
This research aims to propose a pedagogical model that structures the implementation of digital game-based learning (DGBL) in science classes. Design-based research guided the design, development, implementation, and redesign processes of the prototype pedagogical model. The principles that informed the design of the model were gleaned from empirical data on DGBL conditions in junior high schools and science teachers' DGBL practices and perceived barriers to implementing DGBL. Curriculum and science education experts reviewed the model and found it usable, adoptable, implementable, and appropriate for junior high school science classes. The lesson designed based on the model improved junior high school students' motivation to learn and achievement in science. Likewise, science teachers perceived the pedagogical model to be easy to use, useful in science teaching, beneficial for students, and able to enhance their teaching efficiency and productivity. This study is the first to propose a DGBL pedagogical model for science.
Citation: Jun Karren V. Caparoso, Antriman V. Orleans. DiGIBST: An inquiry-based digital game-based learning pedagogical model for science teaching[J]. STEM Education, 2024, 4(3): 282-298. doi: 10.3934/steme.2024017
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This research aims to propose a pedagogical model that structures the implementation of digital game-based learning (DGBL) in science classes. Design-based research guided the design, development, implementation, and redesign processes of the prototype pedagogical model. The principles that informed the design of the model were gleaned from empirical data on DGBL conditions in junior high schools and science teachers' DGBL practices and perceived barriers to implementing DGBL. Curriculum and science education experts reviewed the model and found it usable, adoptable, implementable, and appropriate for junior high school science classes. The lesson designed based on the model improved junior high school students' motivation to learn and achievement in science. Likewise, science teachers perceived the pedagogical model to be easy to use, useful in science teaching, beneficial for students, and able to enhance their teaching efficiency and productivity. This study is the first to propose a DGBL pedagogical model for science.
The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It uses the subdivision of the whole domain into simpler parts, called finite elements, and variational methods from the calculus of variations to solve the problem by minimizing an associated error function. The discontinuous Galerkin finite element method is a variant of the classical finite element method. Its main difference with classical finite element methods is the continuity of the solution across element interfaces. The discontinuous Galerkin (DG) method does not require the continuity of the solution along edges. Since this leads to ambiguities at element interfaces, the technique from finite volume methodology (FVM), namely the choice of numerical fluxes, has been introduced. From this point of view, DG methods combine features of the finite element methods and finite volume methods. Thus DG methods have several advantages. For instance, DG methods are highly parallelizable and very well suited for handling adaptive strategies. However, DG methods have more degrees of freedom than classical finite element methods.
On the other hand, the solution of the convection dominated problem has typically singularity and to resolve it one requires very fine meshes in the domain or very high order polynomials in the approximate spaces, which produces very large degrees of freedom especially in the DG method.
Over the decades, several variants of DG method such as hybridizable DG (HDG), DG with Lagrange multiplier (DGLM), multiscale DG (MDG), have been developed to reduce the degrees of freedom of DG. Concerning MDG, it was first introduced by Hughes et. al. and investigated for advection-diffusion equation [3,6]. They have introduced extra streamline diffusion term in the setting of the MDG to treat the advection term. In [2], MDG was also introduced for elliptic problem without theoretical analysis. In [10], one of the authors has presented the MDG for convection-diffusion-reaction equations without artificial viscosity.
Relating other multiscale methods for convection diffusion equations, we refer the readers to, for example, [4] and the references cited therein.
MDG has the computational structure of the continuous Galerkin (CG) method based on the variational multiscale idea (see [2]), which is indeed a DG method. Storage and computational efforts are reduced significantly in high order approximation.
In this paper, we study computational aspects of the MDG [10]. Especially, we investigate the matrix structure of the MDG. The MDG solution is obtained by composition of the DG and the inter-scale operator. We show that the composition results to the matrix product of the DG matrix and the inter-scale matrix corresponding to the local problem on the element. We apply ILU preconditioned GMRES to effectively solve the resulting global system.
This paper is organized as follows. In Section 2, we introduce the model problem called the convection-diffusion-reaction equation. In Section 3, we introduce finite element spaces for the MDG method. In Section 4, we define the parameters and describe the MDG method for the model problem. In Section 5, we form the computational structure of the MDG. Finally, in Section 6, we show the numerical results with convection dominated problems.
