COMPUTING FRACTIONAL LAPLACIANS ON COMPLEX …...operator through classical spectral theory [15] on...

SIAM J. SCI. COMPUT. c© 2017 Society for Industrial and Applied MathematicsVol. 39, No. 4, pp. A1320–A1344

COMPUTING FRACTIONAL LAPLACIANS ONCOMPLEX-GEOMETRY DOMAINS: ALGORITHMS AND

SIMULATIONS∗

FANGYING SONG† , CHUANJU XU‡ , AND GEORGE EM KARNIADAKIS§

Abstract. We consider a fractional Laplacian defined in bounded domains by the eigen-decomposition of the integer-order Laplacian, and demonstrate how to compute very accurately(using the spectral element method) the eigenspectrum and corresponding eigenfunctions in two-dimensional prototype complex-geometry domains. We then employ these eigenfunctions as trialand test bases to first solve the fractional diffusion equation, and subsequently to simulate two-phaseflow based on the Navier–Stokes equations combined with a fractional Allen–Cahn mass-preservingmodel. A key point to the effectiveness of an exponential convergence of this approach is the use of aweighted Gram–Schmidt orthonormalization of the eigenfunctions that guarantees accurate projec-tion and recovery of spectral accuracy for smooth solutions. We demonstrate that even when onlypart of the eigenspectrum is computed accurately we can still obtain exponential convergence if weemploy the complete set of the eigenvectors of the discrete Laplacian. Accuracy is also verified bycomputing the eigenfunctions on square, disk, and L-shaped domains and obtaining numerical solu-tions of the fractional diffusion equation for different fractional orders. This is accomplished withoutthe need of solving any linear systems as the eigenfunction decomposition leads naturally to a sys-tem of ODEs, and hence no spatial discretization is employed during time stepping. In the secondapplication of the method, we replace the integer-order Laplacian in the Allen–Cahn model with itsfractional counterpart and a similar procedure is followed. However, for the Navier–Stokes equationswe need to solve a linear system, which we invert using an efficient ADI scheme. We demonstratethe effectiveness of the fractional Navier–Stokes/Allen–Cahn solver for the rising bubble problem ina square domain, and compare the results with the integer-order system and also with results by adifferent treatment of the fractional diffusion model using one-dimensional fractional derivatives. Thepresent model yields sharper interface thickness compared to the integer-order model for the sameresolution while it preserves the isotropic diffusion, and hence it is a good candidate for phase-fieldmodeling of multiphase fluid flows.

Key words. eigenvalue problem, spectral element method, fractional Laplacian, fractionaldiffusion, fractional phase-field equations

AMS subject classifications. 35R11, 76D05, 76TXX

DOI. 10.1137/16M1078197

1. Introduction. In the past two decades, fractional calculus has gained consid-erable popularity, mainly due to its potential application in numerous diverse fields,including control theory, biology, electrochemical processes, viscoelastic materials,polymers, finance, etc; see, e.g., [1, 2, 4, 5, 6, 18, 22, 26, 27, 30, 31, 32] and thereferences therein. In particular, fractional diffusion equations have been used to de-scribe anomalous diffusion of particle spreading. Such anomalous behavior can be

∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section June 2,2016; accepted for publication (in revised form) June 5, 2016; published electronically July 27, 2017.

http://www.siam.org/journals/sisc/39-4/M107819.htmlFunding: This work was supported by the OSD/ARO/MURI on “Fractional PDEs for Conser-

vation Laws and Beyond: Theory, Numerics and Applications (W911NF-15-1-0562).” The work ofthe second author is partially supported by NSF of China (grant 11471274).†Division of Applied Mathematics, Brown University, Providence, RI, 02912 (fangying

[email protected]).‡School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Model-

ing and High Performance Scientific Computing, Xiamen University, 361005 Xiamen, China ([email protected]).§Division of Applied Mathematics, Brown University, Providence, RI, 02912 (george karniadakis@

brown.edu).A1320

http://www.siam.org/journals/sisc/39-4/M107819.html

mailto:[email protected]






COMPUTING FPDEs ON COMPLEX-GEOMETRY DOMAINS A1321

represented by Levy processes whose sample paths can be continuous, continuouswith occasional discontinuities, or purely discontinuous. They appear in physics andbiology and are also used in modeling finance, option pricing, and other financial in-struments. Fractional differential equations have been the focus of many theoreticaland numerical studies.

In this paper, we address two open questions in fractional PDEs.• How to compute efficiently the numerical fractional Laplacian in multidimen-

sions with focus on two-dimensional complex-geometry domains [8, 14].• How to obtain exponential convergence for smooth solutions in two dimen-

sions.Considering the entire space Rn, the fractional Laplacian (−∆)s with 0 < s < 1

can be defined in many different but equivalent ways; see, for example, [9, 19, 28, 36,39]. In this paper, we consider bounded domains, where defining and computingthe fractional Laplacian is not well studied. In contrast to the case of the (standard,integer-order) Laplacian operator, where Dirichlet and Neumann boundary conditionsare well understood and have simple interpretations at the particle or probabilisticlevel, physical or probabilistically motivated interpretations of the fractional Laplacianoperator on bounded domains are not well established [11, 44]. The different repre-sentations of the fractional Laplacian may lead to different operators when restrictedto a bounded domain, and this poses challenges for numerical methods, which natu-rally require truncation of the operator to a bounded domain. In the present work,we adopt the spectral decomposition approach to define fractional powers of such anoperator through classical spectral theory [15] on a bounded domain Ω ⊂ Rn. Let(λi, φi)

∞i=1 be the eigenpairs of the Laplacian operator −∆,

(1.1) −∆φi = λiφi,

subject to appropriate boundary conditions, which ensure that all the λi’s are non-negative and that φi is a complete orthonormal basis. Then, if u has the expansionu(x) =

∑∞i=1 ciφi(x), we formally define [19, 24]

(1.2) (−∆)su(x) =

∞∑i=1

λsi ciφi(x).

Specifically, here we focus on eigenvalue problems on complex-geometry domainsunder Dirichlet and Neumann boundary conditions. We use the spectral/spectralelement method [7, 10, 13, 23, 25, 33] for solving the integer-order eigenvalue prob-lem. The numerical results in Appendix A show that the spectral element discreteLaplacian operator (matrix) is an efficient method for approximating the Laplacianoperator. After we obtain the numerical eigenpairs of the eigenvalue problem oncomplex-geometry domains, we can compute the fractional Laplace based on theeigenvalue decomposition [24, 41] on these domains. Based on this construction, wethen obtain numerical solutions of the fractional diffusion equation and the fractionalNavier–Stokes/phase-field equations with the new fractional Laplace operator.

The rest of this paper in organized as follows. In section 2, we present the spectralelement method for solving the eigenvalue problem on prototype complex-geometrydomains. The fractional Laplace operator is defined in section 3. To demonstratethe use of the approximated fractional Laplacian for real applications, we introduce afractional Navier–Stokes/phase-field equations system and present numerical resultsfor the rising bubble problem. We provide a short summary in section 4. In Ap-pendix A, we show the numerical eigenpairs of the problem with different boundary

A1322 F. SONG, C. XU, AND G. E. KARNIADAKIS

Ωl

(a)

Ωl

(b)

Fig. 1. Spectral elements: square domain (left) and L-shaped domain (right).

conditions and demonstrate that the spectral element discrete eigenvalue decomposi-tion is an efficient method to approximate the Laplacian operator. In Appendix B,we present a comparison of the present numerical results with a previous less accurateapproximation of the fractional Laplacian.

