only confirmed numerically. In [2], adiscontinuous least-squares method is introduced for the div-curl system on tetrahedral mesh. Mostof the existing least-squares finite-element methods can only be applied on simplicial mesh such astriangular or quadrilateral mesh.

Recently, a weak Galerkin least-squares finite-element method has been developed in [16] for thesecond-order elliptic problems, which can work on general polytopal mesh. Weak Galerkin methodsrefer to general finite-element techniques for partial differential equations involving novel conceptsof weak function and weak derivative introduced first in [18,19] for second-order elliptic equations.Motivated by the work in [16], we extend the results of the weak Galerkin method to the discontin-uous Galerkin method in this paper. We develop and analyse a discontinuous Galerkin least-squaresmethod on polygonal/polyhedralmesh. To the best of our knowledge, this newmethod is the first dis-continuous Galerkin high-order polygonal least-squares finite-element method for the second-orderelliptic equations with rigorous error estimates.

2. Discontinuous Galerkin least-squares method

Let Th be a partition of the domain � into polygons in 2D or polyhedra in 3D. Assume that Th isshape regular in the sense as defined in [19]. Denote by Eh the set of all edges or flat faces in Th, and letE0h = Eh\∂� be the set of all interior edges or flat faces. Let �h be the subset of Eh of all edges or faces

on � = ∂�. For every element T ∈ Th, we denote by hT its diameter and mesh size h = maxT∈Th hTfor Th.

A mixed form of the problem (1) and (2) can be stated as: Find q = q(x) and u = u(x) satisfyingas follows:

q + a∇u = 0, in �, (3)

∇ · q + cu = f , in �, (4)

u = 0, on ∂�. (5)

Wenow introduce twofinite-element spaces:Vh for the pressure variableu and�h for the flux variableq defined as follows:

Vh = {v ∈ L2(�) : v|T ∈ Pk(T), ∀T ∈ Th},�h = {σ ∈ [L2(�)]d : σ |T ∈ [Pk(T)]d, ∀T ∈ Th},

(6)

where k ≥ 1 is any nonnegative integer.Let us elements T1 and T2 have e as a common edge with n1 and n2 as the unit outward normal,

respectively. We define the jump and the average for a scalar function v on e as follows:

[v]e ={

v|∂T1n1 + v|∂T2n2, e ∈ E0h ,

vn, e ∈ �h,

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 559

{v}e =⎧⎨⎩12(v|∂T1 + v|∂T2), e ∈ E0

h ,

v, e ∈ �h.

We define the jump and the average for a vector function σ on e as follows:

[σ ]e ={

σ |∂T1 · n1 + σ |∂T2 · n2, e ∈ E0h ,

σ · n, e ∈ �h,

{σ }e =⎧⎨⎩12(σ |∂T1 + σ |∂T2), e ∈ E0

h ,

σ , e ∈ �h.

First, we introduce some notations,

(v,w)Th =∑T∈Th

(v,w)T =∑T∈Th

∫T

vw dx,

〈v,w〉∂Th =∑T∈Th

〈v,w〉∂T =∑T∈Th

∫∂T

vw ds,

〈v,w〉Eh =∑e∈Eh

〈v,w〉e =∑e∈Eh

∫evw ds.

Then we define a bilinear form,

A(τ ,w; σ , v) = (∇ · τ + cw, ∇ · σ + cv)Th+ (τ + a∇w, σ + a∇v)Th + s1(w, v) + s2(τ , σ ), (7)

where

s1(w, v) = 〈h−1e [w], [v]〉Eh , s2(τ , σ ) = 〈h−1

e [τ ], [σ ]〉E0h.

Algorithm 2.1: The discontinuous Galerkin least-squares method for the problems (3)–(5) seeks uh ∈Vh and qh ∈ �h satisfying

A(qh, uh; σ , v) = (f ,∇ · σ + cv), ∀σ × v ∈ �h × Vh. (8)

3. Existence and uniqueness

Let T be an element with e as an edge. For any function ϕ ∈ H1(T), the following trace inequalityholds true,

‖ϕ‖2e ≤ C(h−1T ‖ϕ‖2T + hT‖∇ϕ‖2T

). (9)

Define

|v|21,h =∑T∈Th

‖∇v‖2T .