Let
Lu≡−∇⋅(A(x)∇u)+b(x)⋅∇u+c(x)u=f(x), | (2.1) |
where
ζTA(x)ζ>0∀ζ∈Rma.e. x∈¯Ω. | (2.2) |
By
∂Ωo={x∈∂Ω:→n(x)TA(x)→n(x)>0},∂Ω−={x∈∂Ω∖∂Ωo:b(x)⋅→n(x)<0},∂Ω+={x∈∂Ω∖∂Ωo:b(x)⋅→n(x)≥0}. | (2.3) |
The sets
u=gDon ∂ΩD∪∂Ω−,(A∇u)⋅→n=gNon ∂ΩN, | (2.4) |
and adopt the (physically reasonable) hypothesis that
In this section, we introduce the finite element spaces for the MDG method. We recall the space
Let
In consistent DG methods, the solution values are coupled by generalized flux functions across the edges and they appear by the jumps and averages. Let
{ϕ}=12(ϕ1+ϕ2),[[ϕ]]=ϕ1→n1+ϕ2→n2on e∈Eih. |
For a vector valued function
{→τ}=12(→τ1+→τ2),[[→τ]]=→τ1⋅→n1+→τ2⋅→n2on e∈Eih. |
Notice that the jump
[[ϕ]]=ϕ→n,{→τ}=→τ. |
We do not require either of the quantities
We now assign to
Hp(Th)={w∈L2(Ω):w∣E∈Hp(E),∀E∈Th}, |
equipped with the broken Sobolev norm
¯Wh(Ω)={¯v∈H1(Ω):¯v∣E∈Pr(E), ∀E∈Th} |
and then associate with it the discontinuous approximation spaces:
Wh(Ω)={v∈L2(Ω):v∣E∈Pr(E), ∀E∈Th},Wh(E)={μ∈L2(E):μ∈Pr(E), ∀E∈Th}. |
In this section, we introduce the MDG method (see [6,10]). To do that, we start with a DG method for (2.1) and (2.4) given as follows: Find
B(uh,v)=L(gD,gN,f;v), ∀v∈Wh(Ω), | (4.1) |
where
B(uh,v)=∑E∈ThBE(uh,v)+∑e∈EohBoe(uh,v)+∑e∈EihBie(˜uh,˜v), | (4.2) |
where
Now, to define the local problem of the MDG, we decompose
uh=¯uh+u′h. |
Then, (4.1) takes the below forms:
Coarse scale equation:
B(¯uh,¯v)+B(u′h,¯v)=L(gD,gN,f;¯v),∀¯v∈¯Wh(Ω); | (4.3) |
Fine scale equation:
B(u′h,v′)+B(¯uh,v′)=L(gD,gN,f;v′),∀v′∈Wh(Ω). | (4.4) |
By treating the function
B(u′h,v′)=L(gD,gN,f;v′)−B(¯uh,v′),∀v′∈Wh(Ω). | (4.5) |
Take
BE(u′h,v′)+∑e⊆EohBoe(u′h,v′)+∑e⊆EihBie(˜u′h,˜v′)= L(gD,gN,f;v′) −(BE(¯uh,v′)+∑e⊆EohBoe(¯uh,v′)+∑e⊆EihBie(˜¯uh,˜v′)),∀u′h∈Wh(E). | (4.6) |
(4.6) relates fine scales to the coarse scales, but remains coupled to the continuous elements through the numerical flux terms in (4.6). MDG defines the local problem in a way that the fine scales are expressed in terms of the coarse scales that will uncouple (4.6) on inner boundaries.