2. Eigenvalues and eigenfunctions of Laplace operator. We consider thefollowing eigenvalue problem (EVP) for the Laplacian operator:

−∆u− λu = 0, x ∈ Ω,(2.1)

u∣∣∂Ω

= 0, or∂u

∂n

∣∣∂Ω

= 0, or periodic boundary conditions,(2.2)

where Ω ∈ R2 is a bounded domain.The spectral element method (SEM) is used for solving (2.1)–(2.2). Then, (2.1)–

(2.2) can be written in the discretized form

(2.3) ANU − λMNU = 0,

where N represents the number of the degrees of freedom (DoF) of the linear system(2.3) for the given number of elements El and the polynomial degree N in each element.AN is the corresponding matrix of the Laplacian operator under certain boundaryconditions, MN is the mass matrix, and U is the numerical solution of u. Now thecontinuous EVP is approximated by the numerical solution of the eigenpairs (λi, φi)

Ni=1

of the matrix K = M−1N AN , and λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λN .

In Appendix A, we present the numerical results of the EVP (one and two di-mensions) with different boundary conditions. The numerical results show that eventhough the numerical eigenvalues of high frequency are not correct [43], we still obtainexponential convergence for smooth solutions represented in the complete eigenfunc-tion space.

2.1. SEM on square and L-shaped domains. In this subsection, we solvethe EVP (2.1)–(2.2) with SEM on the unit square domain Ω = (−1, 1)2 and theL-shaped domain Ω = (−1, 1)2/[0, 1)2. The domain is divided into El nonoverlap-

ping subdomains Ωl, l = 1, . . . , El, and Ω =⋃Ell=1 Ωl (see Figure 1). Let N be a

nonnegative integer, and let PN (Ω) = p(x)|p(x)|x∈Ωl ∈ PN (Ωl), 1 ≤ l ≤ El, wherePN (Ωl) is the set of all polynomials of degree less than or equal to N defined in Ωl.

We consider the problem (2.1) subject to homogeneous Dirichlet boundary con-ditions. Now, let SN (Ω) = PN (Ω)∩H1

0 (Ω). Then we want to find uN ∈ SN and realnumber λ such that, ∀v ∈ SN we have∫

Ω

∇uN · ∇vdx−∫

Ω

λuN vdx = 0.(2.4)


We will employ spectral element discretization to compute the above formula(2.4). To this end, we introduce the affine mapping

(2.5) ΩlF l−−→ Ω,

where Ω stands for the reference square (−1, 1)2. Let Ωl = (al, a′l)× (bl, b

′l), then F l

is given by

(2.6) ξ = (ξ, η) = F l(x, y) = (F l1(x), F l2(y)) =

(2x− ala′l − al

− 1, 2y − blb′l − bl

− 1

).

Let (ξi, ηj), i, j = 0, . . . , N , are the Gauss–Lobatto–Legendre (GLL) points in the

reference domain Ω, then the corresponding GLL points in Ωl, denoted by ξlij ≡(ξli, η

lj), are defined by (F l)−1(ξi, ηj). Let ωi, i = 0, . . . , N , be the weights of one-

dimensional (1D) GLL quadrature in (−1, 1), then the GLL quadrature weights in Ωlare given by ωl1,iω

l2,j ,

(2.7) ωl1,i = ωihl1/2, ω

l2,j = ωjh

l2/2, 0 ≤ i, j ≤ N,

where hl1, hl2 are the length of the rectangle Ωl in the x and y directions, respectively,

i.e., hl1 = a′l − al, hl2 = b′l − bl. Let G denote the set of the global GLL points, i.e.,G = ξij |ξij ∈ Ω, i, j = 0, . . . , N ; l = 1, . . . , El. Let GD = ∂Ω ∩G, G0 = G/GD.

Now we consider the spectral element discretization problem as follows: finduN ∈ SN such that

(2.8) (∇uN ,∇v)N − λ(uN , v)N = 0, ∀v ∈ SN ,

where for all piecewise continuous functions φ and ψ,

(2.9) (φ, ψ)N =

El∑l=1

N∑i,j=0

φ(ξlij)ψ(ξlij)ωl1,iω

l2,j .

Let Ll1,i; i = 0, . . . , N be the Lagrangian polynomials associated with GLL

points ξli; i = 0, . . . , N, and Ll2,j ; j = 0, . . . , N be associated with ηlj ; j =0, . . . , N. Next we construct a nodal basis (Lagrangian basis) for the space SN ,which are the following piecewise polynomials Llij ∀i, j, such that ξlij ∈ G0, where Llijis defined by

(2.10) Llij(x, y)|Ωk =

Ll1,m(x)Ll2,n(y) if ξlij = ξkmn ∈ Ωk, k = 1, . . . , El,0 otherwise.

By choosing the test functions v to be the Lagrangian basis functions for SN andexpressing uN in terms of these bases, we deduce from problem (2.8) a linear system(2.3). The spectral element stiffness matrix is defined as

(2.11) AN =

(∇Lkij ,∇Llmn)N , ξkij , ξ

lmn ∈ G0

,

and the mass matrix is defined as

(2.12) MN =

(Lkij , Llmn)N , ξ

kij , ξ

lmn ∈ G0

;

U is the vector depending on nodal values of uN .It is clear that AN and MN are N × N symmetric positive definite and MN is

a diagonal matrix. In fact, it has been shown [7, 10] that the mass matrix of thestandard spectral/spectral element method is always diagonal.


Ω

qO

qAqB

qDq

C

-

q q

q q

C B

D A

-

6

Oq

ξ

η

Fig. 2. Disk domain (left) and Gordon–Hall mapping (right).

2.2. SEM on the unit disk domain. In this subsection, we propose an exten-sion of SEM for solving (2.1) on the unit disk domain Ω = (x, y)|x2 + y2 < 1. Inorder to apply SEM in this case, we have to first transform the disk to a square do-main (see Figure 2). Instead of introducing polar coordinates, we prefer (for accuracyreasons) the mapping of Gordon and Hall [20, 21, 23], who developed a fairly simpleprocedure for mapping a square (ξ, η) ∈ S = (−1, 1)2 into a quadrilateral with curvedboundaries.

The mapping from disk to square is given as follows:

(2.13) ξ =1

2

√2 + x2 − y2 + 2

√2x− 1

2

√2 + x2 − y2 − 2

√2x,

(2.14) η =1

2

√2− x2 + y2 + 2

√2y − 1

2

√2− x2 + y2 − 2

√2y,

and from square to disk the mapping is

x = ξ

√1− η2

2, y = η

√1− ξ2

2.

The corresponding Jacobian matrix and determinant are computed from the abovemappings:

J =

√

1− η2

2 − ξη√4−2η2

− ξη√4−2ξ2

√1− ξ2

2

, |J | =

√(1− ξ2

2

)(1− η2

2

)− ξ2η2√

(4− 2ξ2)(4− 2η2).

Now, we are able to apply the spectral method for the transformed equation (2.1) in(ξ, η) coordinates. Let us consider the problem with homogeneous Dirichlet boundaryconditions. We denote by PN (S) the set of all polynomials of degree less than orequal to N defined in S. For a function v(x, y) defined in the unit disk, we denotev(ξ, , η) = v(x(ξ, η), y(ξ, η)). Let SN = PN (S) ∩H1

0 (S). We then find uN such thatfor uN ∈ SN , and real number λ, ∀v ∈ SN we have

(2.15)

∫Ω

(∇uN · ∇v − λuNv

)dx =

∫S

((uN,ξξx + uN,ηηx)(vξξx + vηηx)

+ (uN,ξξy + uN,ηηy)(vξξy + vηηy)− λuN v)|J |dξdη = 0,


where ξx, ηx, ξy, ηy can be derived from the mappings (2.13) and (2.14):

ξx =(2− ξ2)

√2− η2

√2(2− η2 − ξ2)

, ηx =ξη√

2− η2

√2(2− η2 − ξ2)

,

ξy =ξη√

2− ξ2

√2(2− η2 − ξ2)

, ηy =(2− η2)

√2− ξ2

√2(2− η2 − ξ2)

.