560 X. YE AND S. ZHANG

Introduce a norm ‖| · ‖|V for Vh and a norm ‖| · ‖|� for �h as follows:

‖|v‖|2V = |v|21,h + s1(v, v),

‖|σ‖|2� =∑T∈Th

‖∇ · σ‖2T + ‖σ‖2 + s2(σ , σ ).

The following discrete Poincaré inequality has been established in [7],

‖v‖ ≤ C‖|v‖|V , v ∈ Vh. (10)

Lemma 3.1: There exists a constant C such that for all σ × v ∈ �h × Vh one has

A(σ , v; σ , v) ≥ C(‖|σ‖|2� + ‖|v‖|2V). (11)

Proof: First, note that

s1(v, v) ≤ A(σ , v; σ , v), (12)

s2(σ , σ ) ≤ A(σ , v; σ , v). (13)

It follows from the integration by parts,

−(∇v, σ )Th = (v,∇ · σ )Th − 〈v, σ · n〉∂Th= (∇ · σ + cv, v)Th − (cv, v)Th − 〈v, σ · n〉∂Th . (14)

Using the trace and the inverse inequalities, we have that for σ × v ∈ �h × Vh,

〈v, σ · n〉∂Th = 〈[v], {σ }〉Eh + 〈{v}, [σ ]〉E0h

≤ C(s1/21 (v, v)‖σ‖ + s1/22 (σ .σ )‖v‖). (15)

Equation (14) implies

∑T∈Th

‖a1/2∇v‖2T = (a∇v + σ , ∇v)Th − (σ , ∇v)Th

= (a∇v + σ , ∇v)Th + (∇ · σ + cv, v)Th − (cv, v)Th− 〈v, σ · n〉∂Th

≤ (a∇v + σ , ∇v)Th + (∇ · σ + cv, v)Th − 〈v, σ · n〉∂Th ,

and

‖a−1/2σ‖2 = (σ + a∇v, a−1σ )Th − (∇v, σ )Th

= (σ + a∇v, a−1σ )Th + (∇ · σ + cv, v)Th − (cv, v)Th− 〈v, σ · n〉∂Th

≤ (σ + a∇v, a−1σ )Th + (∇ · σ + cv, v)Th − 〈v, σ · n〉∂Th .

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 561

Using Equations (10) and (15), we have

‖|v‖|2V + ‖σ‖2 ≤ C

⎛⎝ ∑

T∈Th‖a1/2∇v‖2T + ‖a−1/2σ‖2

⎞⎠ + s1(v, v)

≤ C((σ + a∇v, ∇v)Th + (σ + a∇v, a−1σ )Th+ 2(∇ · σ + cv, v)Th − 2〈v, σ · n〉Eh) + s1(v, v)

≤ CA1/2(σ , v; σ , v)(‖|v‖|V + ‖σ‖) + A(σ , v; σ , v)

≤ CA(σ , v; σ , v) + 14(‖|v‖|V + ‖σ‖)2 + A(σ , v; σ , v)

≤ CA(σ , v; σ , v) + 12(‖|v‖|2V + ‖σ‖2),

which implies

‖|v‖|2V + ‖σ‖2 ≤ CA(σ , v; σ , v). (16)

Similarly, we have from the estimates (10) and (16) that

‖∇ · σ‖2 ≤ C(‖∇ · σ + cv‖2 + ‖v‖2) ≤ CA(σ , v; σ , v).

Combining the two estimates above with Equation (13) gives

‖|σ‖|2� ≤ CA(σ , v; σ , v),

which, together with Equation (16), completes the proof of the lemma. �

Lemma 3.2: The discontinuous least-squares finite-element scheme (8) has one and only one solution.

Proof: It suffices to prove the uniqueness. If q(1)h × u(1)

h and q(2)h × u(2)

h are two solutions ofEquation (8), then τh = u(1)

h − u(2)h and ηh = q(1)

h − q(2)h would satisfy the following equation:

A(ηh, τh; σ , v) = 0, ∀σ × v ∈ �h × Vh.

