
This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.
Citation: Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions[J]. Electronic Research Archive, 2021, 29(3): 2375-2389. doi: 10.3934/era.2020120
[1] | Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang . A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29(3): 2375-2389. doi: 10.3934/era.2020120 |
[2] | Chunmei Wang . Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes. Electronic Research Archive, 2025, 33(3): 1523-1540. doi: 10.3934/era.2025072 |
[3] | Leilei Wei, Xiaojing Wei, Bo Tang . Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation. Electronic Research Archive, 2022, 30(4): 1263-1281. doi: 10.3934/era.2022066 |
[4] | Guanrong Li, Yanping Chen, Yunqing Huang . A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28(2): 821-836. doi: 10.3934/era.2020042 |
[5] | Victor Ginting . An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, 2021, 29(5): 3405-3427. doi: 10.3934/era.2021045 |
[6] | Jun Pan, Yuelong Tang . Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365 |
[7] | Hongze Zhu, Chenguang Zhou, Nana Sun . A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise. Electronic Research Archive, 2022, 30(6): 2321-2334. doi: 10.3934/era.2022118 |
[8] | Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang . A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28(4): 1487-1501. doi: 10.3934/era.2020078 |
[9] | Suayip Toprakseven, Seza Dinibutun . A weak Galerkin finite element method for parabolic singularly perturbed convection-diffusion equations on layer-adapted meshes. Electronic Research Archive, 2024, 32(8): 5033-5066. doi: 10.3934/era.2024232 |
[10] | Bin Wang, Lin Mu . Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29(1): 1881-1895. doi: 10.3934/era.2020096 |
This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.
For simplicity, we consider Poisson equation with a Dirichlet boundary condition as our model problem.
−Δu=f,inΩ, | (1) |
u=g,on∂Ω, | (2) |
where
Using integration by parts, we can get the variational form: find
(∇u,∇v)=(f,v),∀v∈H10(Ω). | (3) |
Various finite element methods have been introduced to solve the Poisson equations (1)-(2), such as the Galerkin finite element methods (FEMs)[2, 3], the mixed FEMs [15] and the finite volume methods (FVMs) [6], etc. The FVMs emphasis on the local conservation property and discretize equations by asking the solution satisfying the flux conservation on a dual mesh consisting of control volumes. The mixed FEMs is another category method that based on the variable
The classical conforming finite element method obtains numerical approximate results by constructing a finite-dimensional subspace of
(∇uh,∇vh)=(f,vh),∀vh∈V0h, | (4) |
where
One obvious disadvantage of discontinuous finite element methods is their rather complex formulations which are often necessary to ensure connections of discontinuous solutions across element boundaries. For example, the IPDG methods add parameter depending interior penalty terms. Besides additional programming complexity, one often has difficulties in finding optimal values for the penalty parameters and corresponding efficient solvers. Most recently, Zhang and Ye [21] developed a discontinuous finite element method that has an ultra simple weak formulation on triangular/tetrahedal meshes. The corresponding numerical scheme can be written as: find
(∇wuh,∇wvh)=(f,vh),∀vh∈V0h, | (5) |
where
Following the work in [21, 22], we propose a new conforming DG finite element method on rectangular partitions in this work. It can be obtained from the conforming formulation simply by replacing
