A weak Galerkin finite element scheme for the Cahn-Hilliard …szhang/research/p/2019a.pdf ·...

MATHEMATICS OF COMPUTATIONVolume 88, Number 315, January 2019, Pages 211–235https://doi.org/10.1090/mcom/3369

Article electronically published on August 21, 2018

A WEAK GALERKIN FINITE ELEMENT SCHEME

FOR THE CAHN-HILLIARD EQUATION

JUNPING WANG, QILONG ZHAI, RAN ZHANG, AND SHANGYOU ZHANG

Abstract. This article presents a weak Galerkin (WG) finite element methodfor the Cahn-Hilliard equation. The WG method makes use of piecewise poly-nomials as approximating functions, with weakly defined partial derivatives

(first and second order) computed locally by using the information in the in-terior and on the boundary of each element. A stabilizer is constructed andadded to the numerical scheme for the purpose of providing certain weak con-tinuities for the approximating function. A mathematical convergence theoryis developed for the corresponding numerical solutions, and optimal order oferror estimates are derived. Some numerical results are presented to illustratethe efficiency and accuracy of the method.

1. Introduction

This paper is concerned with the development of new numerical methods for theCahn-Hilliard equation using weak Galerkin (WG) finite element techniques. TheCahn-Hilliard equation (1.1)–(1.4) was first introduced by John Cahn and JohnHilliard in 1958 [6]. It describes the phenomenon of phase separation, or spinodaldecomposition. The equation simulates the process that a two-phase alloy fluidseparates into domains of pure of each component. For simplicity, we consider themodel problem that seeks an unknown function u = u(x, t) satisfying

ut −Δw = g, in ΩT := Ω× (0, T ),(1.1)

w = f(u)− γ2Δu, in ΩT := Ω× (0, T ),(1.2)

∂nu = ∂nΔu = 0, on ∂ΩT := ∂Ω× (0, T ),(1.3)

u(x, 0) = u0(x), in Ω,(1.4)

where Δ is the Laplacian operator, Ω is a bounded polygonal or polyhedral domainin R

d for d = 2, 3, n denotes the outward unit normal vector along ∂Ω, f(s) = F ′(s)with F (s) = 1

4 (s2 − 1)2, and g = g(x, t) is given data on ΩT . ∂n denotes the

Received by the editor April 22, 2017, and, in revised form, October 7, 2017, and March 20,2018.

2010 Mathematics Subject Classification. Primary 65N30, 65N15, 65N12, 74N20; Secondary35B45, 35J50, 35J35.

Key words and phrases. Weak Galerkin finite element methods, weak Laplacian, Cahn-Hilliardequation, polytopal meshes.

The research of the first author was supported in part by the NSF IR/D program, while workingat National Science Foundation. However, any opinion, finding, and conclusions or recommenda-tions expressed in this material are those of the author and do not necessarily reflect the views ofthe National Science Foundation.

The research of the third author was supported in part by China Natural National ScienceFoundation (U1530116,91630201,11471141), and by the Program for Cheung Kong Scholars ofMinistry of Education of China, Key Laboratory of Symbolic Computation and Knowledge En-gineering of Ministry of Education, Jilin University, Changchun, 130012, People’s Republic ofChina.

211

Licensed to Univ of Delaware. Prepared on Mon Feb 11 10:13:47 EST 2019 for download from IP 128.175.16.112.

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https://doi.org/10.1090/mcom/3369

212 JUNPING WANG, QILONG ZHAI, RAN ZHANG, AND SHANGYOU ZHANG

directional derivative on ∂Ω in the normal direction n. Without loss of generality,assume that the initial data satisfies

∫Ωu0 = 0.

In the Cahn-Hilliard equation (1.1), the function u ∈ [−1, 1] is used to representthe concentration of each component, and γ is the interface parameter which governsthe width of the transition region. The level set of the function u can be employed toidentify the interface of the two components. Besides the two-phase fluid problem,the Cahn-Hilliard equation also has a wide range of applications in various areas ofphysics and industry, such as multiphase fluid flow, image processing, and planetformation. As to the detailed physical derivation and applications, readers arereferred to [27] and the references therein.

The Cahn-Hilliard equation has no explicit formulation for its solution, and thusnumerical methods are indispensable tools in practical simulation. In the last threedecades, many numerical methods have been developed for the equation, includingthe finite difference method [9, 10, 21, 28], the finite element method [7, 13, 22], andthe spectral method [3]. Due to the simplicity and robustness, the finite elementmethod has been recognized as an efficient approach for the Cahn-Hilliard equation.However, since the Cahn-Hilliard equation is of fourth order, the usual conformingfinite element method requires C1 continuity for the approximation functions. Itis well known that the construction for C1-type elements is quite challenging inpractical computation. There are mainly two ways to conquer this problem: one isto use mixed finite element methods, and the other is to use nonconforming or non-standard finite element methods. The central idea of mixed finite element methodsis to reduce the fourth order equation into two second order equations, which thenrelax the smoothness requirement on finite element functions. The mixed finiteelement method is practical, and a lot of work has been dedicated to the study anddevelopment of this method; cf. [2,11–13,16,19]. However, the mixed finite elementmethod needs the solution of a saddle point problem for which a certain stabilitycondition should be satisfied in the algorithmic development and analysis. Themixed finite element method also introduces new variables, which shall increase thesize of the discrete linear system. Nonstandard finite element methods can solvethe Cahn-Hilliard equation without introducing new variables, and do not usuallyrequire the approximating functions to be C1 continuous. Among several of thenonstandard finite element methods developed for the Cahn-Hilliard equation arethe nonconforming finite element method [14], the discontinuous Galerkin method[15, 16, 18], the local discontinuous Galerkin method [33], and the virtual elementmethod [1].

Recently, the weak Galerkin (WG) finite element method has been developedfor partial differential equations [8, 17, 29, 36]. The WG method employs piecewisepolynomials in the finite element space, which can be discontinuous across elements.The key of WG method is to use locally defined weak derivative operators insteadof the classical derivative operators, plus a stabilizer that ensures a certain weakcontinuity for the approximating function. The WG method has been applied toseveral classes of partial differential equations, including the biharmonic equation[23,24,35], the Stoke’s equation [26,31,32,34], and the Maxwell equation [25]. Manynumerical techniques have also been applied to the WG method, including the aposteriori error estimators [8] and the two-level methods [20].

In this paper, we shall use the WG method to solve the Cahn-Hilliard equation.The main contributions of this paper are twofold. First, a WG finite element



WG FINITE ELEMENT SCHEME FOR CAHN-HILLIARD EQUATION 213

method is devised to show that the WG approach is applicable for polytopal meshesand can be extended to 3D polyhedral partitions without any difficulty. Second,a mathematical convergence theory is established for the corresponding WG finiteelement approximations. As a nonstandard finite element method, WG provides anumerical procedure for the Cahn-Hilliard equation without introducing auxiliaryvariables. On the other hand, the WG method usually consists of degrees of freedomon both interior and boundary of each element, and this results in a discrete systeminvolving a large number of unknowns. To reduce the degrees of freedom, in thispaper, we use a semi-implicit scheme and the Schur complement to eliminate theinterior unknown which reduces the computational cost significantly.