Note that τh ∈ Vh. By letting v = τh and σ = ηh in the above equation, we have

‖|τh‖|2V + ‖|ηh‖|2� ≤ Cas(ηh, τh; ηh, τh) = 0,

which implies τh ≡ 0 and ηh ≡ 0 or equivalently, u(1)h ≡ u(2)

h and q(1)h ≡ q(2)

h . This completes theproof of the lemma. �

4. Error analysis

Let qh × uh ∈ �h × Vh be the discontinuous least-squares finite-element solution arising from (8),and Qhq × Qhu ∈ �h × Vh be the L2 projection of the exact solution q × u defined element-wise.Their differences are referred to as the error functions, and they are denoted as follows:

εh = Qhq − qh, eh = Qhu − uh. (17)

562 X. YE AND S. ZHANG

Lemma 4.1: Assume that Th is shape regular. Then for u ∈ Hk+1(�) and q ∈ [Hk+1(�)]d, we have

|s1(Qhu, v)| ≤ Chk‖u‖k+1‖|v‖|V , (18)

|s2(Qhq, σ )| ≤ Chk‖q‖k+1‖|σ‖|� , (19)

Proof: First, recall that Qh and Qh are the L2 projections defined element-wise onto Pk(T) and[Pk(T)]d, respectively, for each element T ∈ Th. We will use approximation properties of standardL2 projections in the following estimates. Using the trace inequality (9), we obtain

|s1(Qhu, v)| = 〈h−1e [Qhu], [v]〉Eh = 〈h−1

e [Qhu − u], [v]〉Eh

≤ C

⎛⎝ ∑

T∈Th(h−2‖Qhu − u‖2T + ‖∇(Qhu − u)‖2T)

⎞⎠

1/2

‖|v‖|V

≤ Chk‖u‖k+1‖|v‖|V .Similarly, we have

|s2(Qhq, σ )| = 〈h−1e [Qhq − q], [σ ]〉E0

h

≤ Chk‖q‖k+1‖|σ‖|� .�

Theorem 4.1: Let qh × uh ∈ �h × Vh be the discontinuous least-squares finite-element solution ofthe problem (3)–(5) arising from Equation (8). Assume the exact solution u ∈ Hk+1(�) and q ∈[Hk+1(�)]d, then

‖|uh − Qhu‖|V + ‖|qh − Qhq‖|� ≤ Chk(‖u‖k+1 + ‖q‖k+1). (20)

Proof: It is obvious that the solution (u, q) satisfies

A(q, u; σ , v) = (f , ∇ · σ + cv), ∀σ × v ∈ �h × Vh.

The difference of the above equation and Equation (8) gives

A(q − qh, u − uh; σ , v) = 0, ∀σ × v ∈ �h × Vh.

It follows from the equation above that

A(εh, eh; σ , v) = −A(q − Qhq, u − Qhu; σ , v) ∀σ × v ∈ �h × Vh.

Letting v = eh and σ = εh in the equation above, we have

A(εh, eh; εh, eh) = −A(q − Qhq, u − Qhu; εh, eh). (21)

It then follows from Equations (11), (18), (19) and the definitions of Qh andQh that

‖|eh‖|2V + ‖|εh‖|2� ≤ CA(εh, eh; εh, eh) = C|A(q − Qhq, u − Qhu; εh, eh)|≤ C(|(∇ · (q − Qhq) + c(u − Qhu), ∇ · εh + ceh)Th |

+ |(q − Qhq + a∇(u − Qhu), εh + a∇eh)Th |+ |s1(Qhu, eh)| + |s2(Qhq, εh)|)

≤ Chk(‖u‖k+1 + ‖q‖k+1)(‖|eh‖|V + ‖|εh‖|�),

which implies Equation (20). This completes the proof. �

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 563

Remark 4.1: In this paper, both pressure and velocity variables are approximated by kth order poly-nomials for k ≥ 1. For this case, it is difficult to prove an optimal L2 convergence rate of the pressurevariable for the least-squares finite-element method. We cannot find similar results in the literature.In [17], an optimal L2 convergence rate for the pressure is proved when one degree higher polynomi-als are used for the flux variable. However, the numerical results in the next section demonstrate ourmethod produces an approximation for pressure uh with an optimal L2 convergence rate.

5. Numerical test

We numerically solve the boundary value problem (1) and (2) on the unit square � = (0, 1) × (0, 1)with the exact solution

u = 24x(1 − x)y(1 − y), (22)

where a= 1 and c= 1. So the exact gradient solution is

q =(−24(1 − 2x)y(1 − y)

24x(1 − x)(1 − 2y)

). (23)

In the first set of tests, we use the DLS method (6) with polynomial degree k= 1,2,3,4 on theuniform grids shown in Figure 1. We note that the polynomial space of separated degree k, Qk, isused in typical finite-element methods, but here we used Pk polynomials, taking an advantage of the

Figure 1. The first four grids for numerical solutions in Tables 1–4.