We now note that, for
[[τ]]=τi→ni+τo→no=(τi−τo)→ni,uoh=g=¯uhif ∂E∩∂Ω=∅,¯uh=¯uoh=¯uihon ∂E, |
where
Bie(˜u′h,˜v′)=Bie(⟨[[u′h]],[[A∇u′h]],b⋅[[u′h]],{u′h},{A∇u′h}⟩,⟨v′→n,(A∇v′)⋅→n,bv′⋅→n,v′2,A∇v′2⟩). | (4.7) |
Similarly, since
Bie(˜¯uh,˜v′)=Bie(⟨[[¯uh]],[[A∇¯uh]],b⋅[[¯uh]],{¯uh},{A∇¯uh}⟩,⟨v′→n,(A∇v′)⋅→n,bv′⋅→n,v′2,A∇v′2⟩), |
we have that
BE(¯uh,v′)+∑e⊆EohBoe(¯uh,v′)+∑e⊆EihBie(˜¯uh,˜v′)≡0,if v′∈I(¯Wh,0). | (4.8) |
Substituting (4.7)-(4.8) into (4.6), local problem of the MDG is ended up as follows:
bE(u′h,v′)=lE(¯uh,f;v′),∀v′∈Wh(E),∀E∈Th, | (4.9) |
where
bE(u′h,v′)= BE(u′h,v′)+∑e⊆EohBoe(u′h,v′)+∑e⊆EihBie(⟨(u′h)i→n,(A∇u′h)i⋅→n, (bu′h)i⋅→n,(u′h)i2,(A∇u′h)i2⟩,⟨v′→n,(A∇v′)⋅→n,bv′⋅→n,v′2,A∇v′2⟩),lE(¯uh,f;v′)= L(gD,gN,f;v′)+∑e⊆EihBie(⟨¯uh→n,(A∇¯uh)⋅→n,b¯uh⋅→n, −¯uh2,−A∇¯uh2⟩,⟨v′→n,(A∇v′)⋅→n,bv′⋅→n,v′2,A∇v′2⟩). |
We note that (4.9) is a DG formulation for the local problem
We now denote by
I(¯Wh,f)={I(¯uh,f)∣¯uh∈¯Wh(Ω)} | (4.10) |
and
I(¯Wh,0)={I(¯uh,0)∣¯uh∈¯Wh(Ω)}. | (4.11) |
Global MDG method is then given to find
B(uMDGh,v)=L(gD,gN,f;v),∀v∈I(¯Wh,0). | (4.12) |
Remark. Concerning the analysis of the stability and the error estimates of the MDG method, we refer the reader to [10].
In this section we form the matrix equation of the MDG method. To make the presentation simple, we consider the case of standard NIPG (Nonsymmetric Interior Penalty Galerkin) DG method [1,10].
Let
{¯ϕEki}¯Nki=1is a subset of{ϕEki}Nki=1, | (5.1) |
where
By following the process of the previous section, one can obtain the local problem (4.9), which is now given as follows (see [7,10]): Find
bE(uEh,v′h)=b∂E(¯uh;v′h)+FE(v′h), | (5.2) |
where
bE(uEh,v′h)= ∫E(A∇uEh−buEh)⋅∇v′hdx+∫EcuEhv′hdx+∫∂+EuEhv′hb⋅→nds −12∫∂E(A∇uEhv′h−A∇v′huEh)⋅→nds, b∂E(¯uh;v′h)= −∫∂−E¯uhv′hb⋅→nds+12∫∂E(A∇¯uhv′h+A∇v′h¯uh)⋅→nds,FE(v′h)= ∫Efv′hdx. | (5.3) |
Now, let
uMDGh=L∑k=1χEkuEkh=L∑k=1χEkNk∑i=1UkiϕEki,¯uh=N∑r=1¯Ur¯ϕr,fEkh=Nk∑j=1FEkjϕEj, | (5.4) |
where
Now, let
Uk=˜Sk¯Uk | (5.5) |
where
Now, let
Uk=˜Sk¯Uk=Sk¯U. | (5.6) |
Noting
˘ϕEki=¯ϕEki=¯ϕifori=1,⋯,¯NkonEk,(˘Sk¯U)i=(˘Sk)i¯U=0,ifi>¯Nk, | (5.7) |
and from (5.4), we express
uMDGh=L∑k=1χEkNk∑i=1UkiϕEki=L∑k=1χEk¯Nk∑i=1(˜Sk¯Uk)i¯ϕEki=L∑k=1χEk¯Nk∑i=1(Sk¯U)i¯ϕEki,=L∑k=1χEk¯Nk∑i=1(Sk)i¯U¯ϕEki,=L∑k=1N∑i=1(˘Sk)i¯U˘ϕEki. | (5.8) |
Similarly, we consider
vh=χElNl∑j=1ϕElj=χEl¯Nl∑j=1(˜Sl¯Il)j¯ϕElj=χEl¯Nl∑j=1(Sl¯I)j¯ϕElj=χEl¯Nl∑j=1(Sl)j¯I¯ϕElj=N∑j=1(˘Sl)j¯I˘ϕElj, | (5.9) |
where