The paper is organized as follows. In Section 2 we introduce a WG finite elementmethod for the Cahn-Hilliard equation. In Section 3, we present a theoretical frame-work for a modified WG method for the linear biharmonic equation with boundaryconditions that have not been dealt with in existing literature. In particular, a newweak Laplacian operator will be introduced here for an improved estimation of theLaplacian operator. In Section 4, we shall establish an optimal order of convergencefor our WG numerical solutions for the Cahn-Hilliard equation. Finally, in Section5, we report some computational results to demonstrate the stability and accuracyof the numerical approximations.

2. Weak Galerkin finite element scheme

Let Th be a partition of the domain Ω consisting of polygons in 2D or polyhedrain 3D, such that Th is shape regular in the sense as defined in [30]. Denote by Ehthe set of all edges or flat faces in Th, and let E0

h = Eh \ ∂Ω be the set of all interioredges or flat faces.

To devise the WG scheme for (1.1)–(1.4), we introduce the following finite ele-ment space

(2.1) Vh = {{v0, vb, vn} : v0 ∈ Pk(T ) ∀T ∈ Th, vb ∈ Pk(e),

vn ∈ Pk−1(e) ∀e ∈ Eh},

where k ≥ 2 is an integer. Here, the component v0 represents the interior value ofv on each element, and the component vb represents the value of v on each edge.Denote by V 0

h the subspace of Vh with vanishing traces; i.e.,

V 0h = {{v0, vb, vn} ∈ Vh : vn|e = 0 ∀e ∈ ∂T ∩ ∂Ω}.(2.2)

We also introduce the following finite element space

Wh = {v ∈ V 0h ,

∫Ω

v0 = 0},(2.3)

and a set of normal directions

Nh = {ne : ne is unit and normal to e, e ∈ Eh}.

Then, on each edge, the component vn represents the normal derivative of v in thedirection ne. It should be pointed out that vb and vn are single-valued on eachedge, and are irrelevant to the trace of v0.

For any v ∈ Vh, define the weak Laplacian Δwv as follows.




Defintion 2.1 ([23]). For any v ∈ Vh, Δwv is defined as the unique polynomial inPk(T ) on each element T ∈ Th satisfying

(2.4) (Δwv, ϕ)T = (v0,Δϕ)T −〈vb,∇ϕ·n〉∂T +〈vn(ne ·n), ϕ〉∂T ∀ϕ ∈ Pk(T ),

where n is the outward unit normal vector.

It should also be noticed that in the WG methods studied in [23,35], the degreeof Δwv was chosen to be k−2. But in this paper, the degree of Δwv is selected to bek in order to obtain a full order error estimate for the Cahn-Hillard equation. Thus,the properties of the weak Laplacian operator need to be derived again, which isto be accomplished in Section 3.

On each element T ∈ Th, denote by Q0 the L2 projection onto Pk(T ). For eachedge/face e ⊂ ∂T , denote by Qb the L2 projection onto Pk(e), and denote by Qn

the L2 projection onto Pk−1(e). Now for any u ∈ H2(Ω), we shall combine thesethree projections together to define a projection into the finite element space Vh

such that on the element T ,

Qhu = {Q0u,Qbu,Qn(∇u · ne)}.

Next, we introduce the WG algorithm for the equations (1.1)–(1.4). To thisend, we define three bilinear forms s(·, ·), b(·, ·) and c(·, ·) as follows. For anyuh = {u0, ub, un} and vh = {v0, vb, vn} in Vh,

s(uh, vh) =∑T∈Th

h−1T 〈∇u0 · ne − un,∇v0 · ne − vn〉∂T(2.5)

+∑T∈Th

h−3T 〈u0 − ub, v0 − vb〉∂T ,

b(uh, vh) = (Δwuh,Δwvh) + s(uh, vh),(2.6)

c(uh, vh) = −(Δwf(uh), v0),(2.7)

where f(uh) = {f(u0), f(ub), f′(ub)un}.

As to the time direction, we use the backward Euler discretization and at thetime step m, define

dtumh =

umh − um−1

h

τm,

where τm > 0 is the time increment at the time-step m.We are now in a position to introduce the WG algorithm for equations (1.1)–

(1.4).

Weak Galerkin Algorithm 1. Set u0h = Qhu

0 ∈ V 0h . For m = 1, . . . ,M , find

umh ∈ Wh, such that

(2.8) (dtumh , v0) + γ2b(um

h , vh) + c(um−1h , v0) = (g, v0) ∀vh ∈ Wh.

3. Error estimates for a linear problem

In order to study the convergence of the WG Algorithm 1, we need some errorestimates for a linear biharmonic equation with boundary conditions derived fromthe Cahn-Hillard problem. To this end, consider the following boundary value




problem for the biharmonic equation

Δ2u = f, in Ω,(3.1)

∂nu = ∂nΔu = 0, on ∂Ω,(3.2) ∫Ω

u = 0,(3.3)

where Ω is a polygonal or polyhedral domain. The corresponding WG scheme isgiven as follows.

Weak Galerkin Algorithm 2. Find uh ∈ Wh, such that for any vh ∈ Wh,

b(uh, vh) = (f, v0),(3.4)

where b(·, ·) is defined in (2.6).

The error estimate for the numerical scheme (3.4) can be derived by using thetechniques developed in [23] and [35] with slight modification. First, we introducea semi-norm ||| · ||| in V 0

h by setting

|||v|||2 = b(v, v).(3.5)

Lemma 3.1. ||| · ||| defines a norm on Wh.

Proof. It suffices to verify the positivity property for this semi-norm. For anyv ∈ Wh with |||v||| = 0, we have Δwv = 0 on each element T and s(v, v) = 0. Fromthe definition of s(v, v) we obtain v0 = vb and ∇v0 · ne = vn on each edge. Thus,v0 and ∇v0 are continuous in the domain Ω so that v0 ∈ H2(Ω).

From the definition of the weak Laplacian operator, we have that for any φ ∈Pk(T ),

0 = (Δwv, φ)T

= (v0,Δφ)T − 〈vb,∇φ · n〉∂T + 〈vn(n · ne), φ〉∂T= (Δv0, φ)T + 〈v0 − vb,∇φ · n〉∂T − 〈∇v0 · n− vn(n · ne), φ〉∂T= (Δv0, φ),

which implies Δv0 = 0 on Ω. It follows from the conditions∫Ωv0 = 0 and ∇v0 ·ne =

0 on ∂Ω that v0 = 0 on Ω. Then vb = vn = 0 holds true, which completes theproof. �

Lemma 3.2. The Weak Galerkin Algorithm 2 has a unique solution.