Table 1. The errors, eu = Qhu − uh and eq = Qhq − qh , and the order of convergence, by the P1 DLS method (6) on square grids(Figure 1), for Equation (22).

‖eu‖0 hn ‖|eu‖| hn ‖eq‖0 hn ‖|eq‖| hn

2 0.3297 2.1184 1.9957 7.57913 0.1030 1.7 1.3052 0.7 0.7543 1.4 4.5364 0.74 0.0266 2.0 0.6698 1.0 0.3055 1.3 2.4045 0.95 0.0067 2.0 0.3352 1.0 0.1407 1.1 1.2304 1.06 0.0017 2.0 0.1673 1.0 0.0687 1.0 0.6215 1.07 0.0004 2.0 0.0835 1.0 0.0341 1.0 0.3122 1.08 0.0001 2.0 0.0417 1.0 0.0170 1.0 0.1565 1.0

Table 2. The errors, eu = Qhu − uh and eq = Qhq − qh , and the order of convergence, by the P2 DLS method (6) on square grids(Figure 1), for (22).

‖eu‖0 hn ‖|eu‖| hn ‖eq‖0 hn ‖|eq‖| hn

1 1.052684 5.2642 4.721666 8.83212 0.155430 2.8 1.6808 1.6 0.719409 2.7 2.6276 1.73 0.020683 2.9 0.4446 1.9 0.114867 2.6 0.6986 1.94 0.002623 3.0 0.1126 2.0 0.021106 2.4 0.1798 2.05 0.000329 3.0 0.0282 2.0 0.004439 2.2 0.0456 2.06 0.000041 3.0 0.0071 2.0 0.001022 2.1 0.0115 2.0

564 X. YE AND S. ZHANG

discontinuous Galerkin method. The error and the order of convergence are displayed in Tables 1–4,where Qh andQhq are the L2 projections, and uh and qh the numerical solutions. The optimal-orderof convergence is achieved in all cases. In particular, when k= 4, the exact solution is obtained, up to

Table 3. The errors, eu = Qhu − uh and eq = Qhq − qh , and the order of convergence, by the P3 DLS method (6) on square grids(Figure 1), for Equation (22).

‖eu‖0 hn ‖|eu‖| hn ‖eq‖0 hn ‖|eq‖| hn

1 1.129649 6.4021 0.976048 1.45772 0.070223 4.0 0.8147 3.0 0.068320 3.8 0.0974 3.93 0.004366 4.0 0.1022 3.0 0.006246 3.5 0.0076 3.74 0.000273 4.0 0.0128 3.0 0.000546 3.5 0.0006 3.65 0.000017 4.0 0.0016 3.0 0.000048 3.5 0.0001 3.66 0.000001 4.0 0.0002 3.0 0.000004 3.5 0.0000 3.5

Table 4. The errors and the order of convergence, by the P4 DLS method (6) on square grids (Figure 1), for (22).

‖Qhu − uh‖0 hn |Qhu − uh|1 hn ‖Qhq − qh‖0 hn

1 0.000000000 0.000000000 0.0000000002 0.000000000 0.0 0.000000000 0.0 0.000000000 0.03 0.000000000 0.0 0.000000000 0.0 0.000000001 0.0

Figure 2. The finite-element solutions, uh , (qh)1 and (qh)2 of P1 DLS (6) for Equation (22) on level 4 square grid of Figure 1.

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 565

the computer accuracy. To see conformity, i.e. discontinuity, we plot the finite-element solutions onthe level 4 square grid in Figure 2.

In the second set of tests, we use the DLS method (6) with polynomial degree k= 1,2,3,4 again onthe hexagon grids shown in Figure 3. The error and the order of convergence are listed in Tables 5–8,whereQhu andQhq are the L2 projections, and uh and qh the numerical solutions. The optimal orderof convergence is also achieved in all cases.When k= 4, the exact solution is inside the discontinuousfinite-element space, and the numerical solution is exact up to the computer accuracy. We plot thefinite-element solutions on the level 4 hexagon grids in Figure 4.

Figure 3. The first four hexagon grids for numerical solutions in Tables 5–8.

Table 5. The errors, eu = Qhu − uh and eq = Qhq − qh , and the order of convergence, by the P1 DLSmethod (6) on hexagon grids(Figure 3), for Equation (22).