In this paper, we keep the same finite element space as DG method, replace the boundary function with the average of the inner function, and use the weak gradient arising from local Raviart-Thomas (RT) elements [5] to approximate the classic gradient. Moreover, the derivation process in this paper is based on rectangular RT elements [16]. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete
The rest of this paper is organized as follows: In Section 2, we shall present the conforming DG finite element scheme for the Poisson equation on rectangular partitions. Section 3 is devoted to a discussion of the stability and solvability of the new method. In Section 4, we shall prepare ourselves for error estimates by deriving some identities. Error estimates of optimal order in
Throughout this paper, we adopt the standard definition of Sobolev space
H10(Ω)={v∈H1(Ω):v|∂Ω=0}, |
and the space
H(div,Ω)={q∈[L2(Ω)]d:∇⋅q∈L2(Ω)}. |
Assume that the domain
For any interior edge
{v}=12(v|∂T1+v|∂T2),[[v]]=v|∂T1n1+v|∂T2n2, | (6) |
where
{v}=v|eand[[v]]=v|en. | (7) |
We define a discontinuous finite element space
Vh={v∈L2(Ω):v|T∈Qk(T),∀T∈Th}, | (8) |
and its subspace
V0h={v∈Vh:v=0on∂Ω}, | (9) |
where
Definition 2.1. For a given
(∇dv,q)T:=−(v,∇⋅q)T+⟨{v},q⋅n⟩∂T,∀q∈RTk(T), | (10) |
where
The weak gradient operator
(∇dv)|T=∇d(v|T). |
We introduce the following bilinear form:
a(v,w)=(∇dv,∇dw), |
the conforming DG algorithm to solve the problems (1) - (2) is given by
Conforming DG algorithm 1. Find
a(uh,vh)=(f,vh),∀vh∈V0h, | (11) |
where
We will prove the existence and uniqueness of the solution of equation (11). Firstly, we present the following two useful inequalities to derive the forthcoming analysis.
Lemma 3.1 (trace inequality). Let
‖φ‖2e≤C(h−1T‖φ‖2T+hT‖∇φ‖2T), | (12) |
where
Lemma 3.2 (inverse inequality). Let
‖∇φ‖T≤C(n)h−1T‖φ‖T,∀T∈Th. | (13) |
Then, we define the following semi-norms in the discontinuous finite element space
|||v|||2=a(v,v)=∑T∈Th‖∇dv‖2T, | (14) |
‖v‖21,h=∑T∈Th‖∇v‖2T+∑e∈E0hh−1e‖[[v]]‖2e. | (15) |
We have the equivalence between the semi-norms
Lemma 3.3. For any
C1‖v‖1,h≤|||v|||≤C2‖v‖1,h, | (16) |
where
Proof. It follows from the definition of
‖∇dv‖2T1=(∇dv,∇dv)T1=−(v,∇⋅∇dv)T1+⟨{v}n,∇dv⟩∂T1=(∇v,∇dv)T1−⟨(v−{v})n,∇dv⟩∂T1≤‖∇v‖T1‖∇dv‖T1+‖(v−{v})n‖∂T1‖∇dv‖∂T1≤‖∇dv‖T1(‖∇v‖T1+h−12T1‖(v−{v})n‖∂T1). | (17) |
For any
(v−{v})|en1=v|∂T1n1−12(v|∂T1+v|∂T2)n1=12(v|∂T1n1+v|∂T2n2)=12[[v]]e. |
Then we can get
‖(v−{v})n‖2∂T1≤12∑e∈∂T1‖[[v]]‖2e. | (18) |
Substituting (18) into (17) gives
‖∇dv‖2T1≤C2‖∇dv‖T1(‖∇v‖T1+∑e∈∂T1h−12e‖[[v]]‖e), |
this completes the proof of the right-hand of (16).
To prove the left-hand of (16), we consider the subspace of
D(k,T):={q∈RTk(T):q⋅n=0on∂T}. |
Note that
‖∇v‖T=supq∈D(k,T)(∇v,q)T‖q‖T. | (19) |
Using the integration by parts, Cauchy-Schwarz inequality, the definition of
(∇v,q)T=−(v,∇⋅q)T+⟨v,q⋅n⟩∂T=(∇dv,q)T−⟨{v},q⋅n⟩∂T=(∇dv,q)T≤‖∇dv‖T⋅‖q‖T, |
where we have used the fact that
‖∇v‖T≤‖∇dv‖T. | (20) |
We define the space
‖[[v]]‖e=supq∈De(k,T)⟨[[v]],q⋅n⟩e‖q⋅n‖e. | (21) |
Following the integration by parts and the definition of
(∇dv,q)T=(∇v,q)T−⟨v,q⋅n⟩e+⟨{v},q⋅n⟩e. |
Together with (20), we obtain
|⟨[[v]],q⋅n⟩e|=2|(∇dv,q)T−(∇v,q)T|≤2|(∇dv,q)T|+2|(∇v,q)T|≤C(‖∇dv‖T‖q‖T+‖∇v‖T‖q‖T)≤C‖∇dv‖T‖q‖T. |
Substituting the above inequality into (21), by the scaling argument [13], for such
‖[[v]]‖e≤C‖∇dv‖T‖q‖T‖q⋅n‖e≤Ch12‖∇dv‖T. | (22) |
Combining (20) and (22) gives a proof of the left-hand of (16).