Proof. We only need to verify the uniqueness for the homogenous problem. Supposef = 0, by setting vh = uh in (3.4) we have

|||uh|||2 = b(uh, uh) = 0.

It follows from Lemma 3.1 that uh = 0, which completes the proof. �

The relationship between the operators Δ and Δw are revealed in the followinglemma, which plays an essential role in the error analysis.

Lemma 3.3. On each element T ∈ Th and for any u ∈ H2(T ) and φ ∈ Pk(T )there holds:

(ΔwQhu, φ)T = (Q0Δu, φ) + 〈Qn(∇u · n)−∇u · n, φ〉∂T .(3.6)




Proof. For any φ ∈ Pk(T ), from the definition of the discrete weak Laplacian (2.4)we obtain

(ΔwQhu, φ)T = (Q0u,Δφ)T − 〈Qbu,∇φ · n〉∂T + 〈Qn(∇u · n), φ〉∂T= (u,Δφ)T − 〈u,∇φ · n〉∂T + 〈∇u · n, φ〉∂T

+〈Qn(∇u · n)−∇u · n, φ〉∂T= (Δu, φ)T + 〈Qn(∇u · n)−∇u · n, φ〉∂T= (Q0Δu, φ) + 〈Qn(∇u · n)−∇u · n, φ〉∂T ,

which completes the proof. �

The identity (3.6) indicates that the discrete weak Laplacian of the L2 projectionof u is an approximation of the Laplacian of u in the classical sense.

Let u be the exact solution of (3.1)–(3.3), and uh be the numerical solution of(3.4). Denote by eh = Qhu − uh the error between the projection of u and thenumerical approximation uh. Then, we have the following error equation.

Lemma 3.4. Let u ∈ H3(Ω) be the exact solution of (3.1)–(3.3), and let uh be thenumerical solution of (3.4). The following equation holds true:

b(eh, v) = �(u, v) ∀v ∈ Wh,

where

�(u, v) =∑T∈Th

〈Qn(∇u · n)−∇u · n,Δwv〉∂T(3.7)

−∑T∈Th

〈Q0Δu−Δu,∇v0 · n− vn(n · ne)〉∂T

+∑T∈Th

〈∇(Q0Δu−Δu) · n, v0 − vb〉∂T + s(Qhu, v).

Proof. From Lemma 3.3 and the definition of the weak Laplacian operator, we have

(ΔwQhu,Δwv)

= (Q0Δu,Δwv) +∑T∈Th

〈Qn(∇u · n)−∇u · n,Δwv〉∂T

= (ΔQ0Δu, v0) +∑T∈Th


−∑T∈Th

〈∇Q0Δu · n, vb〉∂T +∑T∈Th

〈Q0Δu, vn(n · ne)〉∂T

= (Δu,Δv0) +∑T∈Th


+∑T∈Th

〈∇Q0Δu · n, v0 − vb〉∂T −∑T∈Th

〈Q0Δu,∇v0 · n− vn(n · ne)〉∂T .




By testing (3.1) against v0 we obtain

(f, v0) = (Δ2u, v0)

= (Δu,Δv0)−∑T∈Th

〈Δu,∇v0 · n〉∂T +∑T∈Th

〈∇Δu · n, v0〉∂T

= (Δu,Δv0)−∑T∈Th

〈Δu,∇v0 − vn(n · ne) · n〉∂T

+∑T∈Th

〈∇Δu · n, v0 − vb〉∂T .

It follows from scheme (3.4) that

b(eh, v) = (Δw(Qhu− uh),Δwv) + s(Qhu− uh, v)(3.8)

= (ΔwQhu,Δwv)− (f, v0) + s(Qhu, v)

= �(u, v),


On the regular polytopal partition, the following trace inequality and inverseinequality hold true. The proof can be found in [30]

Lemma 3.5 (Trace inequality). For any ϕ ∈ H1(T ), the following inequality holdstrue on each element T ∈ Th:

‖ϕ‖2∂T ≤ C(h−1T ‖ϕ‖2T + hT ‖ϕ‖21,T ).

Lemma 3.6 (Inverse inequality). There exists a constant C such that on eachelement T ∈ Th, one has

‖∇ψ‖T ≤ Ch−1T ‖ψ‖T ∀ψ ∈ Pk(T ).

The following lemma is devoted to an estimate of the three terms in �(u, v).

Lemma 3.7. Let uh ∈ Vh be the weak Galerkin finite element solution arising from(3.4) with finite element functions of order k ≥ 2. Assume that the exact solutionof the biharmonic equation (3.1)–(3.3) satisfies u ∈ Hk+1(Ω) and Δu ∈ H2(Ω).Then, there exists a constant C such that∣∣∣∣∣

∑T∈Th


∣∣∣∣∣ ≤ Chk−1‖u‖k+1|||v|||,(3.9)

∣∣∣∣∣∑T∈Th

〈Q0Δu−Δu,∇v0 · n− vn(n · ne)〉∂T

∣∣∣∣∣ ≤ Chk−1‖u‖k+1|||v|||,(3.10)

∣∣∣∣∣∑T∈Th

〈∇(Q0Δu−Δu) · n, v0 − vb〉∂T

∣∣∣∣∣ ≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2)|||v|||,

(3.11)

|s(Qhu, v)| ≤ Chk−1‖u‖k+1|||v|||.(3.12)

Proof. The proof of (3.10) and (3.12) can be found in Theorem 6.2 in [23]. Thus,we shall only focus on the treatment of (3.9) and (3.11).




Denote by Q0 the L2 projection onto [Pk−1(T )]

d. Then, from the trace inequalityand the inverse inequality we have∣∣∣∣∣

∑T∈Th


∣∣∣∣∣≤

( ∑T∈Th

h−1T ‖Q0∇u−∇u‖2∂T

) 12( ∑

T∈Th

hT ‖Δwv‖2∂T

) 12

≤ C

( ∑T∈Th

h−2T ‖Q0∇u−∇u‖2T + ‖Q0∇u−∇u‖21,T

) 12

‖Δwv‖

≤ Chk−1‖u‖k+1|||v|||.

Similarly, it follows from the trace inequality and the inverse inequality that∣∣∣∣∣∑T∈Th

〈∇(Q0Δu−Δu) · n, v0 − vb〉∂T

∣∣∣∣∣≤

( ∑T∈Th

h3T ‖∇(Q0Δu−Δu)‖2∂T

) 12( ∑

T∈Th

h−3T ‖v0 − vb‖2∂T

) 12

≤ C

( ∑T∈Th

h2T ‖Q0Δu−Δu‖21,T + h4

T ‖Q0Δu−Δu‖22,T

) 12

s(v, v)12

≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2)|||v|||,


Corollary 3.8. Let uh ∈ Vh be the weak Galerkin finite element solution arisingfrom (3.4) with finite element functions of order k ≥ 2. Assume that the exactsolution of the biharmonic equation (3.1)–(3.3) satisfies u ∈ Hk+1(Ω) and Δu ∈H2(Ω). Then, we have, for �(u, v) defined in (3.7),

|�(u, v)| ≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2)|||v|||.