B(uMDGh,vh)=B(L∑k=1N∑i=1(˘Sk)i¯U˘ϕEki,N∑j=1(˘Sl)j¯I˘ϕElj)=L∑k=1B(N∑i=1(˘Sk)i¯U˘ϕEki,N∑j=1(˘Sl)j¯I˘ϕElj)=L∑k=1N∑i,j=1((˘Sl)j¯I)TB(˘ϕElj,˘ϕEki)(˘Sk)i¯U. | (5.10) |
Let
(Al,k)i,j=B(ϕEki,ϕElj), | (5.11) |
and
˜Al,k={Al,k,onEkandEl,0,otherwise. |
We then by noting (5.7), that (5.10) is rewritten as follows:
B(uMDGh,vh)=L∑k=1N∑i,j=1((˘Sl)i¯I)T(˜Al,k)ij(˘Sk)j¯U=(L∑k=1N∑i,j=1((˘Sl)i¯I)T(˜Al,k)ij(˘Sk)j)¯U. | (5.12) |
Here we have used the fact that
L∑k=1N∑i,j=1((˘Sl)i¯I)T(˜Al,k)ij(˘Sk)j=L∑k=1N∑i,j=1((Sl)i¯I)T(Al,k)ij(Sk)j=L∑k=1N∑i,j=1((˜Sl)i¯I)T(Al,k)ij(˜Sk)j. | (5.13) |
Considering
AMDG=L∑k,l=1(˜Sl)TAl,k˜Sk. | (5.14) |
Remark. We see, by (5.13), that
AMDG=STAS | (5.15) |
where
S=[S1S2⋮SL]andA=[A1,1A1,2⋯A1,LA2,1A2,2⋯A2,L⋮⋱⋮AL,1AL,2⋯AL,L]. | (5.16) |
Here
Remark. We further see, from (5.14)-(5.15), that
STAS=L∑k,l=1(˜Sl)TAk,l˜Sk. | (5.17) |
In the next section, we compute the MDG solution in the following way:
● construct
● construct
● calculate
In this section, we test the MDG with convection dominated problems. When the convection strongly dominates the diffusion (
Err=||u(⋅,⋅)−uh(⋅,⋅)||L2,Conv=logErr(h)Err(h/2)log2, |
where
We solve the problem (2.1) and (2.4) in the case of the diffusion coefficient
We take
u(x,y)=16x(1−x)y(1−y)(12+arctan[2s(x,y)/√k]π), |
where
s(x,y)=142−(x−12)2−(y−12)2. |
The graph of the exact solution and the approximate solution of DG with uniform mesh
Degree of freedom | Convergence order | Degree order | ||
1/32 | 6,144 | 8.66085e-002 | 1 | |
1/64 | 24,576 | 1.28362e-002 | 2.7543 | 1 |
1/128 | 98,304 | 1.77764e-003 | 2.8522 | 1 |
1/32 | 30,720 | 3.22321e-003 | 4 | |
(a) Using DG method | ||||
Degree of freedom | Convergence order | Degree order | ||
1/32 | 1,089 | 8.30570e-002 | 1 | |
1/64 | 4,225 | 1.19648e-002 | 2.7953 | 1 |
1/128 | 16,641 | 1.87018e-003 | 2.6775 | 1 |
1/32 | 10,497 | 3.21210e-003 | 4 | |
1/64 | 41,473 | 5.98660e-004 | 4 | |
(b) Using MDG method |
In this subsection, we consider the case of
We take
In this test, we have also applied the ILU preconditioner to the matrix equation. In Tables 2-3, we compare the results with and without ILU preconditioning for DG and MDG, respectively. We see that ILU preconditioned GMRES effectively solve the large system. As seen in Tables 2-3, DG requires degrees-of-freedom five times more than the ones of MDG to obtain the qualitatively similar solution. CPU time of the MDG solution was also significantly reduced.
Total element num | Degree of freedom | Convergence order | Degree | ||
1/64 | 8,192 | 24,576 | 6.6217e–001 | 1 | |
1/128 | 32,768 | 98,304 | 3.1138e–001 | 1.0885 | 1 |
1/256 | 131,072 | 393,216 | 9.9196e–002 | 1.6503 | 1 |
1/512 | 524,288 | 1,572,864 | 2.1732e–002 | 2.1911 | 1 |
1/1024 | 2,097,152 | 6,291,456 | 1 | ||
1/256 | 131,072 | 1,966,080 | 6.4686e–003 | 4 | |
(a) DG solution | |||||
h | Elapsed time | GMRES iter(O/I) | Elapsed time | PGMRES Iter(O/I) | Degree |
1/64 | 8.2306e+001 | 1/208 | 7.2131e+000 | 1/4 | 1 |
1/128 | 1.4060e+003 | 3/201 | 6.1875e+001 | 1/4 | 1 |
1/256 | 2.5076e+004 | 10/220 | 1.4706e+003 | 1/4 | 1 |
1/512 | 2.1915e+004 | 1/4 | 1 | ||
1/1024 | 1 | ||||
1/256 | 4.9732e+005 | 10/256 | 3.1762e+004 | 1/10 | 4 |
(b) Comparison of GMRES with/without ILU for the DG in (a) |
Total element num | Degree of freedom | convergence order | Degree | ||
1/64 | 8,192 | 4,225 | 6.5406e–001 | 1 | |
1/128 | 32,768 | 16,641 | 3.0008e–001 | 1.1241 | 1 |
1/256 | 131,072 | 66,049 | 9.3895e–002 | 1.6762 | 1 |
1/512 | 524,288 | 263,169 | 2.1614e–002 | 2.1191 | 1 |
1/1024 | 2,097,152 | 1,050,625 | 6.0343e–003 | 1.8417 | 1 |
1/256 | 131,072 | 657,409 | 6.3296e–003 | 4 | |
(a) MDG solution | |||||
h | Elapsed time | GMRES iter(O/I) | Elapsed time | PGMRES iter(O/I) | Degree |
1/64 | 1.2899e+001 | 1/160 | 5.8968e+000 | 1/9 | 1 |
1/128 | 1.1022e+002 | 1/246 | 4.4625e+001 | 1/12 | 1 |
1/256 | 2.4963e+003 | 3/189 | 2.1713e+002 | 1/14 | 1 |
1/512 | 3.0008e+004 | 5/125 | 2.7705e+003 | 1/18 | 1 |
1/1024 | 2.5316e+005 | 1/20 | 1 | ||
1/256 | 6.5098e+004 | 10/256 | 4.8902e+003 | 1/25 | 4 |
(b) Comparison of GMRES with/without ILU for the MDG in (a) |
If the convection strongly dominates, then the linear approximation is not efficient to resolve the spurious oscillations. Figure 9 shows the graphs of the solutions of MDG with
In this subsection, we apply high order approximation only in the region where detailed information is needed. By observing the oscillations occur along the convected direction, we simply apply high order polynomial in the convected direction. We show that MDG is very flexible in increasing the polynomial degree
As seen in the previous tests, oscillation starts to develop at the inner layer and propagates to the gray area in Figure 8. We apply different orders of polynomials in different regions (see [5,8,10,13]). We apply
Matrix structure and
We compare the result with the one of DG with
Ele. num. | Basis num. | Elapsed time | Conv. | Iter.(O/I) | Deg. | ||
1/64 | 8,192 | 4,225 | 1.0632e+001 | 2.0077e–001 | 1/9 | 1 | |
1/128 | 32,768 | 16,641 | 7.3629e+001 | 8.1290e–002 | 1.3044 | 1/11 | 1 |
1/256 | 131,072 | 66,049 | 2.2198e+002 | 2.8772e–002 | 1.4984 | 1/13 | 1 |
1/512 | 524,288 | 263,169 | 2.3875e+003 | 6.5147e–003 | 2.1429 | 1/16 | 1 |
Using mixed polynomials (P1 and P2 elements) |
As a conclusion, the MDG method numerically shows an advantage in adaptive
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1. | Mi-Young Kim, Dongwook Shin, An edgewise iterative scheme for the discontinuous Galerkin method with Lagrange multiplier for Poisson’s equation, 2025, 1017-1398, 10.1007/s11075-025-02017-9 |
Degree of freedom | Convergence order | Degree order | ||
1/32 | 6,144 | 8.66085e-002 | 1 | |
1/64 | 24,576 | 1.28362e-002 | 2.7543 | 1 |
1/128 | 98,304 | 1.77764e-003 | 2.8522 | 1 |
1/32 | 30,720 | 3.22321e-003 | 4 | |
(a) Using DG method | ||||
Degree of freedom | Convergence order | Degree order | ||
1/32 | 1,089 | 8.30570e-002 | 1 | |
1/64 | 4,225 | 1.19648e-002 | 2.7953 | 1 |
1/128 | 16,641 | 1.87018e-003 | 2.6775 | 1 |
1/32 | 10,497 | 3.21210e-003 | 4 | |
1/64 | 41,473 | 5.98660e-004 | 4 | |
(b) Using MDG method |
Total element num | Degree of freedom | Convergence order | Degree | ||
1/64 | 8,192 | 24,576 | 6.6217e–001 | 1 | |
1/128 | 32,768 | 98,304 | 3.1138e–001 | 1.0885 | 1 |
1/256 | 131,072 | 393,216 | 9.9196e–002 | 1.6503 | 1 |
1/512 | 524,288 | 1,572,864 | 2.1732e–002 | 2.1911 | 1 |
1/1024 | 2,097,152 | 6,291,456 | 1 | ||
1/256 | 131,072 | 1,966,080 | 6.4686e–003 | 4 | |
(a) DG solution | |||||
h | Elapsed time | GMRES iter(O/I) | Elapsed time | PGMRES Iter(O/I) | Degree |
1/64 | 8.2306e+001 | 1/208 | 7.2131e+000 | 1/4 | 1 |
1/128 | 1.4060e+003 | 3/201 | 6.1875e+001 | 1/4 | 1 |
1/256 | 2.5076e+004 | 10/220 | 1.4706e+003 | 1/4 | 1 |
1/512 | 2.1915e+004 | 1/4 | 1 | ||
1/1024 | 1 | ||||
1/256 | 4.9732e+005 | 10/256 | 3.1762e+004 | 1/10 | 4 |
(b) Comparison of GMRES with/without ILU for the DG in (a) |
Total element num | Degree of freedom | convergence order | Degree | ||
1/64 | 8,192 | 4,225 | 6.5406e–001 | 1 | |
1/128 | 32,768 | 16,641 | 3.0008e–001 | 1.1241 | 1 |
1/256 | 131,072 | 66,049 | 9.3895e–002 | 1.6762 | 1 |
1/512 | 524,288 | 263,169 | 2.1614e–002 | 2.1191 | 1 |
1/1024 | 2,097,152 | 1,050,625 | 6.0343e–003 | 1.8417 | 1 |
1/256 | 131,072 | 657,409 | 6.3296e–003 | 4 | |
(a) MDG solution | |||||
h | Elapsed time | GMRES iter(O/I) | Elapsed time | PGMRES iter(O/I) | Degree |
1/64 | 1.2899e+001 | 1/160 | 5.8968e+000 | 1/9 | 1 |
1/128 | 1.1022e+002 | 1/246 | 4.4625e+001 | 1/12 | 1 |
1/256 | 2.4963e+003 | 3/189 | 2.1713e+002 | 1/14 | 1 |
1/512 | 3.0008e+004 | 5/125 | 2.7705e+003 | 1/18 | 1 |
1/1024 | 2.5316e+005 | 1/20 | 1 | ||
1/256 | 6.5098e+004 | 10/256 | 4.8902e+003 | 1/25 | 4 |
(b) Comparison of GMRES with/without ILU for the MDG in (a) |
Ele. num. | Basis num. | Elapsed time | Conv. | Iter.(O/I) | Deg. | ||
1/64 | 8,192 | 4,225 | 1.0632e+001 | 2.0077e–001 | 1/9 | 1 | |
1/128 | 32,768 | 16,641 | 7.3629e+001 | 8.1290e–002 | 1.3044 | 1/11 | 1 |