Remark 2.1. We know that uN (ξ(x, y), η(x, y)) is a polynomial in (ξ, η) coordi-nates. However, uN (x, y) = uN (ξ(x, y), η(x, y)) will no longer be a polynomial incoordinates (x, y) as usual. This means that u(x, y) is approximated with a fractionalpolynomial uN (x, y) in (x, y) coordinates. This method is similar to the triangle SEMdeveloped in [13, 25]. Here, we use Legendre polynomials as the basic of the polyno-mial space SN . Finally, we obtain the discretization form with Gauss quadrature inthe (ξ, η) coordinates. The discretization formula can still be written as in (2.3).

3. Fractional Laplace operator. In this section we will show how to computethe fractional Laplace based on the eigenvalue decomposition. Then, we will presentnumerical solutions of the fractional diffusion equation and the fractional Navier–Stokes/phase-field equations with the new fractional Laplace operator.

Usually, the numerical eigenfunctions associated with a repeated eigenvalue arenot perfectly orthogonal with each other, which causes a loss of accuracy if we wantto approximate a function with this set of eigenfunctions as trial basis. To deal withthis problem, we introduce a weighted Gram–Schmidt (W-GS) method to orthogo-nalize the numerical eigenfunctions φ1, φ2, . . . φN of the Laplace operator, and wedenote the new basis as φ1, φ2, . . . φN . Then we expand the solution with the neworthogonalized basis. We will see that there is a significant improvement in accuracywith W-GS as reflected in the numerical results below.

Remark 3.1. The W-GS process is employed here to orthonormalize a set of vec-tors φi in an inner product space with a weight vector w, such that

(3.1) (φi, φj)w = δij ,

where δij is the Kronecker delta and (v1, v2)w =∑nk=1 v1,kv2,kwk with v1, v2 n-

dimensional vectors and the weight w = diag(MN ) for our numerical computation.Here MN is the diagonal mass matrix from the spectral element discretization asdemonstrated in section 2.

3.1. Fractional diffusion equation. We now consider the fractional diffusionequation ∂tu = −µ(−∆)su, x ∈ Ω,

u(x, t)|∂Ω = 0,u(x, 0) = u0(x), x ∈ Ω,

(3.2)

where Ω is a two-dimensional (2D) bounded domain, which can be any shape discussedin section 2. Set u(x, t) =

∑∞n=0

∑∞m=0 cm,n(t)φm,n, where φm,n are the orthogonal

eigenfunctions of the Laplace operator with the same boundary conditions as in (3.2)and λm,n are the corresponding eigenvalues. Then from (3.2) we have

(3.3)

∞∑n=0

∞∑m=0

∂tcm,n(t)φm,n = −µ∞∑n=0

∞∑m=0

cm,n(t)λsm,nφm,n.


By the orthogonality of the eigenfunctions we obtain

(3.4) ∂tcm,n(t) = −µcm,n(t)λsm,n.

Solving the above equations with initial conditions cm,n(0) gives

cm,n(t) = e−µλsm,ntcm,n(0),

where cm,n(0) =∫

Ωu0(x)φm,ndx. Thus we obtain the analytic solution of (3.2)

(3.5) u(x, t) =

∞∑n=0

∞∑m=0

e−µλsm,ntcm,n(0)φm,n.

Using the numerical eigenpairs (λi, φi) of the Laplace operator −∆ from the SEMsolution obtained in section 2, we can approximate the fractional Laplace operator as

(3.6) (−∆)su ≈N∑i=1

ciλsiφi, 0 < s ≤ 1,

where ci = (φi, u)N , and N is the number of the eigenpairs (i.e., DoF). Then, we getthe approximate solution (Ap) of (3.2) as follows:

(3.7) uN (x, t) =

N∑i=1

e−µλsi tci(0)φi,

where ci(0) = (u0, φi)N . In our calculation, the eigenfunctions φi are replaced bytheir orthogonalized counterparts φi via the W-GS method. Figure 3 shows the L2-errors of the numerical solution to diffusion equation (3.2) on the unit square domainΩ = (−1, 1)2 under Dirichlet boundary conditions. The numerical solution uN (x, t) isgiven by (3.7) with different polynomial degree N ; here µ = 1, El = 1, t = 0.2, andthe initial condition is given by u0(x) = 10(x − x3) sin (πx) sin (πy). The analyticalsolution is given by (3.5) with m = n truncated after 32. The approximation (3.7)is exact in time, so all of the error in this scheme is associated with the spatialdiscretization. The lower bound of the error also depends on the truncation error ofthe analytical solution.

Similarly, Figure 4 shows the L2-errors for solutions to the diffusion equation(3.2) on the unit square domain Ω = (−1, 1)2 under periodic boundary conditions.The numerical solution uN (x, t) is given by (3.7) with different polynomial degreeN ; here µ = 0.5, El = 4, t = 0.1, and the initial condition is given by u0(x) =

16(2−sin (πx))(5−sin (2πy)) . The analytical solution is given by (3.5) with m = n truncated

after 48. The approximation (3.7) is exact in time.Finally, Figure 5 shows the L2-errors of solutions to the diffusion equation (3.2) on

the unit disk domain Ω = (x, y)|x2 + y2 < 1 under Dirichlet boundary conditions.The numerical solution uN (x, t) is given by (3.7) with different polynomial degreeN ; here µ = 1, El = 1, t = 0.1, and the initial condition is given by u0(x) =

(e− ex2+y2) sin (π(x− y + 0.3)). The analytical solution is given as (3.5) with m = ntruncated after 30.

To summarize these tests, we conclude that the W-GS orthogonalization signifi-cantly improved the accuracy of the numerical solutions.


4 6 8 10 12 14 16 18 20

N

10-12

10-10

10-8

10-6

10-4

10-2

L2-e

rror

Ap,s=1.0

W-GS,s=1.0

Ap,s=0.75

W-GS,s=0.75

Fig. 3. Fractional diffusion equation (3.2) on the unit square domain with Dirichlet boundaryconditions. Plotted are L2-errors with respect to the number of the eigenmodes N = (N − 1)2 in(3.7) for different fractional orders s = 1.00, 0.75. W-GS (solid line) corresponds to W-GS orthogo-nalization of the eigenfunctions, whereas Ap (dash line) corresponds to the original eigenfunctions(without orthogonalization).

4 6 8 10 12 14 16 18 20

N

10-10

10-8

10-6

10-4

10-2

L2-e

rror

Ap,s=1.0

W-GS,s=1.0

Ap,s=0.75

W-GS,s=0.75

Fig. 4. Fractional diffusion equation (3.2) on the unit square domain with periodic boundaryconditions. Plotted are L2-errors with respect to the number of the eigenmodes N = (2N)2 in (3.7)for different fractional orders s = 1.00, 0.75.

3 5 7 9 11 13 15 17 19 21 23 25 27 29

N

10-14

10-12

10-10

10-8

10-6

10-4

10-2

L2-e

rror

Ap,s=1.0

W-GS,s=1.0

Ap,s=0.75

W-GS,s=0.75

Fig. 5. Fractional diffusion equation (3.2) on the unit disk domain with Dirichlet boundaryconditions. Plotted are L2-errors with respect to the number of the eigenmodes N = (N − 1)2 in(3.7) for different fractional order s = 1.00, 0.75.

3.2. A new formulation of the fractional phase-field equation system.Previously, we have derived a new fractional phase-field equation from the fractionalmixing energy [38]. The fractional Laplace operator was defined by combining theCaputo and Riemann–Liouville derivatives, leading to a system that admits the energy


law. The advantage of the fractional model is that it can yield sharper interfacesthan the standard (integer-order) phase-field model without increasing the resolution.In this subsection we introduce the new fractional Laplace operator (3.6) into thefractional Allen–Cahn equation.

