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DiGIBST: An inquiry-based digital game-based learning pedagogical model for science teaching


  • 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 Ω be a bounded open domain in Rm, m=2,3 with a polyhedral boundary Ω. We consider the convection-diffusion-reaction equation

    Lu(A(x)u)+b(x)u+c(x)u=f(x), (2.1)

    where fL2(Ω) and cL(Ω) are real valued, b is the solenoidal vector field defined on ¯Ω, and A is a symmetric matrix whose entries are bounded, piecewise continuous real valued functions defined on ¯Ω, with

    ζTA(x)ζ>0ζRma.e. x¯Ω. (2.2)

    By n(x) we denote the unit outward normal vector to Ω at xΩ. We divide the boundary as follows: Let

    Ωo={xΩ:n(x)TA(x)n(x)>0},Ω={xΩΩo:b(x)n(x)<0},Ω+={xΩΩo:b(x)n(x)0}. (2.3)

    The sets Ω and Ω+ will be referred to as the inflow and outflow boundaries, respectively. Clearly we see that Ω=ΩoΩΩ+. If Ω0 is nonempty, we shall further divide it into disjoint subsets ΩD and ΩN whose union is Ωo with ΩD is nonempty and relatively open in Ω. We supplement (2.1) with the boundary conditions:

    u=gDon ΩDΩ,(Au)n=gNon ΩN, (2.4)

    and adopt the (physically reasonable) hypothesis that bn0 on ΩN whenever ΩN is nonempty. We assume that the values of the diffusion coefficient A and the velocity field b ensure well-posedness of (2.1) and (2.4). Additional assumptions on these coefficients will be set later on (see [10]).

    In this section, we introduce the finite element spaces for the MDG method. We recall the space L2(Ω) of all square integrable functions and the space H1(Ω) of all functions in L2(Ω) that have square integrable derivatives. To define approximation spaces, we consider a uniformly regular partition Th of Ω into finite elements E.

    Let Th be a regular family of triangulations of Ω in the sense that there exists a κ>0 such that hhmin<κ, where hmin=min(hE)|ETh, hE is the diameter of ETh, and h=max(hE)|ETh. We assume that Th contains only regular nodes, that is, each element vertex is also a vertex to all adjacent elements and there are no hanging nodes. The elements ETh are either triangles and/or quadrilaterals in two dimensional space or tetrahedra and/or hexahedra in three dimensional space. The vector nE(x) is also used as the outward unit normal to E at xE, which coincides with the one on Ω. We denote by Eh the set of all edges of Th, by Eih the set of all interior edges, and by Eoh=EhEih the set of all outer edges.

    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 e be an interior edge shared by two elements E1 and E2 and let n1 and n2 be the unit normal vectors on e pointing exterior to E1 and E2, respectively. With ϕj:=ϕEj, following [1,2,10], we define

    {ϕ}=12(ϕ1+ϕ2),[[ϕ]]=ϕ1n1+ϕ2n2on eEih.

    For a vector valued function τ which is piecewise smooth on T, with analogous meaning for τ1 and τ2, we define

    {τ}=12(τ1+τ2),[[τ]]=τ1n1+τ2n2on eEih.

    Notice that the jump [[ϕ]] of the scalar function ϕ across eEih is a vector parallel to the normal to e, and the jump [[τ]] of the vector function τ is a scalar quantity. For eEoh, the set of boundary edges, we let

    [[ϕ]]=ϕn,{τ}=τ.

    We do not require either of the quantities {ϕ} or [[τ]] on boundary edges, and leave them undefined there.

    We now assign to Th the broken Sobolev space of order p,

    Hp(Th)={wL2(Ω):wEHp(E),ETh},

    equipped with the broken Sobolev norm (EThw2Hp(E))1/2. When p=0, we denote by E the norm in L2(E) and similarly by E the norm in L2(E). For the construction of the MDG, we first consider the continuous finite element space

    ¯Wh(Ω)={¯vH1(Ω):¯vEPr(E), ETh}

    and then associate with it the discontinuous approximation spaces:

    Wh(Ω)={vL2(Ω):vEPr(E), ETh},Wh(E)={μL2(E):μPr(E), ETh}.