‖eu‖0 hn ‖|eu‖| hn ‖eq‖0 hn ‖|eq‖| hn

1 0.30867 1.9875 2.30605 4.44362 0.17965 0.8 1.5004 0.4 1.09749 1.1 4.1366 0.13 0.06923 1.4 1.0490 0.5 0.65407 0.7 2.6785 0.64 0.02153 1.7 0.5866 0.8 0.31632 1.0 1.5249 0.85 0.00615 1.8 0.3009 1.0 0.13172 1.3 0.8107 0.96 0.00167 1.9 0.1512 1.0 0.05180 1.3 0.4174 1.07 0.00044 1.9 0.0756 1.0 0.02009 1.4 0.2117 1.08 0.00011 2.0 0.0378 1.0 0.00784 1.4 0.1066 1.0

Table 6. The errors, eu = Qhu − uh and eq = Qhq − qh , and the order of convergence, by the P2 DLSmethod (6) on hexagon grids(Figure 3), for Equation (22).

‖eu‖0 hn ‖|eu‖| hn ‖eq‖0 hn ‖|eq‖| hn

1 0.088752 1.0551 0.90923 2.75022 0.068734 0.4 0.8140 0.4 0.40785 1.2 2.4527 0.23 0.014699 2.2 0.2802 1.5 0.09190 2.1 0.8278 1.64 0.002319 2.7 0.0840 1.7 0.02314 2.0 0.2338 1.85 0.000326 2.8 0.0230 1.9 0.00594 2.0 0.0618 1.96 0.000044 2.9 0.0060 1.9 0.00152 2.0 0.0159 2.0

Table 7. The errors, eu = Qhu − uh and eq = Qhq − qh , and the order of convergence, by the P3 DLSmethod (6) on hexagon grids(Figure 3), for Equation (22).

‖eu‖0 hn ‖|eu‖| hn ‖eq‖0 hn ‖|eq‖| hn

1 0.061983 0.0 0.65491 0.0 0.13334 0.0 0.14701 0.02 0.034715 0.8 0.39334 0.7 0.06547 1.0 0.07382 1.03 0.003257 3.4 0.07242 2.4 0.01132 2.5 0.01180 2.64 0.000239 3.8 0.01054 2.8 0.00159 2.8 0.00161 2.95 0.000016 3.9 0.00141 2.9 0.00021 2.9 0.00021 2.96 0.000001 4.0 0.00018 3.0 0.00003 3.0 0.00003 3.0

566 X. YE AND S. ZHANG

Table 8. The errors and the order of convergence, by the P4 DLS method (6) on hexagon grids (Figure 3), for Equation (22).

‖Qhu − uh‖0 hn |Qhu − uh|1 hn ‖Qhq − qh‖0 hn

1 0.00000000 0.0 0.0000000 0.0 0.00000 0.02 0.00000000 0.0 0.0000000 0.0 0.00000 0.03 0.00000000 0.0 0.0000000 0.0 0.00000 0.0

Figure 4. The finite-element solutions, uh , (qh)1 and (qh)2 of P1 DLS (6) for (22) on the level 4 hexagon grid of Figure 3.

6. Conclusion

In this work, a discontinuous Galerkin least-squares finite-element method is designed and provedto be of optimal-order convergence. Previously such a method works only for Pk triangular elementsor Qk quadrilateral elements. The new theory here ensures the optimal order of convergence for Pkelements on any polygonal/polyhedralmesh. Because the gradient of a traditional finite-element solu-tion is discontinuous, the advantage of the discontinuous Galerkin method is apparent, with a bettermatch in the least-squares formulation. Numerically, we can see, for example, the P3 discontinu-ous Galerkin least-squares finite-element converges independently of mesh type. This is not possiblefor the traditional finite-element method as there is no P3 continuous finite element on hexagonalmeshes.

Disclosure statementNo potential conflict of interest was reported by the authors.

FundingThe research of the first author was supported in part by National Science Foundation Division of MathematicalSciences (Grant no. DMS-1620016). The research of the second author was supported by National Natural ScienceFoundation of China (NSFC) (Grant no. 11571023).

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 567

ORCIDXiu Ye http://orcid.org/0000-0002-1559-3836Shangyou Zhang http://orcid.org/0000-0002-1114-4179

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