3.2.1. Fractional phase-field equation system. We consider a low densityratio mixture of two immiscible and incompressible fluids with densities ρ1, ρ2 andviscosities µ1, µ2. The fractional phase-field system is given by

(3.8) φt + (u · ∇)φ = −γ((−∆)sφ+ f(φ)− ξ(t)

),

(3.9) ρ(φ) =ρ1 − ρ2

2φ+

ρ1 + ρ2

2, µ(φ) =

µ1 − µ2

2φ+

µ1 + µ2

2,

(3.10) ρ(ut + (u · ∇)u

)= ∇ · µD(u)−∇p+ λ

((−∆)sφ+ f(φ)

)∇φ+ g(ρ),

(3.11) ∇ · u = 0,

where φ is a phase-field function which identifies the regions occupied by the twofluids, such that

(3.12) φ(x, t) =

1, fluid 1,−1, fluid 2

with the smooth transition layer of thickness η connecting the two fluids; the interfaceof the mixture can be described by Γt = x : φ(x, t) = 0, and ξ(t) = 1

|Ω|∫

Ωf(φ)dx

is a nonlocal Lagrange multiplier to ensure that the mass is conserved. Let F (φ) =1

4η2 (φ2−1)2 be the Ginzburg–Landau double-well potential, and f(φ) = F ′(φ). In the

momentum equation (3.10), where D(u) = ∇u+∇uT , λ is the mixing energy density,and g(ρ) is an additional gravitational force to account for the density difference.

To be specific, we consider the homogeneous Dirichlet boundary conditions for u:

(3.13) u|∂Ω = 0,

and homogeneous Neumann boundary conditions for φ:

∇φ · n∣∣∣∂Ω

= 0.(3.14)

The above fractional Laplacian is defined by

(3.15) (−∆)sv =

N∑i=1

viλsiφi,

where (λi, φi) are the eigenpairs of the Laplace operator −∆ with homogeneous Neu-mann boundary condition. φi forms a complete set of orthonormal eigenfunctions,and vi is the spectrum of v in the φi-expansion, i.e.,

v =

N∑i=1

viφi, vi = (v, φi)N .


3.2.2. A proof of the energy law. We give a proof of the energy law of ournew phase-field system (3.8)–(3.11). First it is readily seen from (3.15) that, for allfunctions u, v ∈ Hs(Ω), it holds

((−∆)su, v

)=

∞∑i=1

uiviλsi =

(u, (−∆)sv

)=((−∆)s/2u, (−∆)s/2v

).

We denote the norm ‖ · ‖s by

(3.16) ‖v‖s =

( ∞∑i=1

|vi|2λsi

)1/2

.

Then,

(3.17) ‖v‖2s =((−∆)sv, v

)=((−∆)s/2v, (−∆)s/2v

)= ‖(−∆)s/2v‖20.

The energy law of the new Allen–Cahn phase-field model is then given in thefollowing theorem.

Theorem 3.1. If u, φ, and ξ are solutions of (3.8)–(3.10)–(3.11) subject to suit-able boundary condition, then

(3.18)

d

dt

(∫Ω

(1

2ρ|u|2 + λF (φ)

)dx+

λ

2‖φ‖2s

)

=

∫Ω

g · udx−∫

Ω

(µ

2|D(u)|2 + λγ

∣∣(−(−∆)sφ− f(φ) + ξ(t)∣∣2)dx,

where ‖φ‖s is defined in (3.16).

Proof. Taking the inner product of (3.8) with λ(−(−∆)sφ − f(φ) + ξ(t)), weobtain(3.19)(

φt + (u · ∇)φ, λ(−(−∆)sφ− f(φ) + ξ(t)))

= γλ∥∥− (−∆)sφ− f(φ) + ξ(t)

∥∥2

0.

For the terms on the left-hand side, we have the following basic facts:(φt, f(φ)

)=

d

dt

∫Ω

F (φ)dx,((u · ∇)φ, f(φ)

)= −(φf(φ),∇ · u) +

∫∂Ω

φf(φ)u · ndσ = 0,

(φt, ξ(t)) = ξ(t)d

dt

∫Ω

φdx = 0,((u · ∇)φ, ξ(t)

)= −ξ(t)(φ,∇ · u) + ξ(t)

∫∂Ω

φu · ndσ = 0.

Bringing all these into (3.19), we get

(3.20) λ(φt+(u ·∇)φ,−(−∆)sφ

)− d

dt

∫Ω

λF (φ)dx = γλ∥∥− (−∆)sφ−f(φ)+ξ(t)

∥∥2

0.

Next, taking the inner product of (3.10) with u, and using the boundary conditionsfor u, we get

ρ(ut + (u · ∇)u,u) = (∇ · µD(u),u)− (∇p,u) + λ((−∆)sφ∇φ+∇F (φ),u

)+ (g,u).


Taking into account the fact that

(3.21)

ρ(ut + (u · ∇)u,u) =d

dt

∫Ω

1

2ρ|u|2dx,

(∇ · µD(u),u) = −∫

Ω

µ

2|D(u)|2dx,

and

−(∇p,u) +(∇F (φ),u

)= 0,

we obtain

(3.22)d

dt

∫Ω

1

2ρ|u|2dx = −

∫Ω

µ

2|D(u)|2dx+ λ

((−∆)sφ∇φ,u

)+ (g,u).

Subtracting (3.22) from (3.20) gives

(3.23)

d

dt

∫Ω

1

2ρ|u|2dx+ λ

(φt, (−∆)sφ

)+d

dt

∫Ω

λF (φ)dx

= −∫

Ω

µ

2|D(u)|2dx+ (g,u)− γλ

∥∥∆sφ− f(φ) + ξ(t)∥∥2

0.

Finally, we derive from (3.17)

(3.24) λ(φt, (−∆)sφ

)=λ

2

d

dt‖φ‖2s.

Bringing (3.24) into (3.23) gives (3.18).

3.2.3. Time discretization scheme for the phase field in two dimensions.We reformulate the system (3.8)–(3.11) into an equivalent form, which is more conve-nient for numerical approximation. Using the phase-field equation (3.8), and the factthat ξ(t)∇φ = ∇(ξ(t)φ), we have

(3.25) ∇p− λ((−∆)sφ∇φ+ f(φ)∇φ

)= ∇

(p− λξ(t)φ

)+λ

γ

(φt + u · ∇φ

)∇φ.

Therefore, if we define the modified pressure as Π = p − λξ(t)φ, we can rewrite thesystem (3.8)–(3.11) as

(3.26) φt + (u · ∇)φ = γ(−(−∆)sφ− f(φ) + ξ(t)

),

(3.27) ρ(φ) =ρ1 − ρ2

2φ+

ρ1 + ρ2

2, µ(φ) =

µ1 − µ2

2φ+

µ1 + µ2

2,

(3.28) ρ(ut + (u · ∇)u

)= ∇ · µD(u)−∇Π− λ(φt + u · ∇φ)∇φ+ g(ρ),

(3.29) ∇ · u = 0.

Then, let L be the number of time steps to integrate up to final time T , with ∆t = T/L.We denote by superscripts the time levels and set initial conditions φ0 = φ(x, 0),


φ−1 = φ0, u0 = u(x, 0), u−1 = u0, Π−12 = p0, ψ−

12 = 0, and ρ = min(ρ1, ρ2).

(Alternatively, we can simulate the first time step with a first-order scheme.) We look

for solutions of (φn+1,un+1,Πn+ 12 , ψn+ 1

2 ) for n = 0, . . . , L − 1. We introduce thefollowing notation for convenience:

φn+ 12 =

1

2(φn+1 + φn), φ∗,n+ 1

2 =1

2(3φn − φn−1), φn+1 =

1, φn+1 > 1,φn+1, |φn+1| ≤ 1,−1, φn+1 < −1,

ρn+ 12 =

ρn+1 + ρn

2, ρ∗,n+ 1

2 =1

2(3ρn − ρn−1), µn+ 1

2 =µn+1 + µn

2,

un+ 12 =

1

2(un+1 + un), u∗,n+ 1

2 =1

2(3un − un−1),

p?,n+ 12 = Πn− 1

2 + ψn−12 .

• Phase field:

(3.30)

φn+1 − φn

∆t+ γ((−∆)sφn+ 1

2 + f(φ∗,n+ 12 )− ξ(t∗,n+ 1

2 ))

+ (u∗,n+ 12 · ∇)φ∗,n+ 1

2 +S∆t

η2(φn+1 − φn) = 0,

with boundary conditions defined in (3.14). Here, S∆tη2 (φn+1−φn) is an extra stability

term, which was introduced in [16] for the integer-order case. If the parameter S

satisfies S ≥ l2

4η2 , then the above scheme is unconditionally stable, where l is the

upper bound of |f ′(φ)| with F (φ) modified as [35, (4.9)].• Velocity:

(3.31) ρn+1 =ρ1 − ρ2

2φn+1 +

ρ1 + ρ2

2, µn+1 =

µ1 − µ2

2φn+1 +

µ1 + µ2

2,

(3.32)

ρn+ 12 (un+1 − un)

∆t−∇ · µn+ 1

2D(un+ 12 ) + ρ∗,n+ 1

2 (u∗,n+ 12 · ∇)u∗,n+ 1

2

+ ∇p?,n+ 12 +

λ

γ

(φn+1 − φn

∆t+ u∗,n+ 1

2 · ∇φ∗,n+ 12

)∇φn+ 1

2 = gn+ 12 ,

un+1|∂Ω = 0.

• Pressure correction:

(3.33)

−∇2ψn+ 12 = − ρ

∆t∇ · un+1,

∇ψn+ 12 · n|∂Ω = 0,

Πn+ 12 = Πn− 1

2 + ψn+ 12 − χµn+ 1

2∇ · un+ 12 ,

where χ ∈ [0, 1] is a user-defined coefficient; the choice χ = 0 yields the standard formof the algorithm, whereas χ = 1 yields the rotational form [42].

In order to solve the incompressible Navier–Stokes equations, we will employ thespectral direction splitting method [12] based on pressure stabilization. The velocityis represented as Legendre expansions of order N whereas the pressure is representedas a Legendre expansion of order N−2.


10-3

10-2

∆ t

10-5

10-4

10-3

10-2

10-1

Density e

rror

2-2s=10e-1

2-2s=10e-2

2-2s=10e-3

2-2s=10e-4

2-2s=10e-5

2-2s=10e-6

2-2s=10e-7

2-2s=0.0

(a)

10-3

10-2

∆ t

10-5

10-4

10-3

10-2

Velo

city e

rror

2-2s=10e-1

2-2s=10e-2

2-2s=10e-3

2-2s=10e-4

2-2s=10e-5

2-2s=10e-6

2-2s=10e-7

2-2s=0.0

(b)

10-3

10-2

∆ t

10-3

10-2

10-1

Pre

ssure

err

or

2-2s=10e-1

2-2s=10e-2

2-2s=10e-3

2-2s=10e-4

2-2s=10e-5

2-2s=10e-6

2-2s=10e-7

2-2s=0.0

(c)

Fig. 6. Fractional Navier–Stokes/Allen–Cahn equations on the square domain. Density (phasefield) (a), velocity (b), and pressure (c) errors in L2-norm as a function of ∆t in log-log scale withdifferent fractional order 2s of the phase-field equations. The different curves correspond to differentvalues of (2− 2s). The space polynomial degree is N = 64.

3.2.4. Simulations. In order to investigate if the fractional phase-field equationis converging to the integer-order model as the fractional order 2s is approaching 2,we consider the system in Ω = [−1, 1]2 with µ = 1. We also set the forcing term of thephase-field equation equal to the one derived from the exact solutions correspondingto s = 1 such that the exact solutions of phase field φ, density ρ, velocity u, andpressure p are given by

φ(x, t) = sin(t) cos(πx) cos(πy),ρ(x, t) = φ+ 2,u(x, t) = πsin(t)

(sin(2πy) sin2(πx),− sin(2πx) sin2(πy)

),

p(x, t) = sin(t) cos(πx) sin(πy),

and the densities of the two fluids are ρ1 = 3, ρ2 = 1, while the viscosities are µ1 =µ2 = 1. Here we employ the fixed parameter χ = 0.5. We used 652 GLL points so thespatial discretization errors are negligible compared with the time discretization error.This corresponds toN = 652 eigenfunctions computed as shown in Appendix A for thefractional phase-field equation under Neumann boundary conditions. In Figure 6 wesee that the solutions of the fractional phase-field model are approaching the integer-order phase-field solutions with linear convergence with respect to the parameter(1− s) at time T = 1.0.

Next, we add the gravity term in the Navier–Stokes equation to conduct numericalsimulations of a rising bubble for a low density ratio case. We also want to comparethe new results with our previous fractional Laplacian model. In this case all fourboundaries are walls with a no-slip flow velocity as boundary condition. The densitiesare ρ1 = 0.5, ρ2 = 3ρ1, and the viscosities are µ1 = µ2 = 0.1, g = 10, λ = 0.1, γ = 1,∆t = 10−3, T = 1.8, η = 0.04, El = 1, N = 256, while the external body force g is


(a) s=1.00.

(b) s = 0.90.

(c) s = 0.90, (PM).

(d) s = 0.85.

(e) s = 0.85, (PM).

(f) s = 0.75.

(g) s = 0.75, (PM).

Fig. 7. Phase-field evolution from left to right at t = 0.1,0.7,1.1,1.12,1.16,1.28,1, 80, frombottom to top for fractional orders s = 0.75,0.85,0.90,1.00. (PM) corresponds to the fractionalLaplacian defined in Appendix B.

the buoyancy force. This corresponds to 2572 eigenfunctions computed as shown inAppendix A for Neumann boundary conditions.

Initially the bubble is described by φ(x, 0) = tanh(

√x2+(y+0.4)2−0.3

η ). It startsas a circular bubble near the bottom of the domain and then it rises as shownin Figure 7. Specifically, Figure 7 shows the bubble rising for fractional orderss = 0.75, 0.85, 0.90, 1.00. The interface between the bubble and the background fluid(solvent) is sharper for smaller fractional order, but we also observe that the risingspeed has a weak dependence on the fractional order. However, the previous model(PM) [38] had a much stronger dependence on the fractional order s resulting inchanges of the rising speed but also of the shape of the bubble (See Figure 7(g)).Figures 7(a), (b), (d), and (f) show that our new model does not affect the shape ofthe bubble and only yields sharper interfaces. In addition, Figure 8 shows the profiles


-1 -0.5 0 0.5 1

x

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

φ

s=0.75

s=0.85

s=0.90

s=1.00

(a) y = 1.

-1 -0.5 0 0.5 1

y

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

φ

s=0.75

s=0.85

s=0.90

s=1.00

0.7 0.8 0.9 10.7

0.8

0.9

1

(b) x = 0.

Fig. 8. Phase-field profiles along the lines (left) y = 1 (upper wall) and (right) x = 0 (center-line) at t = 1.8. The inset shows a zoom in around the bubble.

-1 -0.5 0 0.5 1

x

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

φ

s=0.75

s=0.75 (PM)

s=1.00

-1 -0.98 -0.96-1

-0.95

(a) y = 1.

-1 -0.5 0 0.5 1

y

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

φ

s=0.75

s=0.75 (PM)

s=1.00

-1 -0.98 -0.96-1

-0.95

(b) x = 0.

Fig. 9. Phase-field profiles along the lines (left) y = 0.1947 (bubble centerline) and (right)x = 0 (centerline), s = 0.75 at t = 0.7. The inset of the left figure shows the details around sidewall (x = −1) and the insert of the right figure shows the details around the bottom (y = −1).

of the phase-field at time T = 1.8. We can see that the phase-field profiles along thehorizontal direction at y = 1 (upper wall) are similar for different fractional orders.Moreover, our new model can overcome the erroneous boundary layer, which is anartifact of the previous nonlocal fractional Laplace operator (see Figures 9 and 10).This is avoided with the new model because from the definition of the new fractionalLaplace we always satisfy the homogeneous Neumann boundary n · ∇φ|∂Ω = 0 for allfractional orders s. Figure 10 shows the profiles comparison of the phase field withdifferent models at T = 1.8 and s = 0.75.

4. Summary. Fractional calculus has provided us with a new powerful approachfor modeling nonlocal phenomena in space time but significant new progress is requiredto properly formulate the basic elements of multidimensional vector fractional calcu-lus. A fundamental such issue is the definition and numerical approximation of thefractional Laplacian in more than one dimension, with multiple definitions currentlyin the literature, which unfortunately do not seem to be equivalent on bounded do-mains. In the current work, we adopted a natural definition through the eigenfunctiondecomposition, also used in previous works by Liu and collaborators [24, 41].

In numerical computations, the success of this definition is determined largely byour ability to compute very accurately the eigenfunctions of the integer-order Lapla-


-1 -0.5 0 0.5 1

x

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

φ

s=0.75

s=0.75 (PM)

s=1.00

(a) y = 1.

-1 -0.5 0 0.5 1

y

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

φ

s=0.75

s=0.75 (PM)

s=1.00

0.7 0.8 0.9 10.95

1

(b) x = 0.

Fig. 10. Phase-field profiles along the lines (left) y = 1 (upper wall) and (right) x = 0 (cen-terline), s = 0.75 at t = 1.8. The inset on the right figure shows the details around the upper wall(x = 0).

cian on complex geometry domains. This can be accomplished by the SEM and h-p re-finement for nonsmooth domains, as we demonstrated here, but there is an additionalissue, which we encounter also for the eigenfunctions of the singular Sturm–Liouvilleproblem even when we obtain them analytically, namely, the loss of orthogonality asthe number of eigenfunctions increases. Hence, simply computing these eigenfunctionsand employing them in a Galerkin projection (as Liu and collaborators have done inthe past) would lead to a severe loss of accuracy, especially for realistic applicationsin nontrivial domains where high numerical resolution is required. To this end, wehave introduced a W-GS orthonormalization that leads to exponential accuracy inthe numerical approximation of (smooth) functions and solutions to fractional PDEsfor any order of the fractional Laplacian.

We demonstrated our numerical results on square, disk, and L-shaped domainsby computing the solution of fractional diffusion and comparing with analytical solu-tions wherever possible. In addition to the accuracy, the approach we developed is alsoefficient as it does not require the solution of any linear systems since the eigenfunc-tion decomposition leads naturally to a system of ODEs, which can be readily solvedwith standard methods. As a second application of the method, we also consideredmultiphase flow modeling and fractional energy laws, by replacing the integer-orderLaplacian in the Allen–Cahn model with its fractional counterpart and following asimilar procedure as in the numerical approximation of the fractional diffusion equa-tion. However, for the Navier–Stokes equations we needed to solve a linear system,which we inverted using an efficient ADI scheme. Specifically, we demonstrated theeffectiveness of the fractional Navier–Stokes/Allen–Cahn solver for the rising bubbleproblem in a square domain, and compared the results with the integer-order systemand also with results by a different treatment of the fractional diffusion model using 1Dfractional derivatives. The surprising advantage of the fractional Allen–Cahn modelis that it yields sharper interfaces compared to the numerical results obtained by theinteger-order models for the same resolution while it preserves the isotropic diffusion.To the best of our knowledge, this is the first application of fractional Laplacians tomultiphase flows, and it would be interesting to expand such studies in the future todocument the modeling advantages of fractional descriptions of diffusion in diverseapplications in multidimensions.

Appendix A. Eigenvalue problems. In this appendix, we will present somenumerical results to demonstrate that the approximations (3.6) and (3.7) are efficient


10 20 30 40 50 60

i

101

102

103

104

105

106

λi

Exact

N=16

N=32

N=64

2N/3

10 20 30 40 50 6010

-10

100

106

|λ

i,0

-λ

i|

(a) SM (El=1).

10 20 30 40 50 60

i

101

102

103

104

105

106

λi

Exact

El=1

El=4

El=16

El=32

10 20 30 40 50 6010

-10

100

106

|λ

i,0

-λ

i|

(b) SEM, fixing the DoF N = 65.

Fig. 11. The eigenvalues of the 1D EVP under Neumann boundary conditions: (a) spectralmethod (El = 1) with different N = 16, 32, 64; the square symbol marks two-thirds N ; (b) SEMfixing N = 65 with different El = 1, 4, 16, 32. Here DoF N = El × N + 1 corresponding to theNeumann boundary conditions. The inset figures show the error of the eigenvalues |λi,0−λi|, whereλi,0 correspond to the exact eigenvalues of the 1D EVP.

and highly accurate. Zhang [43] has shown that when using hp methods to approx-imate eigenvalues of 2m-order elliptic problems, the number of reliable numericaleigenvalues can be estimated in terms of the total number of DoF N in resultingdiscrete systems. However, we found that in our numerical experiments this esti-mate is rather optimistic, and the actual number of correctly computed eigenvalues isless.

Here, we present eigenvalues and eigenfunctions computed on different shape do-mains in 2 dimensions under different boundary conditions.

A.1. Numerical results. In this subsection, we first show the accuracy of theeigenvalues in a 1D domain. Then we show the spectral convergence of the approxi-mation (3.6) and (3.7) for given functions.

A.1.1. Numerical accuracy in 1 dimension. We first consider the 1D EVP(2.1) under Neumann boundary conditions. Figure 11 plots the exact eigenvaluesand numerical eigenvalues with different polynomial degree N. We observe that onlythe first few leading eigenvalues are accurate for both spectral method (SM, El = 1)and SEM (SEM, El = 4, 16, 32). In the following numerical examples, we want todemonstrate that every numerical function plays an important role for our approx-imation scheme (3.7), even though the accuracy of the high frequencies is worsen-ing with index number. We first introduce an incomplete approximation (IAp) asfollows:

(A.1) un(x) =

n∑i=1

ciφi, n ≤ N ,

where ci = (u, φi)N . The fractional Laplacian operator is approximated as

(A.2) − (−∆)α2 u ≈

n∑i=1

ciλα/2i φi, 1 < α ≤ 2, n ≤ N .


1 5 9 13 17 21 25 29 33 37 41 45 47

n

10 -14

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

L2-e

rro

r

α=2.0

N=16, IAp

N=32, IAp

N=48, IAp

N=16, W-GS

N=32, W-GS

N=48, W-GS

Spectral Convergence

(a) The error to solution (A.4).

1 5 9 13 17 21 25 29 31

n

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

L2-e

rror

α=2.0

N=4, IAp

N=8, IAp

N=16, IAp

N=32, IAp

N=4, W-GS

N=8, W-GS

N=16, W-GS

N=32, W-GS

10 -14

(b) The error to solution (A.5).

Fig. 12. The L2-error as a function of n with different polynomial degree N, El = 1, and α = 2.0.

Then let us consider the Helmholtz equation as follows:

(A.3) u+ µ(−∆)α2 u = f, α = 2.0,

where Ω = [−1, 1] and µ = 1. We test the accuracy for the following three analyticsolutions:

u(x) =5

4 + sin(πx)− 5

4,(A.4)

u(x) = 1− x2,(A.5)

with homogeneous Dirichlet boundary conditions u(±1) = 0, and

u(x) =

15∑i=1

cos(iπx)(A.6)

with homogeneous Neumann boundary conditions ux(±1) = 0.Figure 12 shows that the approximation accuracy can be significantly improved

with W-GS. It also shows that every numerical eigenfunction is critical to obtainhigh accuracy. In particular, exponential convergence is achieved if and only if theapproximation (IAp) is complete (i.e., n = N ). Figure 13 shows that the completeapproximation can even achieve spectral convergence for a highly oscillatory function.Finally, we test the convergence of (IAp) with the eigenmodes obtain from the spectralelement discretization, where the number of DoF N is constant but we change theelement number El and polynomial order N . The numerical results in Figure 14 showthe same property as the one element case.

A.1.2. Numerical eigenvalues and eigenfunctions on different shapeddomains. Here, we will present the numerical eigenvalues and eigenfunctions on dif-ferent shaped domains under different boundary conditions, and compare the resultsto the analytic solutions.

If we consider (2.1) on the unit square domain Ω = (−1, 1)2 under homogeneousNeumann boundary conditions, we have the following analytical solutions for theeigenvalues:

(A.7) λm,n =π2

4(n2 +m2), m, n = 0, 1, 2, 3, . . . ,


-1 -0.5 0 0.5 1

x

-4

-2

0

2

4

6

8

10

12

14

16

(a) The profile of solution (A.6).

10 20 30 40 50 60 70 80

n

10 -10

10 -8

10 -6

10 -4

10 -2

10 0

L2-e

rro

r

α=2.0

N=36, IAp

N=49, IAp

N=64, IAp

N=81, IAp

N=36, W-GS

N=49, W-GS

N=64, W-GS

N=81, W-GS 50 60 70 80

N

10 -15

10 -10

10 -5

10 0

(b) The error to solution (A.6).

Fig. 13. The L2-error as a function of n with different polynomial degree N, El = 1 andα = 2.0. The inset shows the spectral convergence for the complete interpolation.

10 20 30 40 50 60

n

10-12

10-10

10-8

10-6

10-4

10-2

L2-e

rro

r

α=2.0

El=1, W-GS

El=4, W-GS

El=16, W-GS

El=32, W-GS

(a) The error to (A.4) with different El.

10 20 30 40 50 60

n

10-12

10-10

10-8

10-6

10-4

10-2

L2-e

rror

α=2.0

El=1, W-GS

El=4, W-GS

El=16, W-GS

El=32, W-GS

(b) The error to (A.5) with different El.

Fig. 14. The L2-error as a function of n with different element number El and α = 2.0. Wefix the DoF N = El ×N − 1 = 63.

and for the eigenfunctions

(A.8) φm,n(x) = cos(mπ(x+ 1)

2

)cos(nπ(y + 1)

2

).

Table 1 shows the leading 20 eigenvalues error |λi − λm,n| with different polynomialdegree N under homogeneous Neumann boundary conditions; here we have set El =16 for computing the EVP.

Next, we show the leading 20 eigenvalues error |λi − λm,n| with different poly-nomial degree N on a unit disk domain under homogeneous Dirichlet boundary con-ditions in Table 2; here we have set El = 1 in the computations. For the sakeof comparison, we also list the results presented first in [34], which were obtainedby the Legendre–Galerkin method in polar coordinates. The analytic eigenvaluesλm,n = τm,n, where τm,n are the roots of the first kind Bessel functions [40].

The results in Tables 1 and 2 show that we can obtain spectral convergence underdifferent boundary conditions both on the unit square and the unit disk domains.This implies that SEM is an efficient method for solving EVPs on regular domains.However, we need higher polynomial degree N for improving the accuracy of thehigher eigenmodes.


Table 1The leading 20 eigenvalues error |λi−λm,n| with different polynomial degree N on unit square

domain under homogeneous Neumann boundary condition. Here, N = (4N + 1)2.

i N = 3 N = 4 N = 5 N = 7 N = 9 N = 111 3.983e-15 3.776e-14 7.450e-15 3.286e-14 1.153e-13 3.433e-14

2,3 1.915e-06 3.500e-09 4.334e-12 1.376e-14 1.136e-13 4.218e-144 3.831e-06 7.000e-09 8.681e-12 5.329e-15 1.110e-13 2.930e-14

5,6 4.927e-04 3.539e-06 1.754e-08 1.243e-13 1.367e-13 1.776e-147,8 4.946e-04 3.543e-06 1.754e-08 1.527e-13 1.119e-13 8.881e-159 9.855e-04 7.079e-06 3.508e-08 3.161e-13 4.618e-14 7.815e-14

10,11 1.303e-02 2.006e-04 2.214e-06 1.088e-10 8.526e-14 0.000e-0012,13 1.303e-02 2.006e-04 2.214e-06 1.089e-10 8.171e-14 2.131e-1414,15 1.352e-02 2.041e-04 2.231e-06 1.090e-10 1.918e-13 2.131e-1416,17 7.223e-01 2.197e-02 7.155e-04 1.564e-07 1.151e-11 7.105e-1518,19 7.223e-01 2.197e-02 7.155e-04 1.564e-07 1.136e-11 8.526e-14

20 2.606e-02 4.012e-04 4.428e-06 2.175e-10 2.415e-13 7.815e-14N 169 289 441 841 1369 2025

Table 2The leading 20 eigenvalues error |λi − λm,n| with different polynomial degree N on the unit

disk domain under homogeneous Dirichlet boundary condition. Here, N = (N − 1)2.

i N = 6 N = 8 N = 12 N = 16 N = 201 [34] - 6.5e-5 3.9e-9 1.1e-14 1.2e-16

1 1.860e-04 4.126e-07 2.269e-10 1.545e-13 6.217e-152,3 1.569e-02 7.190e-05 2.605e-11 1.492e-13 3.375e-144,5 2.301e+00 5.127e-02 8.466e-07 5.755e-13 7.105e-146 3.368e+00 1.009e-01 2.793e-06 5.123e-12 1.065e-14

7,8 2.521e+00 5.017e-02 5.120e-06 5.339e-11 1.918e-139,10 1.592e+00 5.919e-01 1.981e-04 2.935e-09 1.421e-1311,12 3.520e+00 4.620e+00 3.897e-03 1.458e-07 7.531e-1313,14 9.276e+00 1.176e+01 6.009e-02 4.939e-06 4.776e-11

15 1.469e+01 3.326e+00 6.620e-02 6.561e-06 7.821e-1116,17 1.264e+01 5.378e+00 8.924e-03 5.451e-07 3.524e-1218,19 1.185e+01 9.154e+00 1.707e-02 2.130e-05 1.038e-09

20 2.721e+01 7.958e+00 4.274e-01 2.117e-04 1.112e-08N 25 49 121 225 361

Finally, we solve the EVP on the L-shaped domain Ω = (−1, 1)2/[0, 1)2 under Dirichlet boundary conditions. Since, the domain is nonsmooth at thecorner, the rate of convergence of the solution from the Galerkin method is restrictedby the vertex singularities [3]. In order to increase the convergence, we refine geomet-rically the mesh near the corner as in Figure 15. The mesh sizes near the singularcorner in Figure 15 are ha = 0.5, hb = ha

2 , and hc = hb2 .

Here, we do not know the exact solutions of the EVP on the L-shaped domain,however, for the sake of comparison, we list available results from the references [17, 29]in Table 3. We fix the polynomial degree N = 8 in each element for this computation.We list four cases corresponding to different element number El = 27, 75, 300, 675.The first leading 10 numerical eigenvalues are all accurate up to the ninth significantdigit (i.e., all the eigenvalues are within the bounds of references [17, 29] when El =675). We also list the number of DoF (N ) for each computation.

We plot the first 9 leading eigenfunctions on different shaped domains at the endof this subsubsection. In Figures 16 (a), (b), (c), and (d), we plot the contours of thenumerical eigenfunctions φi, i = 1, 2, . . . , 9 corresponding to the eigenvalues λi. We


...

(a) El = 12

...

(b) El = 27

...

(c) El = 48

Fig. 15. Geometric h-refinement in the L-shaped domain around the 270 degrees angle.

Table 3The leading 10 eigenvalues λi with different element number El on the L-shaped domain under

homogeneous Dirichlet boundary conditions. The polynomial degree is N = 8 in each element. Thethird exact eigenpair is known: λ3 = 2π2, φ3(x) = sin (πx) sin (πy).

i El = 27 El = 75 El = 300 El = 675 [17]UpperLower [29]Upper

Lower

1 9.64115769 9.63994969 9.63972606 9.63972386 9.639723884059.66999.5585

2 15.1972544 15.1972519 15.1972519 15.1972519 15.172520118415.22514.950

3 19.7392088 19.7392088 19.7392088 19.7392088 19.73920918519.78719.326

4 29.5214821 29.5214811 29.5214811 29.5214811 29.52148180429.62628.605

5 31.9161352 31.9131874 31.9126413 31.9126360 31.91263883132.05830.866

6 41.4771420 41.4749246 41.4745139 41.4745099 41.474515903941.68039.687

7 44.9485007 44.9484881 44.9484877 44.9484877 44.948509467 -

8,9 49.3480220 49.3480220 49.3480220 49.3480220 49.34803509 -

10 56.7125467 56.7100728 56.7096144 56.7096099 56.70961802 -

N 1633 4641 18881 42721 - -

normalize the eigenfunctions (φi, φi)N = 1 in each figure. In Figure 16(b) the firsteigenvalue is equal to 0 and the eigenfunction is a constant c = 0.5.

Appendix B. Comparison of different eigenvalue models. In this ap-pendix, we recall our previous model (PM) of the nonlocal fractional Laplace operatordefined by combining the Caputo derivative and Riemann–Liouville derivative. Thefractional EVP is given as follows:

(B.1) −D2su− λu = 0, x ∈ Ω = (−1, 1)2,

the boundary condition is u|∂Ω = 0, and the fractional Laplace operator is defined asD2s := 1

2 (RLD2sx +RL

xD2s +RLD2s

y +RLyD

2s), where RLD2s∗ and RL

∗D2s are the left and

right Riemann–Liouville fractional derivatives, respectively.We still use the spectral method for solving the above fractional EVP. The details

of the discretization are given in reference [37]. We set the polynomial degree N = 64for solving problem (B.1). Following the definition of the fractional Laplace in (3.6),we know analytically the eigenvalues of (3.6) under homogeneous Dirichlet boundary

conditions, given as λsm,n = ( (m2+n2)4 π2)s. Table 4 and Figures 17 and 18 show that

the eigenvalues of the problem (B.1) are the same as in (3.6) if and only if s = 1, but

different for 0 < s < 1. Here, we denote the difference Ei,s =|λi,s−λsm,n|

λi,sin the table,


λ1=4.9348

-1 0 1

-1

-0.5

0

0.5

1

0.2

0.4

0.6

0.8

λ2=12.337

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=12.337

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=19.7392

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ5=24.674

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

1

λ6=24.674

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ7=32.0762

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ8=32.0762

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ9=41.9458

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

(a)

λ1=1.3445e-11

-1 0 1

-1

-0.5

0

0.5

1

-0.500000000001

-0.5000000000005

-0.5

-0.4999999999995

λ2=2.4674

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=2.4674

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=4.9348

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ5=9.8696

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ6=9.8696

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ7=12.337

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ8=12.337

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ9=19.7392

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

(b)

λ1=5.7832

-1

0

1

λ2=14.682

-1

0

1

λ3=14.682

-1

0

1

λ4=26.3746

-1

0

1

λ5=26.3746

-1

0

1

λ6=30.4713

-1

0

1

λ7=40.7065

-1

0

1

λ8=40.7065

-1

0

1

λ9=49.2185

-1

0

1

(c)

λ1=9.6397

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.8

-0.6

-0.4

-0.2

λ2=15.1973

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=19.7392

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=29.5215

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ5=31.9127

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ6=41.4745

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

λ7=44.9485

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ8=49.348

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ9=49.348

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

(d)

Fig. 16. The first 9 leading eigenfunctions of EVP (2.1): (a) on the unit square domain withDirichlet boundary condition. El = 100, N = 20; (b) on the unit square domain with Neumannboundary condition. El = 100, N = 20; (c) on the unit disk domain with Dirichlet boundarycondition. El = 1, N = 62; (d) on the L-shaped domain with Dirichlet boundary condition. El =75, N = 20.

Table 4Comparison of eigenvalues for the two different fractional Laplacian models.

is = 1.00 s = 0.75 s = 0.60

λi,s Ei,s λi,s Ei,s λi,s Ei,s

1 4.9348 2.8437e-14 2.2593 3.1762e-01 0.8016 6.9237e-012,3 12.337 8.0632e-14 4.7076 2.8486e-01 1.4784 6.7261e-014 19.739 2.2030e-13 7.1559 2.3587e-01 2.1551 6.4002e-01

5,6 24.674 6.7673e-14 7.9142 2.8513e-01 2.2278 6.7451e-017,8 32.076 2.2285e-13 10.362 2.3118e-01 2.9045 6.3745e-019,10 41.946 6.7928e-14 11.750 2.8711e-01 3.0384 6.7713e-0111 44.413 2.7853e-13 13.569 2.1130e-01 3.6540 6.2481e-01

12,13 49.348 2.0921e-13 14.198 2.3742e-01 3.7151 6.4190e-0114,15 61.685 2.8544e-13 16.114 2.6793e-01 3.8907 6.7197e-0116,17 64.152 6.4018e-14 17.405 2.3218e-01 4.4645 6.3234e-0118,19 71.555 1.8053e-13 18.562 2.4553e-01 4.5674 6.4772e-01

20 78.957 3.2577e-13 20.954 2.0890e-01 4.7812 6.5238e-01


0 5 10 15 20

i

0

10

20

30

40

50

60

70

80

eige

nval

ue

λm,n

1.00

λi,1.00

λm,n

0.75

λi,0.75

λm,n

0.60

λi,0.60

Fig. 17. Comparison of eigenvalues using two different definitions of the Laplacian for s =1.00, 0.75, 0.60 on the unit square domain with Dirichlet boundary conditions. The lines correspondto analytic solutions given by (3.6) for s = 1.0 (–), s = 0.75 (- -), and s = 0.60 (··), whereasthe symbols correspond to eigenvalues of the fractional Laplacian defined in (B.1) for s = 1.0 (O),s = 0.75 (), and s = 0.60 (♦).

λ1=4.9348

-1 0 1

-1

-0.5

0

0.5

1

0.2

0.4

0.6

0.8

λ2=12.337

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=12.337

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=19.7392

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ5=24.674

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

1

λ6=24.674

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ7=32.0762

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ8=32.0762

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ9=41.9458

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

(a) s = 1.00,

λ1=3.897

-1 0 1

-1

-0.5

0

0.5

1

0.2

0.4

0.6

0.8

λ2=9.0844

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=9.0844

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=14.2719

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ5=16.9752

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ6=16.9752

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

λ7=22.1627

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ8=22.1627

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

λ9=27.365

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

(b) s = 0.90,

λ1=2.2593

-1 0 1

-1

-0.5

0

0.5

1

0.2

0.4

0.6

0.8

λ2=4.7076

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=4.7076

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=7.1559

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ5=7.9142

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ6=7.9142

-1 0 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

λ7=10.3625

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ8=10.3625

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ9=11.75

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

(c) s = 0.75,

λ1=0.80166

-1 0 1

-1

-0.5

0

0.5

1

0.2

0.4

0.6

0.8

λ2=1.4784

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ3=1.4784

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ4=2.1551

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ5=2.2278

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ6=2.2278

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

1

λ7=2.9045

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ8=2.9045

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

λ9=3.0384

-1 0 1

-1

-0.5

0

0.5

1

-0.5

0

0.5

(d) s = 0.60.

Fig. 18. The first 9 leading eigenfunctions of the fractional EVP with Dirichlet boundaryconditions on the unit square domain and the fractional orders s = 1.00, 0.90, 0.75, 0.60.


where (λi,s)pi=1 are the numerical solutions of the problem (B.1). This means that the

two Laplace operator definitions (B.1) and (3.6) are equivalent for s = 1 but differentfor 0 < s < 1 on bounded domains.

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