1/256 | 131,072 | 66,049 | 2.2198e+002 | 2.8772e–002 | 1.4984 | 1/13 | 1 |
1/512 | 524,288 | 263,169 | 2.3875e+003 | 6.5147e–003 | 2.1429 | 1/16 | 1 |
Using mixed polynomials (P1 and P2 elements) |
Degree of freedom | Convergence order | Degree order | ||
1/32 | 6,144 | 8.66085e-002 | 1 | |
1/64 | 24,576 | 1.28362e-002 | 2.7543 | 1 |
1/128 | 98,304 | 1.77764e-003 | 2.8522 | 1 |
1/32 | 30,720 | 3.22321e-003 | 4 | |
(a) Using DG method | ||||
Degree of freedom | Convergence order | Degree order | ||
1/32 | 1,089 | 8.30570e-002 | 1 | |
1/64 | 4,225 | 1.19648e-002 | 2.7953 | 1 |
1/128 | 16,641 | 1.87018e-003 | 2.6775 | 1 |
1/32 | 10,497 | 3.21210e-003 | 4 | |
1/64 | 41,473 | 5.98660e-004 | 4 | |
(b) Using MDG method |
Total element num | Degree of freedom | Convergence order | Degree | ||
1/64 | 8,192 | 24,576 | 6.6217e–001 | 1 | |
1/128 | 32,768 | 98,304 | 3.1138e–001 | 1.0885 | 1 |
1/256 | 131,072 | 393,216 | 9.9196e–002 | 1.6503 | 1 |
1/512 | 524,288 | 1,572,864 | 2.1732e–002 | 2.1911 | 1 |
1/1024 | 2,097,152 | 6,291,456 | 1 | ||
1/256 | 131,072 | 1,966,080 | 6.4686e–003 | 4 | |
(a) DG solution | |||||
h | Elapsed time | GMRES iter(O/I) | Elapsed time | PGMRES Iter(O/I) | Degree |
1/64 | 8.2306e+001 | 1/208 | 7.2131e+000 | 1/4 | 1 |
1/128 | 1.4060e+003 | 3/201 | 6.1875e+001 | 1/4 | 1 |
1/256 | 2.5076e+004 | 10/220 | 1.4706e+003 | 1/4 | 1 |
1/512 | 2.1915e+004 | 1/4 | 1 | ||
1/1024 | 1 | ||||
1/256 | 4.9732e+005 | 10/256 | 3.1762e+004 | 1/10 | 4 |
(b) Comparison of GMRES with/without ILU for the DG in (a) |
Total element num | Degree of freedom | convergence order | Degree | ||
1/64 | 8,192 | 4,225 | 6.5406e–001 | 1 | |
1/128 | 32,768 | 16,641 | 3.0008e–001 | 1.1241 | 1 |
1/256 | 131,072 | 66,049 | 9.3895e–002 | 1.6762 | 1 |
1/512 | 524,288 | 263,169 | 2.1614e–002 | 2.1191 | 1 |
1/1024 | 2,097,152 | 1,050,625 | 6.0343e–003 | 1.8417 | 1 |
1/256 | 131,072 | 657,409 | 6.3296e–003 | 4 | |
(a) MDG solution | |||||
h | Elapsed time | GMRES iter(O/I) | Elapsed time | PGMRES iter(O/I) | Degree |
1/64 | 1.2899e+001 | 1/160 | 5.8968e+000 | 1/9 | 1 |
1/128 | 1.1022e+002 | 1/246 | 4.4625e+001 | 1/12 | 1 |
1/256 | 2.4963e+003 | 3/189 | 2.1713e+002 | 1/14 | 1 |
1/512 | 3.0008e+004 | 5/125 | 2.7705e+003 | 1/18 | 1 |
1/1024 | 2.5316e+005 | 1/20 | 1 | ||
1/256 | 6.5098e+004 | 10/256 | 4.8902e+003 | 1/25 | 4 |
(b) Comparison of GMRES with/without ILU for the MDG in (a) |
Ele. num. | Basis num. | Elapsed time | Conv. | Iter.(O/I) | Deg. | ||
1/64 | 8,192 | 4,225 | 1.0632e+001 | 2.0077e–001 | 1/9 | 1 | |
1/128 | 32,768 | 16,641 | 7.3629e+001 | 8.1290e–002 | 1.3044 | 1/11 | 1 |
1/256 | 131,072 | 66,049 | 2.2198e+002 | 2.8772e–002 | 1.4984 | 1/13 | 1 |
1/512 | 524,288 | 263,169 | 2.3875e+003 | 6.5147e–003 | 2.1429 | 1/16 | 1 |
Using mixed polynomials (P1 and P2 elements) |