Remark 3.1. The problem (3.1)–(3.3) can be written in a mixed form as follows:

−Δw = f, in Ω,

−Δu = w, in Ω,

∂w

∂n=

∂u

∂n= 0, on ∂Ω,∫

Ω

u = 0.

Thus, the variable w = −Δu satisfies the Poisson equation with homogeneousNeumann boundary condition. It follows that w = −Δu ∈ H2(Ω) when Ω isconvex polygonal.

With these preparations, we are ready to establish an optimal order error esti-mate for the error function eh in the trip-bar norm.




Theorem 3.9. Let uh ∈ Vh be the weak Galerkin finite element solution arisingfrom (3.4) with finite element functions of order k ≥ 2. Assume that the exactsolution of biharmonic equation (3.1)–(3.3) satisfies u ∈ Hk+1(Ω) and Δu ∈ H2(Ω).Then, there exists a constant C such that

|||uh −Qhu||| ≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2).(3.13)

Proof. By setting v = eh in Lemma 3.4, we arrive at

|||eh|||2 = �(u, eh).

From Lemma 3.7, we have

|||eh|||2 = �(u, eh)

≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2)|||eh|||,


To obtain an error estimate in the L2 norm, we consider the following dualproblem

Δ2ψ = e0, in Ω,(3.14)

∂ψ

∂n=

∂Δψ

∂n= 0, on ∂Ω,(3.15) ∫

Ω

ψ = 0.(3.16)

Assume that the solution of the dual problem (3.14)–(3.16) has the following reg-ularity estimate:

‖ψ‖3 + ‖Δψ‖2 ≤ C‖e0‖.(3.17)

For a high order error estimate in L2, the following assumption is required:

‖ψ‖4 ≤ C‖e0‖.(3.18)

Remark 3.2. For the clamp boundary condition, when the polygonal domain Ω isconvex, the solution u belongs to H3(Ω). When all the corners of Ω are less than126.2839◦, the solution u belongs to H4(Ω); cf. Theorem 2 in [4].

For the Cahn-Hilliard type boundary condition, when all the corners w in Ωsatisfy w ≤ π

2 , the solution u belongs to H3(Ω). When all the corners w in Ω satisfyπ3 < w < π

2 or w = π2 , the solution u belongs to H4(Ω). Especially, u ∈ H4(Ω)

when Ω is a rectangular domain; cf. p. 2107, Appendix A in [5].

Lemma 3.10. Assume that the solution of (3.1)–(3.2) satisfies u ∈ H3(Ω), and theregularity assumption (3.17) holds true for the dual problem (3.14)–(3.15). Thenwe have the following identity,

‖e0‖2 = �(u,Qhψ)− �(ψ, eh),

where �(·, ·) is defined in (3.7).




Proof. From Lemma 3.3 and the definition of the weak Laplacian operator, we have

(ΔwQhψ,Δweh)

= (Q0Δψ,Δweh) +∑T∈Th

〈Qn(∇ψ · n)−∇ψ · n,Δweh〉∂T

= (ΔQ0Δψ, e0) +∑T∈Th


−∑T∈Th

〈∇Q0Δψ · n, eb〉∂T +∑T∈Th

〈Q0Δψ, en(n · ne)〉∂T

= (Δψ,Δe0) +∑T∈Th


+∑T∈Th

〈∇Q0Δψ · n, e0 − eb〉∂T −∑T∈Th

〈Q0Δψ,∇e0 · n− en(n · ne)〉∂T .

Testing (3.14) by e0, it follows from Lemma (3.4) that

‖e0‖2 = (Δ2ψ, e0)

= (Δψ,Δe0)−∑T∈Th

〈Δψ,∇e0 · n〉∂T +∑T∈Th

〈∇Δψ · n, e0〉∂T

= (Δψ,Δe0)−∑T∈Th

〈Δψ,∇e0 − en(n · ne) · n〉∂T

+∑T∈Th

〈∇Δψ · n, e0 − eb〉∂T

= b(Qhψ, eh)− �(ψ, eh)

= �(u,Qhψ)− �(ψ, eh). �

For the term �(u,Qhψ), we have the following estimate.

Lemma 3.11. Assume that u ∈ Hk+1(Ω) is the solution of (3.1)–(3.2) satisfyingΔu ∈ H2(Ω). Let ψ ∈ H3+s(Ω) be the solution of (3.14)–(3.15) with s ∈ [0, 1].Then we have the following estimate:

�(u,Qhψ) ≤ Chk+s(‖u‖k+1 + hδk,2‖Δu‖2)‖ψ‖3+s.

Proof. We estimate �(u,Qhψ) term by term. From the trace inequality and theCauchy-Schwarz inequality, it follows that for the first term, we have

∣∣∣∣∣∑T∈Th

〈Qn(∇u · n)−∇u · n,ΔwQhψ〉∂T

∣∣∣∣∣≤

( ∑T∈Th

‖Qn(∇u · n)−∇u · n‖2∂T

) 12( ∑

T∈Th

‖Q0(Δψ)−Qn(Δψ)‖2∂T

) 12

≤ Chk+s‖u‖k+1‖ψ‖3+s.




For the second term, we have∣∣∣∣∣∑T∈Th

〈Q0Δu−Δu,∇Q0ψ · n−Qn(∇ψ · n)〉∂T

∣∣∣∣∣≤

( ∑T∈Th

‖Q0Δu−Δu‖2∂T

) 12( ∑

T∈Th

‖∇Q0ψ · n−Qn(∇ψ · n)‖2∂T

) 12

≤ Chk+s‖u‖k+1‖ψ‖3+s.

For the third term, it can be seen that∣∣∣∣∣∑T∈Th

〈∇(Q0Δu−Δu) · n, Q0ψ −Qbψ〉∂T

∣∣∣∣∣≤

( ∑T∈Th

‖∇(Q0Δu−Δu) · n‖2∂T

) 12( ∑

T∈Th

‖Q0ψ −Qbψ‖2∂T

) 12

≤ Chk+s(‖u‖k+1 + hδk,2‖Δu‖2)‖ψ‖3+s.

As to the last term, we have

s(Qhu,Qhψ) =∑T∈Th

h−1T 〈∇Q0u · n−Qn(∇u · n),∇Q0ψ · n−Qn(∇ψ · n)〉∂T

+∑T∈Th

h−3T 〈Q0u−Qbu,Q0ψ −Qbψ〉∂T

≤( ∑

T∈Th

h−1T ‖∇Q0u · n−Qn(∇u · n)‖2∂T

) 12

( ∑T∈Th

h−1T ‖∇Q0ψ · n−Qn(∇ψ · n)‖2∂T

) 12

+

( ∑T∈Th

h−3T ‖Q0u−Qbu‖2∂T

) 12( ∑

T∈Th

h−3T ‖Q0ψ −Qbψ‖2∂T

) 12

≤ Chk+s‖u‖k+1‖ψ‖3+s. �

Using the Nitsche’s technique, we obtain the following error estimate in L2.

Theorem 3.12. Let uh ∈ Vh be the weak Galerkin finite element solution arisingfrom (3.4) with finite element functions of order k ≥ 2. Assume that the exactsolution of the biharmonic equation (3.1)–(3.3) satisfies u ∈ Hk+1(Ω), Δu ∈ H2(Ω)and the dual problem (3.14)–(3.16) satisfies the assumption (3.17). Then, thereexists a constant C such that

‖u0 −Q0u‖ ≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2).(3.19)

Furthermore, if k ≥ 3 and the dual problem (3.14)–(3.16) satisfies the H4-regularity assumption (3.18), then the following optimal order of error estimateholds true:

‖u0 −Q0u‖ ≤ Chk+1‖u‖k+1.(3.20)




Proof. From Lemma 3.10, we have

‖e0‖2 = �(u,Qhψ)− �(ψ, eh).

It follows from (3.17) that ψ ∈ H3(Ω) is the solution of (3.14)–(3.16) satisfyingΔψ ∈ H2(Ω). From Lemma 3.7 we have

�(ψ, eh) ≤ Chk−1(‖ψ‖k+1 + hδk,2‖Δψ‖2)|||eh|||.

Since ‖ψ‖ ∈ H3(Ω) and Δψ ∈ H2(Ω), by taking k = 2 in Lemma 3.7 and usingTheorem 3.9 we obtain

�(ψ, eh) ≤ Ch(‖ψ‖3 + h‖Δψ‖2)|||eh|||≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2)(‖ψ‖3 + h‖Δψ‖2).

Taking s = 0 in Lemma 3.11 we have

�(u,Qhψ) ≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2)‖ψ‖3.

Thus, from assumption (3.17) we conclude that

‖e0‖2 ≤ |�(u,Qhψ)|+ |�(ψ, eh)|≤ Chk(‖ψ‖3 + h‖Δψ‖2)(‖u‖k+1 + hδk,2‖Δu‖2)≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2)‖e0‖,

which implies

‖e0‖ ≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2).

Similarly, under the assumption (3.18) and k ≥ 3. By taking k = 3 in Lemma3.7 and using Theorem 3.9 we obtain

�(ψ, eh) ≤ Ch2(‖ψ‖4)|||eh|||≤ Chk+1‖u‖k+1‖ψ‖4.

Taking s = 1 in Lemma 3.11 we have

�(u,Qhψ) ≤ Chk+1(‖u‖k+1)‖ψ‖4.

Thus, from assumption (3.18) we conclude that

‖e0‖2 ≤ |�(u,Qhψ)|+ |�(ψ, eh)|≤ Chk+1(‖u‖k+1)‖ψ‖4≤ Chk+1‖u‖k+1‖e0‖,

which implies

‖e0‖ ≤ Chk+1‖u‖k+1,


Remark 3.3. In Theorem 3.12, we have a sub-optimal order of convergence fork = 2 and optimal order of convergence for k ≥ 3. For the quadratic element, theconvergence order is O(h2), and for the higher order element of Pk, k ≥ 3, theconvergence order is O(hk+1).




For any v ∈ Vh, define the edge-based norms as follows:

‖vb‖2Eh=

∑e∈Eh

he‖vb‖2L2(e),

‖vn‖2Eh=

∑e∈Eh

he‖vn‖2L2(e).

Similar to Theorem 5.3 in [35], we have the following estimates.

Theorem 3.13. Let uh ∈ Vh be the weak Galerkin finite element solution arisingfrom (3.4) with finite element functions of order k ≥ 2. Assume that the exactsolution of the biharmonic equation (3.1)–(3.3) satisfies u ∈ Hk+1(Ω), Δu ∈ H2(Ω)and the dual problem (3.14)–(3.16) satisfies the assumption (3.17). Then, thereexists a constant C such that

‖ub −Qbu‖Eh≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2),(3.21)

‖un −Qn(∇u · ne)‖Eh≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2).(3.22)

Furthermore, if k ≥ 3 and the dual problem (3.14)–(3.16) satisfies the H4-regularity assumption (3.18), the following estimate holds true:

‖ub −Qbu‖Eh≤ Chk+1‖u‖k+1,(3.23)

‖un −Qn(∇u · ne)‖Eh≤ Chk‖u‖k+1.(3.24)

Proof. Denote eh = Qhu− uh. From Theorem 3.9 and Theorem 3.12, we have

‖ub −Qbu‖Eh

≤ ‖u0 −Q0u‖Eh+ ‖e0 − eb‖Eh

≤ C(‖Q0u− u0‖+ h2s(eh, eh))

≤ Chk(‖u‖k+1 + hδk,2‖Δu‖2).

From Theorem 3.9, Theorem 3.12, and the inverse inequality, we obtain

‖un −Qn(∇u · ne)‖Eh

≤ ‖∇(Q0u− u0) · ne‖Eh+ ‖∇e0 · ne − en‖Eh

≤ C(h−1‖Q0u− u0‖+ hs(eh, eh))

≤ Chk−1(‖u‖k+1 + hδk,2‖Δu‖2).

When k ≥ 3 and the dual problem (3.14)–(3.16) satisfies the H4-regularity as-sumption (3.18), the proof is similar and thus omitted. �

4. Error estimates for the Cahn-Hilliard equation

Based on the results established in Section 3, we shall establish an error estimatefor the WG-FEM solution uh arising from (2.8).

Since the nonlinear term c(·, ·) is explicit in (2.8), the solvability of (2.8) is astraightforward application of Lemma 3.2.




Denote by Ehu the elliptic projection of u satisfying∫ΩEhu =

∫Ωu and

b(Ehu, vh) = (Δ2u, v0) ∀vh ∈ Wh.(4.1)

Let umh be the numerical solution of the Cahh-Hilliard problem at the m-th step of

the WG algorithm (2.8). Define error functions η and ξ as follows:

ηm = um − Ehum, ξm = Ehu

m − umh .(4.2)

The estimation of ηm has been derived in Theorems 3.9–3.13 in various Sobolevnorms. It remains to estimate the second term ξm. To this end, we first derive theerror equation for the WG finite element solution obtained from (2.8).