    Wh(Ω) is then viewed as being obtained from ¯Wh(Ω) by releasing interelement continuity constraints. Also Wh(Ω) is a formal union of the local spaces Wh(E).

    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 uhWh(Ω) such that

    B(uh,v)=L(gD,gN,f;v), vWh(Ω), (4.1)

    where B(,) is a bilinear form defined on H2(Th)×H2(Th) and L(gD,gN,f;) is a linear functional defined on H2(Th). We assume that the values of B(,) on the interior edges are determined by averages and jumps. That is,

    B(uh,v)=EThBE(uh,v)+eEohBoe(uh,v)+eEihBie(˜uh,˜v), (4.2)

    where BE(,) is a bilinear element form defined on ETh, Boe(,) is a boundary edge bilinear form defined on the boundary edges, Bie(,) is an interior edge bilinear form that depends only on the values from the two elements that share interior edge e in Eih, and (˜uh,˜v)=([[uh]],[[Auh]],b[[uh]],{uh},{Auh},[[v]],[[av]],b[[v]],[[v]],{v},{Av}). We here note that a large class of consistent DGs satisfies (4.2) (see [1,6]).

    Now, to define the local problem of the MDG, we decompose uh of ¯uh and uh. That is, let

    uh=¯uh+uh.

    Then, (4.1) takes the below forms:

    Coarse scale equation:

    B(¯uh,¯v)+B(uh,¯v)=L(gD,gN,f;¯v),¯v¯Wh(Ω); (4.3)

    Fine scale equation:

    B(uh,v)+B(¯uh,v)=L(gD,gN,f;v),vWh(Ω). (4.4)

    By treating the function ¯uh as data, we rewrite (4.4) as follows: Find uhWh(Ω) such that

    B(uh,v)=L(gD,gN,f;v)B(¯uh,v),vWh(Ω). (4.5)

    Take vWh(E) and, assume, without loss of generality, that v is extended to be zero outside of E. Using (4.2), we then see that (4.5) is equivalent to the following: For each ETh, find uhWh(E) such that, vWh(E),

     BE(uh,v)+eEohBoe(uh,v)+eEihBie(˜uh,˜v)= L(gD,gN,f;v) (BE(¯uh,v)+eEohBoe(¯uh,v)+eEihBie(˜¯uh,˜v)),uhWh(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 τWh(E),

    [[τ]]=τini+τono=(τiτo)ni,uoh=g=¯uhif EΩ=,¯uh=¯uoh=¯uihon E,

    where vi and vo are the values of v from the inside and the outside of E, respectively. We then see that, uhWh(E),

     Bie(˜uh,˜v)=Bie([[uh]],[[Auh]],b[[uh]],{uh},{Auh},vn,(Av)n,bvn,v2,Av2). (4.7)

    Similarly, since

     Bie(˜¯uh,˜v)=Bie([[¯uh]],[[A¯uh]],b[[¯uh]],{¯uh},{A¯uh},vn,(Av)n,bvn,v2,Av2),

    we have that

     BE(¯uh,v)+eEohBoe(¯uh,v)+eEihBie(˜¯uh,˜v)0,if vI(¯Wh,0). (4.8)

    Substituting (4.7)-(4.8) into (4.6), local problem of the MDG is ended up as follows: ¯uh¯Wh, find uhEThWh(E) such that

    bE(uh,v)=lE(¯uh,f;v),vWh(E),ETh, (4.9)

    where bE(,) and {lE(,;)} are linear with respect to its first argument and affine with respected to its second and third arguments (see [7]):

    bE(uh,v)= BE(uh,v)+eEohBoe(uh,v)+eEihBie((uh)in,(Auh)in, (buh)in,(uh)i2,(Auh)i2,vn,(Av)n,bvn,v2,Av2),lE(¯uh,f;v)= L(gD,gN,f;v)+eEihBie(¯uhn,(A¯uh)n,b¯uhn, ¯uh2,A¯uh2,vn,(Av)n,bvn,v2,Av2).

    We note that (4.9) is a DG formulation for the local problem (A(x)uh) +b(x)uh+c(x)uh=f on E having weakly imposed boundary condition on E, given that gD=¯uh on ΩD and gN=(A¯uh)n on ΩN.

    We now denote by I:¯Wh(Ω)×L2(Ω)Wh(Ω) the operator which associates to each (¯uh,f)¯Wh(Ω)×L2(Ω) the solution uh of the local problem on each ETh. Let

    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 uMDGhI(¯Wh,f) such that

    B(uMDGh,v)=L(gD,gN,f;v),vI(¯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 Th={Ek}Lk=1. Let {ϕEki}Nki=1 be the nodal basis of Wh(Ek) and let {¯ϕi}Ni=1 be the nodal basis functions of ¯Wh(Ω). We then see that

    {¯ϕEki}¯Nki=1is a subset of{ϕEki}Nki=1, (5.1)

    where ¯Nk is the number of degrees of freedom of the continuous finite element on Ek.

    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 uEhWh(E), such that, for vhWh(E), ETh,

    bE(uEh,vh)=bE(¯uh;vh)+FE(vh), (5.2)

    where

     bE(uEh,vh)= E(AuEhbuEh)vhdx+EcuEhvhdx++EuEhvhbnds 12E(AuEhvhAvhuEh)nds, bE(¯uh;vh)= E¯uhvhbnds+12E(A¯uhvh+Avh¯uh)nds,FE(vh)= Efvhdx. (5.3)

    Now, let uMDGhI(¯uh;f) be the solution to the MDG method (4.12) for (2.1) and (2.3) and let

    uMDGh=Lk=1χEkuEkh=Lk=1χEkNki=1UkiϕEki,¯uh=Nr=1¯Ur¯ϕr,fEkh=Nkj=1FEkjϕEj, (5.4)

    where ¯U=(¯U1,¯U2,,¯UN)T, Uk=(Uk1,Uk2,,UkNk)T, Fk=(Fk1,Fk2,,FkNk)T, for k=1,,L, and χE is the characteristic function having value one on ¯E and zero otherwise. Let ¯Uk=(¯Uk1,¯Uk2,,¯Uk¯Nk)T be the vector of the continuous nodal values on Ek.

    Now, let ˜Sk be the NkׯNk inter-scale matrix of the local problem (5.2), that is,

    Uk=˜Sk¯Uk (5.5)

    where ˜Sk=(Lk,m)1¯Rm, (Lk,m)ji=bE(ϕEki,ϕEmj), and (¯Rm)jr=bE(¯ϕr,ϕEmj). We denote by (Λ)i and (Λ)ij the i-th row vector and the ij component of matrix Λ, respectively.

    Now, let Sk be the Nk×N zero extension of ˜Sk and let ˘Sk be the N×N zero extension of Sk. We denote by ˘ϕEki the zero extension of ¯ϕEki into Ω. We then see that

    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 in terms of ¯U and we have that

    uMDGh=Lk=1χEkNki=1UkiϕEki=Lk=1χEk¯Nki=1(˜Sk¯Uk)i¯ϕEki=Lk=1χEk¯Nki=1(Sk¯U)i¯ϕEki,=Lk=1χEk¯Nki=1(Sk)i¯U¯ϕEki,=Lk=1Ni=1(˘Sk)i¯U˘ϕEki. (5.8)

    Similarly, we consider vhI(¯uh;0) such that, for 1lL,

    vh=χElNlj=1ϕElj=χEl¯Nlj=1(˜Sl¯Il)j¯ϕElj=χEl¯Nlj=1(Sl¯I)j¯ϕElj=χEl¯Nlj=1(Sl)j¯I¯ϕElj=Nj=1(˘Sl)j¯I˘ϕElj, (5.9)

    where ¯Il is the ¯Nl×1 vector such that ¯Ilp=δlp and ¯I is the N×1 zero extension of ¯Il. Substituting (5.8)-(5.9) into (4.12) and noting (5.1) and (5.7), we find that

    B(uMDGh,vh)=B(Lk=1Ni=1(˘Sk)i¯U˘ϕEki,Nj=1(˘Sl)j¯I˘ϕElj)=Lk=1B(Ni=1(˘Sk)i¯U˘ϕEki,Nj=1(˘Sl)j¯I˘ϕElj)=Lk=1Ni,j=1((˘Sl)j¯I)TB(˘ϕElj,˘ϕEki)(˘Sk)i¯U. (5.10)

    Let Al,k be the Nl×Nk matrix such that

    (Al,k)i,j=B(ϕEki,ϕElj), (5.11)

    and ˜Al,k be the N×N zero extension of Al,k such that

    ˜Al,k={Al,k,onEkandEl,0,otherwise.

    We then by noting (5.7), that (5.10) is rewritten as follows:

    B(uMDGh,vh)=Lk=1Ni,j=1((˘Sl)i¯I)T(˜Al,k)ij(˘Sk)j¯U=(Lk=1Ni,j=1((˘Sl)i¯I)T(˜Al,k)ij(˘Sk)j)¯U. (5.12)

    Here we have used the fact that ˘Sl and ˜Al,k are the N×N matrices and ¯I and ¯U are the N×1 vectors. We now note, by the definition of ˘Sl,Sl, and A, that

    Lk=1Ni,j=1((˘Sl)i¯I)T(˜Al,k)ij(˘Sk)j=Lk=1Ni,j=1((Sl)i¯I)T(Al,k)ij(Sk)j=Lk=1Ni,j=1((˜Sl)i¯I)T(Al,k)ij(˜Sk)j. (5.13)

    Considering l=1,,L and also from (5.12)-(5.13), we see that the MDG matrix of (4.12), AMDG, is given by

    AMDG=Lk,l=1(˜Sl)TAl,k˜Sk. (5.14)

    Remark. We see, by (5.13), that

    AMDG=STAS (5.15)

    where

    S=[S1S2SL]andA=[A1,1A1,2A1,LA2,1A2,2A2,LAL,1AL,2AL,L]. (5.16)

    Here S is the M×N inter-scale matrix corresponding to the inter-scale operator I(¯uh;f), where M=Lk=1Nk, and A is the M×M DG matrix. We note that the MDG matrix is a product of the interscale matrices of local problems and the DG matrix.

    Figure 1.  Schematic illustration of the basis functions in the local problem. The left hand side figure is a 16-node bicubic quadrilateral element. Its boundary nodes are identified on the one of the right hand side. The corresponding basis functions satisfy ¯ϕj=ϕj, j=1,2,,12. The internal degrees-of-freedom, corresponding to ϕ13, ϕ14, ϕ15, ϕ16 are eliminated by the solution of the local problem. Only the unique, shared, boundary degrees-of-freedom are retained in the global problem (see [6]).

    Remark. We further see, from (5.14)-(5.15), that

    STAS=Lk,l=1(˜Sl)TAk,l˜Sk. (5.17)

    In the next section, we compute the MDG solution in the following way:

    1. Do for all ETh:

    ● construct Ak,l: compute the local mass and stiffness matrices of usual DG.

    ● construct ˜Sk: solve the local problem on Ek.

    ● calculate (˜Sl)TAk,l˜Sk.

    2. Sum up the computations following the global numbering.

    3. Solve the global system by ILU preconditioned GMRES.

    In this section, we test the MDG with convection dominated problems. When the convection strongly dominates the diffusion (k=106), a fast linear solver is needed to solve the large system. We apply ILU preconditioned GMRES method to solve the matrix equation effectively. We compute L2-error and convergence order defined by

    Err=||u(,)uh(,)||L2,Conv=logErr(h)Err(h/2)log2,

    where u(,) denotes the exact solution and uh is an approximate solution.

    We solve the problem (2.1) and (2.4) in the case of the diffusion coefficient k=104 with the following data: A=kI (here I is the 2 by 2 identity matrix), b=(2,3), c=1, Ω=(0,1)×(0,1) and g=0 (see [9]).

    We take f such that the exact solution is given by

    u(x,y)=16x(1x)y(1y)(12+arctan[2s(x,y)/k]π),

    where

    s(x,y)=142(x12)2(y12)2.

    The graph of the exact solution and the approximate solution of DG with uniform mesh h=1/32 are given in Figure 2 and Figure 3, respectively. The DG solution shows spurious oscillation on the inner layer and in the direction of convection. To reduce these errors, we take smaller meshes or use high order degree of the approximate polynomials. We have tested the MDG and DG to compare the results. The results are given in Table 1. When P1 elements are used, the DG solution requires the degrees-of-freedom six times more than the one of MDG solution. The graphs of the approximate solutions with h=1/128 are given in Figure 4. Similarly, when P4 elements are used, the DG solution requires the degrees-of-freedom three times more than the one of MDG solution. The graphs of the approximate solutions with h=1/32 are given in Figure 5.

    Figure 2.  Exact solution when k=104.
    Figure 3.  DG solution with h=1/32 and d.o.f =6,144. Oscillation occurs due to convection dominant to k=104 (diffusion coefficient).
    Table 1.  Cases of P1 and P4 elements with k=104 using GMRES with tol = 106.
    h Degree of freedom L2 error 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
    h Degree of freedom L2 error 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

     | Show Table
    DownLoad: CSV
    Figure 4.  k=104, h=1/128, and P1 element.
    Figure 5.  k=104, h=1/32, and P4 element.

    In this subsection, we consider the case of k=106 in diffusion coefficient.

    We take h=1/512 and P1 element. The DG solution shows spurious oscillation as seen in Figure 6. It is numerically shown that P1 approximation requires very small h to resolve the spurious oscillation and the computation is not efficient. Figure 7 shows the approximate solutions without oscillation when P4 and h=1/256. DG solution requires degrees-of-freedom three times more than the one of MDG to produce similar results.

    Figure 6.  k=106, h=1/256, and P1 element. DG solution with d.o.f =393,216.
    Figure 7.  k=106, h=1/256, and P4 element.

    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.

    Table 2.  DG approximation with P1 and P4 elements, k=106, tol = 108 using GMRES with/without ILU (see [11,12]).
    h Total element num Degree of freedom L2 error 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)

     | Show Table
    DownLoad: CSV
    Table 3.  MDG approximation with P1 and P4 elements, k=106, tol = 108 using GMRES with/without ILU.
    h Total element num Degree of freedom L2 error 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)

     | Show Table
    DownLoad: CSV

    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 h=1/32 and 1/256.

    Figure 8.   .
    Figure 9.  MDG solution with k=106 and P1 element.

    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 p where it is needed.

    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 P1 in the grid part and P4 in the shadow part, respectively, in Figure 10.

    Figure 10.  Domain with uniform mesh h=1/64. In grid and shadow parts, P1 and P4 elements are used, respectively.

    Matrix structure and P1 and P4 elements are given in Figure 11. 3×3, 15×15 block matrices corresponding to P1 and P4 elements, respectively, are placed on the diagonal.

    Figure 11.  LHS: P1, P4 elements and triangle numbers. RHS: Matrix structure of DG corresponding to the triangles numbered on LHS.

    We compare the result with the one of DG with P4 elements. MDG solution exhibits similar behavior of DG with much less degrees of freedom. The figures are given in Figure 12. We compute the L2-error for the MDG solution with polynomials of mixed degrees. The results are given in Table 4.

    Figure 12.  MDG solution ¯uMDGh with k=106 using P1 and P4 elements.
    Table 4.  MDG solution ¯uMDGh when k=106, tol = 108, using ILU GMRES.
    h Ele. num. Basis num. Elapsed time L2 error 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)

     | Show Table
    DownLoad: CSV

    As a conclusion, the MDG method numerically shows an advantage in adaptive p refinements. ILU GMRES is applied to the resulting system. Combination of P1 element in the smooth region and higher order element in region where more detailed information are needed, can be a reasonable candidate in refinement. Further research on the adaptive refinement will be done in the future.



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  • Author's biography Jun Karren V. Caparoso holds a doctorate degree in Science Education from Philippine Normal University-Manila. He is an Associate Professor at the Department of Science and Mathematics Education, College of Education, Mindanao State University-Iligan Institute of Technology, Iligan City, Philippines; Antriman V. Orleans holds a doctorate degree in Education (Science Education) from Hiroshima University, Japan. He is a Professor of Science Education and the current dean of the College of Advanced Studies at Philippine Normal University- Manila, Manila, Philippines
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