Lemma 3.4. The semi-norm
Proof. We shall only verify the positivity property for
The above two lemmas imply the well posedness of the scheme (11). We prove the existence and uniqueness of solution of the conforming DG method in Theorem 3.1.
Theorem 3.1. The conforming DG scheme (11) has and only has one solution.
Proof. To prove the scheme (11) is uniquely solvable, it suffices to verify that the homogeneous equation has zero as its unique solution. To this end, let
a(uh,uh)=0, |
which leads to
In this section, we will derive an error equation which will be used for the error estimates. For any
(∇⋅q,v)T=(∇⋅Πhq,v)T,∀v∈Qk(T). | (23) |
For any
‖Πh(∇w)−∇w‖≤Chk‖w‖1+k. | (24) |
Moreover, it is easy to verify the following property holds true.
Lemma 4.1. For any
∑T∈Th(−∇⋅q,v)T=∑T∈Th(Πhq,∇dv)T,∀v∈V0h. | (25) |
Proof.
∑T∈Th⟨{v},Πhq⋅n⟩∂T=0. | (26) |
By the definition of
∑T∈Th(−∇⋅q,v)T=∑T∈Th(−∇⋅Πhq,v)T=∑T∈Th(−∇⋅Πhq,v)T+∑T∈Th⟨{v},Πhq⋅n⟩∂T=∑T∈Th(Πhq,∇dv)T. |
This completes the proof of the lemma.
Before establishing the error equation, we define a continuous finite element subspace of
˜Vh={v∈H1(Ω):v|T∈Qk(T),∀T∈Th}. | (27) |
so as a subspace of
˜V0h:={v∈˜Vh:v|∂Ω=0}. | (28) |
Lemma 4.2. For any
∇dv=∇v. |
Proof. By the definition of
(∇dv,q)T=−(v,∇⋅q)T+⟨{v},q⋅n⟩∂T=−(v,∇⋅q)T+⟨v,q⋅n⟩∂T=(∇v,q)T, |
which gives
(∇dv−∇v,q)T=0,∀q∈RTk(T). |
Letting
Let
‖Ihu−u‖≤Chk+1‖u‖k+1, | (29) |
‖∇Ihu−∇u‖≤Chk‖u‖k+1. | (30) |
It is obvious that
Lemma 4.3. Denote
a(eh,vh)=lu(vh), | (31) |
where
lu(vh)=∑T∈Th(∇Ihu−Πh∇u,∇dvh). | (32) |
Proof. Since
∑T∈Th(∇dIhu,∇dvh)T=∑T∈Th(∇Ihu,∇dvh)T=∑T∈Th(∇Ihu−Πh∇u+Πh∇u,∇dvh)T=∑T∈Th(∇Ihu−Πh∇u,∇dvh)T+∑T∈Th(Πh∇u,∇dvh)T=lu(vh)−∑T∈Th(∇⋅∇u,vh)T=lu(vh)+(f,vh). |
By the definition of the scheme (11), we have
∑T∈Th(∇dIhu−∇duh,∇dvh)T=lu(vh). |
This completes the proof of the lemma.