Lemma 4.1. Let um and umh ∈ Vh be the solution of (1.1)–(1.4) and (2.8), respec-

tively. Then the error functions ηm and ξm satisfy the following equation:

(4.3) (dtξm, v0) + γ2b(ξm, vh)

= (Rm, v0)− (dtηm, v0) + (Δf(um)−Δwf(u

m−1h ), v0)

for all v ∈ Vh, where

Rm = dtum − um

t = −τ−1m

∫ tm

tm−1

(t− tm−1)utt(t)dt.

Proof. For any vh ∈ Vh, we have

(dtξm, v0) + γ2b(ξm, vh)

= (dtEhum, v0) + γ2b(Ehu

m, vh)− (dtumh , v0)− γ2b(um

h , vh)

= (dtEhum, v0) + γ2b(Δ2um, v0)− (g, v0)− (Δwf(u

m−1h ), v0)

= (dtEhum, v0)− (um

t , v0) + (Δf(um), v0)− (Δwf(um−1h ), v0),

which leads to the error equation (4.3). �

Next, we shall establish some estimates for all three terms on the right-hand sideof the error equation (4.3).

The nonlinear function f(u) is not globally Lipschitz on R. However, on anygiven interval [−L,L] the function f is Lipschitz continuous. It is also trivial to seethat f, f ′, and f ′′ are continuous and uniformly bounded on [−L,L] and

|f(x)− f(y)| ≤ CL|x− y| ∀x, y ∈ [−L,L].

In the application to the Cahn-Hilliard equation, the size of the interval willbe taken to be L = 2max0≤t≤T ‖u(t)‖∞ + C0 in Theorem 4.2. Note that L is aconstant depending upon the exact solution u. In particular, it will be shown thatfor sufficiently small meshsize h and time step τ ≤ h

12 , the value of the numerical

solution uh lies in [−L,L]. Thus, the effective domain for the nonlinear functionwill be bounded by the interval [−L,L] on which the function f is smooth andLipschitz continuous.




Assume that the solution u of (1.1)–(1.4) satisfies the following regularity as-sumptions:

u ∈ C(0, T ;L2(Ω)),

u, ut ∈ L2(0, T ;H3(Ω) ∩Hk+1(Ω)),

utt ∈ L2(0, T ;L2(Ω)).

Theorem 4.2. Assume that the solution u ∈ Hk+1 is the exact solution of (1.1)–(1.4) satisfying Δu ∈ H2(Ω) and um

h is the WG solution arising from Algorithm 1with k ≥ 2. Assume τ = max1≤m≤M τm and h = maxT∈Th

hT are sufficiently small

and τ ≤ minT∈Thh

12

T , then the following estimates hold true for the error functionξm = Ehu

m − umh :

‖ξm0 ‖2 ≤ CeC2T (τ2 + h2k),(4.4)

γ2m∑

n=1

τn|||ξn|||2 ≤ CeC2T (τ2 + h2k),(4.5)

where the constant C2 is proportional to C2L.

Furthermore, if k ≥ 3 and u ∈ H4(Ω), then the following estimate holds true:

‖ξm0 ‖2 ≤ CeC2T (τ2 + h2k+2),

γ2m∑

n=1

τn|||ξn|||2 ≤ CeC2T (τ2 + h2k+2).

Proof. By letting vh = ξm in the error equation (4.3), we have

1

2dt‖ξm0 ‖2 + τm

2‖dtξm0 ‖2 + γ2|||ξm|||2

= (Rm, ξm0 )− (dtηm, ξm0 ) + (Δf(um)−Δwf(u

m−1h ), ξm0 ).(4.6)

The rest of the proof is devoted to the estimation of the terms on the right-handside of (4.6). For the first term, we use the Cauchy-Schwarz inequality to obtain

(Rm, ξm0 ) =∑T∈Th

(Rm, ξm0 )T ≤ ‖Rm‖‖ξm0 ‖.(4.7)

From the Cauchy-Schwarz inequality we have

‖Rm‖2 =

∫Ω

(τ−1m

∫ tm

tm−1

(t− tm−1)uttdt

)2

dx

≤∫Ω

τm

∫ tm

tm−1

u2ttdtdx

= τm

∫ tm

tm−1

∫Ω

u2ttdxdt

= τm

∫ tm

tm−1

‖utt‖2dt.




Substituting the above inequality into (4.7) gives

(Rm, ξm0 ) ≤ Cτ12m(Am

1 )12 ‖ξm0 ‖,(4.8)

where Am1 =

∫ tm

tm−1

‖utt‖2dt.

From the definition of Eh (4.1), for any time interval (tm−1, tm) and vh ∈ Vh wehave

(Eh

∫ tm

tm−1

utdt, vh

)

=

∫Ω

Δ2

(∫ tm

tm−1

utdt

)vhdx

=

∫ tm

tm−1

∫Ω

Δ2utvhdxdt

=

∫Ω

∫ tm

tm−1

Ehutdtvhdx

=

(∫ tm

tm−1

Ehutdt, vh

),

which leads to

Eh

∫ tm

tm−1

utdt =

∫ tm

tm−1

Ehutdt.

For the second term, from Theorem 3.12 we have

(dtηm, ξm0 ) = τ−1

m (um − um−1 − Eh(um − um−1), ξm0 )

= τ−1m

(∫ tm

tm−1

ut − Ehutdt, ξm0

)

≤ τ−1m

⎛⎝∫

Ω

(∫ tm

tm−1

ut − Ehutdt

)2

dx

⎞⎠

12

‖ξm0 ‖

≤ Cτ− 1

2m

(∫ tm

tm−1

‖ut − Ehut‖2dt) 1

2

‖ξm0 ‖

≤ Cτ− 1

2m hk(Am

2 )12 ‖ξm0 ‖,(4.9)

where Am2 =

∫ tm

tm−1

‖ut‖2kdt.




The third term can be estimated by using the Cauchy-Schwarz inequality asfollows:

(Δf(um)−Δwf(um−1h ), ξm0 )

= (f(um)− f(um−10 ),Δξm0 )−

∑T∈Th

⟨f(um)− f(um−1

b ),∂ξm0∂n

⟩∂T

+∑T∈Th

⟨f ′(um)

∂um

∂n− f ′(um−1

b )um−1n (n · ne), ξ

m0

⟩∂T

= (f(um)− f(um−10 ),Δξm0 )−

∑T∈Th

⟨f(um)− f(um−1

b ),

(∂ξm0∂n

− ξmn

)(n · ne)

⟩∂T

+∑T∈Th

⟨(f ′(um)− f ′(um−1

b ))∂um

∂n

−f ′(um−1b )

(∂um

∂n− um−1

n (n · ne)

), ξm0 − ξmb

⟩∂T

≤ C‖f(um)− f(um−10 )‖‖Δξm0 ‖

+C

( ∑T∈Th

‖f(um)− f(um−1b )‖2∂T

) 12( ∑

T∈Th

‖∂ξm0

∂ne− ξmn ‖∂T

) 12

+C

( ∑T∈Th

‖f ′(um)− f ′(um−1b )‖2∂T

+‖∂um

∂ne− um−1

n ‖2∂T) 1

2

( ∑T∈Th

‖ξm0 − ξmb ‖∂T

) 12

≤ C‖um − um−10 ‖|||ξm|||+ Ch

12

( ∑T∈Th

‖um − um−1b ‖2∂T

) 12

|||ξm|||

+Ch32

( ∑T∈Th

∥∥∥∥∂um

∂ne− um−1

n

∥∥∥∥2

∂T

) 12

|||ξm|||.

Notice that

‖um − um−10 ‖(4.10)

≤ ‖um − um−1‖+ ‖um−1 − Ehum−10 ‖+ ‖Ehu

m−10 − um−1

0 ‖= ‖um − um−1‖+ ‖ηm−1

0 ‖+ ‖ξm−10 ‖,

‖um − um−1b ‖∂T(4.11)

≤ ‖um − um−1‖∂T + ‖um−1 − (Ehum−1)b‖∂T

+‖ξm−10 ‖∂T + ‖ξm−1

0 − ξm−1b ‖∂T

≤ C(h− 12 ‖um − um−1‖T + h

12 ‖um − um−1‖1,T )

+C(‖ηm−1b ‖∂T + h− 1

2 ‖ξm−10 ‖T + ‖ξm−1

0 − ξm−1b ‖∂T )




and ∥∥∥∥∂um

∂ne− um−1

n

∥∥∥∥∂T

(4.12)

≤∥∥∥∥∂um

∂ne− ∂um−1

∂ne

∥∥∥∥∂T

+

∥∥∥∥∂um−1

∂ne− (Ehu

m−1)n

∥∥∥∥∂T

+

∥∥∥∥∂ξm−10

∂ne

∥∥∥∥∂T

+

∥∥∥∥∂ξm−10

∂ne− ξm−1

n

∥∥∥∥∂T

≤ C(h− 1

2 ‖um − um−1‖1,T + h12 ‖um − um−1‖2,T

)+C

(‖ηm−1

n ‖∂T + h− 32 ‖ξm−1

0 ‖T +

∥∥∥∥∂ξm−10

∂ne− ξm−1

n

∥∥∥∥∂T

).

Then we conclude from (4.10)–(4.12) that(Δf(um)−Δwf(u

m−1h ), ξm0

)≤ C(‖um − um−1‖+ h‖um − um−1‖1 + h2‖um − um−1‖2)|||ξm|||+ C(‖ηm−1

0 ‖+ ‖ηm−1b ‖Eh

+ h‖ηm−1n ‖Eh

+ h2|||ξm−1|||+ ‖ξm−10 ‖)|||ξm|||

≤ C(τ12m(Am

3 )12 + hk‖um−1‖k+1 + ‖ξm−1

0 ‖)|||ξm|||+ Ch2|||ξm−1||||||ξm|||,(4.13)

where Am3 =

∫ tm

tm−1

‖ut‖22 dt.

Combining the above estimates with the error equation (4.3) we arrive at:

1

2dt‖ξm0 ‖2 + τm

2‖dtξm0 ‖2 + γ2|||ξm|||2

≤ C(τ

12m(Am

3 )12 + hk‖um−1‖k+1 + hk‖ξm−1

0 ‖)|||ξm|||+ Cτ

12m(Am

1 )12 ‖ξm0 ‖

+Ch2|||ξm−1||||||ξm|||+ Cτ− 1

2m hk(Am

2 )12 ‖ξm0 ‖

≤ γ2

2|||ξm0 |||2 + 1

2‖ξm0 ‖2 + C

γ2‖ξm−1

0 ‖2 + C

γ2h2|||ξm−1|||2 + CτmAm

1

+C

γ2τmAm

3 +C

γ2h2k‖um−1‖2k+1 + Cτ−1

m h2kAm2 .

Multiplying both sides of the above inequality by 2τm, we obtain

(1− τm)‖ξm0 ‖2 + γ2τm|||ξm|||2(4.14)

≤ (1 +C

γ2τm)‖ξm−1

0 ‖2 + C

γ2τmh2|||ξm−1|||2 + Γm,

where

Γm =C

γ2

(τ2mAm

3 + τmh2k‖um−1‖2k+1

)+ Cτ2mAm

1 + Ch2kAm2 .

Denote αm =1+ C

γ2 τm

1−τm, C1 = 2( C

γ2 + 1), and C2 = C2γ2 . Then

eC1τm < αm < eC2τm for 0 < τm <1

4.




From the equation (4.14), we get

‖ξm0 ‖2 ≤ αm‖ξm−10 ‖2 + 2Γm +

C

γ2τmh2|||ξm−1|||2

≤ αm(αm−1‖ξm−20 ‖2 + 2Γm−1(4.15)

+C

γ2τm−1h

2|||ξm−2|||2) + 2Γm +C

γ2τmh2|||ξm−1|||2

≤m∏

n=1

αn‖ξ00‖2 + 2m∑

n=1

m∏j=n

αj

(Γn +

C

γ2τnh

2|||ξn−1|||2)

≤ eC2T

(‖ξ00‖2 + 2

m∑n=1

(Γn +

C

γ2τnh

2|||ξn−1|||2))

≤ CeC2T(τ2 + h2k

)+

C

γ2h2eC2T

m∑n=1

τn|||ξn|||2.

From the regularity assumptions,∑m

l=1Al1,

∑ml=1 A

l2, and

∑ml=1 A

l3 are all

bounded. Summing the equation (4.14) from 1 to m, we obtain

γ2m∑

n=1

τn|||ξn|||2 ≤ −‖ξm0 ‖2 + ‖ξ00‖2 +C

γ2

m∑n=1

τn‖ξn0 ‖2(4.16)

+C

γ2h2eC2T

m∑n=1

τn|||ξn|||2 +m∑

n=1

Γn

≤ CeC2T (τ2 + h2k) +C

γ4h2eC2T

m∑n=1

τn|||ξn|||2.

Combining (4.15) and (4.16) we have

‖ξm0 ‖2 + γ2m∑

n=1

τn|||ξn|||2 ≤ CeC2T (τ2 + h2k),(4.17)

provided that the meshsize h is sufficiently small.By setting hmin = minT∈Th

hT , from the inverse inequality and the error estimatein L2 we arrive at

‖umh ‖L∞ ≤ ‖um

h − Ehum‖L∞ + ‖um − Ehu

m‖L∞ + ‖um‖L∞

≤ Ch− 1

2

min (‖ξm0 ‖+ ‖ηm0 ‖) + ‖um‖L∞

≤ Ch− 1

2

min(τ + hk) + ‖um‖L∞ ,

when τ ≤ h12

min and h is sufficiently small, we have ‖umh ‖L∞ ≤ 2‖um‖L∞ + C0,

where C0 is a constant independent of h and τ . It follows that the range of the

numerical solution uh lies in [−L,L] when τ ≤ h12 and h is sufficiently small. On

the interval [−L,L], there exists a constant CL such that

|f(x)− f(y)| ≤ CL|x− y| ∀x, y ∈ [−L,L],

i.e., f is Lipschitz continuous on [−L,L].For k ≥ 3 and u ∈ H4(Ω), the corresponding error estimate can be derived

analogously by following a similar procedure. Details are thus omitted. �




5. Numerical results

In this section, we use four numerical examples to verify the theoretical resultsderived in previous sections.

5.1. P2-WG for the biharmonic equation. We solve the linear biharmonicproblem (3.1)–(3.2) with the constraint (3.3) being replaced by∫

Ω

u =4

π2.

The exact solution is chosen as

u(x, y) = cos(πx) cos(πy),(5.1)

defined in the unit square domain Ω = (0, 1)× (0, 1). The right-hand side functionf in equation (3.1) is calculated by matching the exact solution u.

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�

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��

�

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��

�

��

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�

Figure 5.1. The level 1, 2, and 3 uniform grids.

We apply the P2 weak Galerkin algorithm, i.e., k = 2 in the finite element spaceV 0h (2.1) and (2.2), and in the computation of the weak-Laplacian (2.4). The first

three levels of grid are depicted in Figure 5.1. The optimal order of convergenceof the method is confirmed by the numerical results shown as in Table 5.1 wherethe discrete norms | · |s1 and | · |s2 are induced by the two bilinear forms in s(·, ·),(2.5). It can be seen that the weak Galerkin finite element method does achievethe optimal order of convergence in the H2-like discrete norm. It is interestingto note that the numerical solutions seem to be super-convergent in the L2 norm.A theoretical investigation for such a superconvergence could be taken as a futureresearch topic.

Table 5.1. The errors, where eh = Qhu − uh, and the orders ofconvergence (n in the O(hn) error), by the P2 element (2.1) on theuniform grids (Figure 5.1) for (5.1).

level ‖Q0u− u0‖ n |||eh||| n |eh|s1 n |eh|s2 n1 2.17481 7.48116 9.99672 12.648852 0.23105 3.2 9.96075 0.0 5.07280 1.0 2.92707 2.13 0.01054 4.5 5.37375 0.9 1.02005 2.3 0.30465 3.34 0.00061 4.1 2.73367 1.0 0.14057 2.9 0.06932 2.15 0.00006 3.4 1.37275 1.0 0.02645 2.4 0.01859 1.9




5.2. P2-WG for the Cahn-Hilliard equation. In the present test, we solve thenonlinear Cahn-Hilliard equation (1.1) by using the P2 weak Galerkin finite elementmethod, where the exact solution is given by

u(x, y) = e−4π4t cos(πx) cos(πy).(5.2)

The function g in (1.1) is computed by matching the exact solution u with γ = 1.Table 5.1 illustrates the numerical errors on the fourth grid computed by usingthree time-step sizes:

τ = 0.002, 0.001 and 0.0005.

From Table 5.1, we see a first order of convergence in the time discretization:

‖Q0u− u0‖ = O(τ ), |||Qhu− uh||| = O(τ ).

Table 5.2. The errors, where eh = (Qhu − uh)(0.01), and theorders of convergence (n in the O(τn) error), by the P2 element(2.1) on the uniform grids (Figure 5.1) for (5.2).

τ ‖Q0u− u0‖ order n |||eh||| order n0.0020 0.02728 0.551730.0010 0.01365 1.00 0.29534 0.900.0005 0.00679 1.01 0.18049 0.71

5.3. P2-WG for the two-phase Cahn-Hilliard equation. In this numericalexperiment, we solve the nonlinear Cahn-Hilliard equation (1.1) by using the P2

weak Galerkin finite element method. The exact solution for this test case is notknown. This example is taken from [18] for which we have g = 0 in (1.1) andγ = 0.01. The initial condition u0 is plotted in Figure 5.2. The time step in ourcomputational is given by τ = 0.0001.

The WG solution on the grid level 5 (cf. Fig. 5.1) is plotted in the right figureof Figure 5.2. On the fifth grid, the number of unknowns is

2n2(6) + (2n(n+ 1) + n2)(3 + 2) = 7072, where n = 25−1.

At each time level, we use the conjugate gradient iterative algorithm to solve the im-plicit finite element equation. We also use some adaptive grids with hanging nodes(local refinement and coarsening after 100 time-steps) to compute the solution.Figure 5.3 illustrates the final grid (at time t = 0.3) after the local refinement. Wenote that the computation on adaptive grids is much more efficient than on uniformgrids, due to the sharp change of the solution near the interface.

5.4. P4-WG for the two-phase Cahn-Hilliard equation. In our final numeri-cal test, we solve the nonlinear Cahn-Hilliard equation (1.1) by using the P4 weakGalerkin finite element method. In this setting, the initial condition is shown inthe left diagram of Figure 5.4. This example is also taken from [18] with g = 0 in(1.1) and

γ = 0.01, and time-step τ = 0.0001.

As the initial condition is a smooth function, the process is expected to takea longer time to reach the steady-state of the solution. We first try to solve this




Figure 5.2. The WG initial solution u0(0) and the WG steady-state solution u0(0.3).

Figure 5.3. The final adaptive grid (at t = 0.3) for WG methodsolving problem of Figure 5.2.

problem by using P2-WG finite element method. But the computational results arenot satisfactory; the iteration often stalls at some discrete steady-state solutions inwhich the discrete interface appears to be rough and the discrete concentration forcef(u) seems to balance the smoothing force of the discrete biharmonic operator. Wethen moved to P4-WG finite element method on grid level 6; i.e., k = 4 in (2.1) and(2.4). On the grid level 6, the number of unknowns for P4-WG is

2n2(15) + (2n(n+ 1) + n2)(5 + 4) = 14880, where n = 26−1.




At each time level, we apply the conjugate gradient iterative scheme to solve theimplicit finite element equations. Figure 5.4 illustrates the numerical solutionsat three times levels. It shows that the numerical solutions take a longer time(than the last testing case) to reach its steady-state. Figure 5.5 demonstratesmore computational results at other time levels till the steady-state solution is wellapproached. This test shows the accuracy and stability of high order WG finiteelement methods in practical scientific computing.

Figure 5.4. The WG solution u0(0), and solutions u0(0.15), u0(0.3).

Figure 5.5. The WG solution u0(0.4) and u0(0.5).

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Division of Mathematical Sciences, National Science Foundation, Arlington, Vir-

ginia 22230

Email address: [email protected]

Department of Mathematics, Jilin University, Changchun, China


Department of Mathematics, Jilin University, Changchun, China


Department of Mathematical Sciences, University of Delaware, Newark, Delaware

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