The goal of this section is to derive the error estimates in
Theorem 5.1. Let
|||eh|||≤Chk|u|k+1. | (33) |
Proof. Letting
|||eh|||2=lu(eh). | (34) |
From the Cauchy-Schwarz inequality, the triangle inequality, the definition of
lu(vh)=∑T∈Th(∇Ihu−Πh(∇u),∇dvh)T≤∑T∈Th‖∇Ihu−Πh(∇u)‖T‖∇dvh‖T≤(∑T∈Th‖∇Ihu−Πh(∇u)‖2T)12(∑T∈Th‖∇dvh‖2T)12=(∑T∈Th‖∇Ihu−∇u+∇u−Πh(∇u)‖2T)12|||vh|||≤(∑T∈Th‖∇Ihu−∇u‖2T+‖∇u−Πh(∇u)‖2T)12|||vh|||≤Chk|u|k+1|||vh|||. |
Then, we have
lu(eh)≤Chk|u|k+1|||eh|||. | (35) |
Substituting (35) to (34), we obtain
|||eh|||2≤Chk|u|k+1|||eh|||, |
which completes the proof of the lemma.
It is obvious that
(∇˜uh,∇v)=(f,v),∀v∈˜V0h. | (36) |
For any
(∇duh−∇˜uh,∇v)=0,∀v∈˜V0h. | (37) |
In the rest of this section, we derive an optimal order error estimate for the conforming DG approximation (11) in
−∇⋅(∇Φ)=uh−˜uh,inΩ. | (38) |
Assume that the dual problem satisfies
‖Φ‖2≤C‖uh−˜uh‖. | (39) |
In the following of this paper, we note
Theorem 5.2. Assume
‖u−uh‖≤Chk+1|u|k+1. | (40) |
Proof. First, we shall derive the optimal order for
a(Φh,v)=(εh,v),∀v∈V0h. | (41) |
Since
(∇duh−∇˜uh,∇IhΦ)=0,∇dIhΦ=∇IhΦ, |
which gives
(∇duh−∇˜uh,∇dIhΦ)=0. | (42) |
Setting
‖εh‖2=a(Φh,εh)=∑T∈Th(∇dΦh,∇dεh)T=∑T∈Th(∇d(Φh−IhΦ),∇duh−∇˜uh)T≤|||Φh−IhΦ|||(|||uh−Ihu|||+‖∇(Ihu−˜uh)‖). |
Then, by the Cauchy-Schwarz inequality, (33) and (39), we obtain
‖εh‖2≤Ch|Φ|2hk|u|k+1≤Chk+1|u|k+1‖εh‖, |
which gives
‖εh‖≤Chk+1|u|k+1. | (43) |
Combining the error estimate of finite element solution, the triangle inequality and (43) yields (40), which completes the proof of the theorem.
In this section, we shall present some numerical results for the conforming discontinuous Galerkin method analyzed in the previous sections.
We solve the following Poisson equation on the unit square domain
−Δu=2π2sin(πx)sin(πy)in Ω | (44) |
u=0on ∂Ω. | (45) |
The exact solution of the above problem is
We first use the
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
The same test case is also computed using the
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
To test the superconvergence of
−Δu+u=fin Ωu=0on ∂Ω, |
where
u=(x−x2)(y−y3). | (46) |
Uniform square grids as shown in Figure 1 are used for numerical computation. The numerical results are listed in Table 3. Surprising, for this problem, the
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
To test further the superconvergence of
−∇(a∇u)=fin Ωu=0on ∂Ω, |
where
u=(x−x3)(y2−y3). | (47) |
Uniform square grids as shown in Figure 1 are used for computation. The numerical results are listed in Table 4. Surprising, again, the
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |
In this paper, we establish a new numerical approximation scheme based on the rectangular partition to solve second order elliptic equation. We derived the numerical scheme and then proved the optimal order of convergence of the error estimates in
[1] |
Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. (2001/02) 39: 1749-1779. ![]() |
[2] |
The finite element method with penalty. Math. Comp. (1973) 27: 221-228. ![]() |
[3] |
Finite element methods for elliptic equations using nonconforming elements. Math. Comp. (1977) 31: 45-59. ![]() |
[4] |
Basic principles of virtual element methods. Math. Models Methods Appl. Sci. (2013) 23: 199-214. ![]() |
[5] | On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO (1974) 8: 129-151. |
[6] |
A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions. Numer. Math. (1999) 82: 409-432. ![]() |
[7] |
An HDG method for distributed control of convection diffusion PDEs. J. Comput. Appl. Math. (2018) 343: 643-661. ![]() |
[8] |
Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations. J. Sci. Comput. (2014) 59: 129-157. ![]() |
[9] |
Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs. J. Sci. Comput. (2018) 74: 1651-1676. ![]() |
[10] |
The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. J. Comput. Phys. (2011) 230: 305-328. ![]() |
[11] |
Simplified weak Galerkin and new finite difference schemes for the Stokes equation. J. Comput. Appl. Math. (2019) 361: 176-206. ![]() |
[12] | Y. Liu and J. Wang, A locking-free P0 finite element method for linear elasticity equations on polytopal partitions, preprint, arXiv: 1911.08728, 2019. |
[13] |
L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, Numerical Solution of Partial Differential Equations: Theory, Algorithms, and their Applications, in: Springer Proceedings in Mathematics and Statistics, 45 (2013), 247-277. doi: 10.1007/978-1-4614-7172-1_13
![]() |
[14] |
A ![]() |
[15] | P.-A. Raviart and J. M. Thomas, A Mixed Finite Element Method for Second Order Elliptic Problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, in: Lecture Notes in Math., vol. 606, Springer-Verlag, New York, 1977. Technical Report LA-UR-73-0479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973. |
[16] |
M. Stynes, Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements, Math. Comp., 83 (2014), 2675-2689. doi: 10.1090/S0025-5718-2014-02826-3
![]() |
[17] |
A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J. Comput. Appl. Math. (2016) 307: 346-366. ![]() |
[18] |
A weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. (2016) 302: 171-185. ![]() |
[19] |
A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. (2013) 241: 103-115. ![]() |
[20] |
A Weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comp. (2014) 83: 2101-2126. ![]() |
[21] | X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J. Numer. Anal. and Model., 17 (2020), 110-117. arXiv: 1904.03331. |
[22] | X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part Ⅱ, arXiv: 1907.01397. |
[23] | Weak Galerkin finite element method for second order parabolic equations. Int. J. Numer. Anal. Model. (2016) 13: 525-544. |
1. | Xiu Ye, Shangyou Zhang, A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes, 2022, 178, 01689274, 155, 10.1016/j.apnum.2022.03.017 | |
2. | Jun Zhou, Da Xu, Wenlin Qiu, Leijie Qiao, An accurate, robust, and efficient weak Galerkin finite element scheme with graded meshes for the time-fractional quasi-linear diffusion equation, 2022, 124, 08981221, 188, 10.1016/j.camwa.2022.08.022 | |
3. | Xiu Ye, Shangyou Zhang, A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes, 2021, 93, 0271-2091, 1913, 10.1002/fld.4959 | |
4. | Xiu Ye, Shangyou Zhang, Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes, 2023, 2096-6385, 10.1007/s42967-023-00330-5 | |
5. | Yan Yang, Xiu Ye, Shangyou Zhang, A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids, 2024, 32, 2688-1594, 3413, 10.3934/era.2024158 | |
6. | Xiu Ye, Shangyou Zhang, A superconvergent CDG finite element for the Poisson equation on polytopal meshes, 2023, 0044-2267, 10.1002/zamm.202300521 | |
7. | Xiu Ye, Shangyou Zhang, Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes, 2024, 2096-6385, 10.1007/s42967-024-00444-4 | |
8. | Xiu Ye, Shangyou Zhang, Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations, 2024, 50, 1019-7168, 10.1007/s10444-023-10100-9 | |
9. | Fuchang Huo, Weilong Mo, Yulin Zhang, A locking-free conforming discontinuous Galerkin finite element method for linear elasticity problems, 2025, 465, 03770427, 116582, 10.1016/j.cam.2025.116582 |
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |