An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

DOI:10.1051/m2an/2012012 www.esaim-m2an.org

AN OPERATOR-SPLITTING GALERKIN/SUPG FINITE ELEMENT METHOD FOR POPULATION BALANCE EQUATIONS: STABILITY

AND CONVERGENCE

Sashikumaar Ganesan

Abstract. We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates.

The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations.

In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity,i.e.the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.

Mathematics Subject Classification. 35K20, 65M60, 65M12.

Received March 16, 2011. Revised November 28, 2011.

Published online May 31, 2012.

1. Introduction

The numerical solution of a population balance equation (PBE) is highly demanded in many industrial applications such as crystallization, polymerization etc., see for example [3,25,26]. The PBE depends not only on time and space but also contains derivatives with respect to the properties of the particles (internal variables) such as the size, length, etc. Thus, the PBE is posed on a high-dimensional (more than three) domain.

A population balance equation describing the particle size distributionuin a population balance system (PBS)

Keywords and phrases. Population balance equations, operator-splitting method, error analysis, streamline Upwind Petrov Galerkin finite element methods, backward Euler scheme.

1 Numerical Mathematics and Scientific Computing, Supercomputer Education and Research Centre, Indian Institute of Science, 560012 Bangalore, India.sashi@serc.iisc.ac.in

Article published by EDP Sciences c EDP Sciences, SMAI 2012

of a crystallization process can be defined as:

∂u

∂t −Δ_xu+b· ∇xu+g· ∇u=f in (0, T]×Ω, u(t, x, ) = 0 on (0, T]×∂Ω, u(0, x, ) =u₀(x, ) inΩ.

(1.1)

Here, the computational domainΩ is the Cartesian product of the physical space (X-direction) domainΩ_X⊂ R^d, d= 2,3 and the internal property coordinate (L-direction) domainΩ_L ⊂R^s, s≥1,i.e., Ω:= (Ω_X×ΩL)⊂ R^d+s with a polyhedral boundary ∂Ω. Further, ∇x and ∇ denote the gradient operators in Ω_X and Ω_L, respectively, whereasΔ_xdenotes the Laplace operator inΩ_X. The fluid transport velocityb(t, x) and the growth rateg(t, x) are givend- ands-dimensional vector functions, respectively, >0 is a constant diffusion coefficient inΩ_X,u₀is a given initial distribution ofu, andT is a given final computational time. The source termfmay be considered as a term arising from the aggregation and breakage. For simplicity, we assumef ∈C⁰(0, T;L²(Ω)) and use the homogeneous Dirichlet boundary condition. Further, to reduce the technicalities as much as possible, we assume that the fluid transport velocityb(t, x) is divergence free, that is,

∇x·b= 0. (1.2)

In a crystallization process the growth rateg(t, x) is often assumed to be independent of the particle size∈Ω_L. Therefore, we naturally have the property

∇·g(t, x) = 0. (1.3)

One of the main challenges in the finite element solution of high-dimensional PBE is to discretize it spatially with the standard finite elements, especially when d+s >3. Further, developing a fully-practical numerical scheme to solve a high-dimensional PBE will become more challenging, when the PBE is coupled with the flow and species concentration equations [8,15]. Since the dimension of the PBE will be higher than the dimension of all other equations in a population balance system (PBS), a special care has to be taken in order to handle the coupling conditions [5,13,15,18].

Several numerical methods, method of moments and its variants, discretization methods, finite difference, least square method, spectral and finite element methods, have been proposed and used to solve PBEs by several authors, see for example [16,17,22–26] and the references therein. However, most of these methods are restricted to one-dimensional (spatially) PBEs or applied to a system of one-dimensional (1D) PBEs which are obtained by splitting the high-dimensional PBE. For example, in [4] the PBE inR²has been split into two 1D equations and a mixed Euler-Lagrange method has been applied. In the high-resolution finite volume computations [10,19–21]

of PBEs, the dimensional splitting has been applied to the PBE to obtain a system of 1D equations. Handling of the coupling conditions between the high-dimensional PBE and the 3D flow and species concentration equations will be more challenging when the PBE is split into a system of 1D equations. Further, a special technique is needed for the load balancing in parallel computations of the system of operator-split 1D equations [11].

In this paper, we present a heterogeneous finite element method for a high-dimensional population balance equation. In our approach, we split the PBE into two equations, that is, we split the (d+s)-dimensional PBE (1.1) into d- and s-dimensional equations. Then, we solve thed- and s-dimensional equations with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite element methods, respectively. The SUPG discretization suppress the spurious oscillations in the numerical solution due to the absence of the diffusion in theΩ_L. A rigorous numerical analysis of the operator-splitting (first order in time) Galerkin/SUPG finite element method for the population balance equation is presented in this paper. An advantage in our splitting approach is that the shape of the physical domain (Ω_X) can be arbitrary and it is not the case when the PBE is split into a system of 1D equations. Further, the coupling conditions between the PBE and the flow equations can be handled easily in our splitting [8]. Since the decomposition of the physical domain is sufficient

for parallel computations, load balancing in the proposed splitting approach can be scheduled in an usual way [8]. Although the splitting is quite natural, to the best of the author’s knowledge, the operator-splitting finite element method for high-dimensional PBEs has not been proposed before in the literature.

The paper is organized as follows. In the next section, we briefly discuss the standard discrete and algebraic form of the population balance equation. Then, we present the operator-splitting finite element method for the population balance equation in Section3. After that, in Section4, the stability anda priorierror estimates of the operator-splitting Galerkin/SUPG finite element method for the population balance equation are presented.

Finally, we present the numerical results in Section 5.

2. Finite element method for population balance equations

2.1. Preliminaries

LetΩ:=Ω_X×Ω_L⊂R^d+sbe a bounded domain. Assume that the particle size distribution functionu(t, x, ) in (1.1) is measurable,i.e., _T

0

ΩX×ΩL

|u(t, x, )|d(x, ) dt <∞ then, due to Fubini’s theorem we have

_T

0

ΩX×ΩL

u(t, x, )d(x, ) dt = _T

0

ΩX

ΩL

u(t, x, )d

dxdt

= _T

0

ΩL

ΩX

u(t, x, )dx

ddt.

Moreover, ifu(t, x, ) =g(t, x)h(t, ), then we have _T

0

ΩX×ΩL

g(t, x)h(t, )d(x, ) dt= _T

0

ΩX

g(t, x) dx

ΩL

h(t, ) d

dt.

2.2. Finite element spaces for the operator-splitting method

Let (·,·) and|| · || be theL²-inner product and norm overΩ, respectively, that is, (v, q) :=

Ω

v(x, )q(x, ), ||v||²= (v, v) ∀v, q∈L²(Ω).

Further, letH^m(Ω_X) andH^m(Ω_L) be the usual Sobolev spaces with the weak derivatives of orderm. We now define

H^m,m(Ω) := (H^m(Ω_X;H^m(Ω_L)))∩(H^m(Ω_L;H^m(Ω_X))). (2.1) For each integerm≥0, the associated norm of a functionu∈H^m(Ω_X;H^m(Ω_L) can be defined as

u ²_Hm(ΩX;H^m(ΩL):=

|β|≤m,

|α|≤m

∂^β∂_x^αu ²_L2(Ω).

Hence, the associated norm and seminorm for a functionu∈H^m,m(Ω) can be defined as u ²_Hm,m(Ω):=

|β|≤m,

|α|≤m

∂^β∂_x^αu ²_L2(Ω), |u|²_Hm,m(Ω):=

|β|=m,

|α|=m

∂^β∂_x^αu ²_L2(Ω).

Note that the space H^m,m(Ω) is slightly more regular than the usual Sobolev spaceH^m(Ω), i.e., the mixed partial derivatives of the functions from the spaceH^m,m(Ω) are bounded. This additional regularity is necessary in the analysis of operator-splitting finite element method.

Now, let V :=H₀¹(Ω_X) and Q:=H₀¹(Ω_L) be the usual Sobolev spaces, whose function values are zero on their respective boundaries. LetT_handS_hbe the triangulation ofΩ_XandΩ_Lrespectively, and are assumed to be shape regular. Suppose V_h ⊂V and Q_h⊂Qare conforming finite element (finite dimensional) spaces. We denote the diameter of the cells K ∈ Th and K ∈ Sh by h_K and h_K, respectively. Further, the global mesh size in each domain is defined ash_x:= max{hK : K∈ Th} andh:= max{hK : K∈ Sh}, respectively, and the global mesh sizehinΩ is defined ash:= max{hx, h}. Letφ_h:=φ_i(x),i= 1,2, . . . ,M,andψ_h:=ψ_k(), k= 1,2, . . . ,N, be the basis functions ofV_handQ_h, respectively,i.e.,

V_h= span{φ_i(x)}, Q_h= span{ψ_k()}. (2.2) We then define the discrete finite element spaceW_h such that

W_h=V_h⊗Q_h=

⎧⎨

⎩ξ_h: ξ_h= M j=1

N l=1

ξ_j,lφ_j(x)ψ_l(); ξ_j,l∈R

⎫⎬

⎭⊂H₀^1,1(Ω), where

H₀^1,1(Ω) =

ξ_h: ξ_h∈H^1,1(Ω); ξ(x, ) = 0, ∀ (x, )∈∂Ω

⊂H₀^1,1(Ω).

Also, the finite element functions are defined as follows u_h=

M j=1

N l=1

u_j,lφ_jψ_l, v_h = M i=1

N k=1

v_i,kφ_iψ_k,

∇_xu_h= M j=1

N l=1

u_j,l(∇_xφ_j)ψ_l, ∇_xv_h = M i=1

N k=1

v_i,k(∇_xφ_i)ψ_k,

∇u_h= M j=1

N l=1

u_j,lφ_j(∇ψ_l), ∇v_h = M i=1

N k=1

v_i,kφ_i(∇ψ_k).

(2.3)

Further, for allK∈ ThandK∈ Sh, we define the mesh-dependent norm v_h ²_0,K,K:=

K

K

v²_h, v_h ²_1,1,K,K:=

K

K

(∂∂_xv_h)².

2.3. Galerkin/SUPG stabilized discretization

It is well known that the standard Galerkin discretization of convection diffusion equations is not stable for small diffusion coefficients (in comparison with convection), and induce spurious oscillations in the solution.

In the considered problem (1.1), even if we assume a sufficiently large , the standard Galerkin discretization still induce spurious oscillations due to the absence of the diffusion in the L-direction. One possibility to circumvent the instability and suppress spurious oscillations is to use the SUPG method [6,12]. SUPG is one of the most popular stabilization methods for finite element discretization, and it adds artificial diffusion along the streamlines of the solution, see for example [2,14] and the references there in. Since we assumed that is sufficiently large, it is sufficient to stabilize the equation (1.1) only in the L-direction. Therefore, we use the standard Galerkin and the consistent SUPG stabilized discretizations in theX- andL-directions, respectively.

Since we use different discretizations in different directions, we call it as a heterogeneous discretization method.

Applying the Galerkin/SUPG discretization, the semi-discrete form of (1.1) reads:

For a givenu_h(0) =u_h,0, findu_h(t)∈W_h such that for allt∈(0, T] ∂u_h

∂t , v_h

+a_LS(u_h, v_h) +

ΩX

K∈Sh

δ_K ∂u_h

∂t , g_h· ∇v_h

K

= (f, v_h) +F_S(f, v_h), v_h∈W_h, (2.4)

where

a_LS(u_h, v_h) :=

Ω

(∇xu_h· ∇xv_h +b_h· ∇xu_h v_h+g_h· ∇u_h v_h) +

ΩX

K∈Sh

δ_K(g_h· ∇u_h,g_h· ∇v_h)_K, F_S(f, v_h) :=

ΩX

K∈Sh

δ_K(f, g_h· ∇v_h)_K.

Here, u_h,0 ∈ W_h is a L²-projection of the initial value u₀ onto W_h and (·,·)K denotes the L²-inner product in the mesh cell K ∈ Sh. Further,{δK} are the local stabilization parameters and in order to guarantee the coercivity of the bilinear forma_LS(·,·) we assume

0< δ_K ≤δ₀h²_K≤1, (2.5)

whereδ₀>0 is a constant.

Lemma 2.1 (coercivity of a_LS(·,·)). Let the discrete form of the assumptions (1.2)and (1.3)be satisfied and the assumption (2.5) be fulfilled. Then, the bilinear form associated with the Galerkin/SUPG discretization satisfies

a_LS(u_h, u_h)≥ |||uh|||². (2.6) Here, the mesh-dependent heterogeneous norm

|||uh|||²:=

K∈Th

K∈Sh

∇xu_h ²_0,K,K +δ_K g_h· ∇u_h ²_0,K,K

. (2.7)

Proof. Using the definition of thea_LS, and applying integration by parts we obtain a_LS(u_h, u_h) =

ΩL

ΩX

∇_xu_h· ∇_xu_h +1

2b_h· ∇_xu²_h

+1 2

ΩX

ΩL

g_h· ∇u²_h +

ΩX

K∈Sh

δ_K

K

g_h· ∇u_h g_h· ∇u_h,

=

ΩL

ΩX

∇xu_h· ∇xu_h −1

2∇x·b_hu²_h

−1 2

ΩX

ΩL

∇·g_hu²_h

+

ΩX

K∈Sh

δ_K

K

(g_h· ∇u_h)²,

≥

K∈Th

K∈Sh

∇xu_h ²_0,K,K +δ_K g_h· ∇u_h ²_0,K,K

,

for allu_h∈V_h.

2.4. Temporal discretization

Let 0 = t⁰ < t¹ < · · · < t^N = T be a decomposition of the considered time interval [0, T]. Let us denote τ =τⁿ =tⁿ -tⁿ⁻¹, 1≤n≤N, be an uniform time step, and denote uⁿ_h be the approximation of u(tⁿ, x, ) in W_h. Further, we denoted the one step finite difference operator

∂u¯ ⁿ_h= uⁿ_h−uⁿ⁻¹_h

τ ·

After applying the backward Euler time discretization in (2.4), the heterogeneous discrete form of (1.1) can be written as:

For givenfⁿ,u⁰_h=u_h,0, finduⁿ_h∈W_h in the time interval (tⁿ⁻¹, tⁿ) such that for allv_h∈W_h ∂u¯ ⁿ_h, v_h

+a_LS(uⁿ_h, v_h) +

ΩX

K∈Sh

δ_K∂u¯ ⁿ_h, g_h· ∇v_h

K = (fⁿ, v_h) +F_S(fⁿ, v_h). (2.8) Now, using the definition of finite element functions (2.3), the algebraic form of the discrete equation (2.8) can be written as

M+M^S+τ A+τ A^SU¯ⁿ=τ Fⁿ+τ F^S,n+

M +M^SU¯ⁿ⁻¹. (2.9)

Here, ¯Uⁿ = vec(Uⁿ) is the vectorization of the solution matrix Uⁿ = [uⁿ_j,l]_M×N. Further, the mass, stiffness and stabilization matrices are defined as follows:

M :=M_x⊗M, M^S:=M_x⊗S A:=A_x⊗M+M_x⊗A, A^S :=M_x⊗G, (2.10) where⊗denotes the Kronecker product of two matrices. Further,

A_x:=

ΩX

∇xφ_j· ∇xφ_i+

ΩX

φ_j b_h· ∇xφ_i, M_x:=

ΩX

φ_jφ_i,

A:=

ΩL

g_h· ∇ψ_l ψ_k, M:=

ΩL

ψ_lψ_k, S:=

K∈Sh

δ_K(ψ_l, g_h· ∇ψ_k)_K, G :=

K∈Sh

δ_K(g_h· ∇ψ_l, g_h· ∇ψ_k)_K,

F^S,n:=

ΩX

K∈Sh

δ_K(fⁿ,g_h· ∇v_h)_K, Fⁿ :=

Ω

fⁿv_h.

(2.11)

Here, the entries in the mass matrix which are obtained by the Kronecker product of the matricesM_xandM, are given by:

M =M_x⊗M:=

⎡

⎢⎢

⎢⎣

[Φ_1,1]_k,l · · · [Φ_1,M]_k,l

· · ·

· · ·

· · ·

[Φ_M,1]_k,l· · ·[Φ_M,M]_k,l

⎤

⎥⎥

⎥⎦

MN ×MN

,

where the entries in theN × N block matrix [Φ_i,j]_k,l, 1≤k, l≤ N are evaluated by

[Φ_i,j]_k,l:=

ΩX

φ_iφ_j

ΩL

ψ_kψ.

Note that solving an algebraic system of sizeMN × MN in each time step will be very expensive. Moreover, when the PBE is coupled with the flow equations in a population balance system, this large system has to be solved repeatedly several times in each time step to decouple the system of equations. This, requires enormous computing power.

3. Operator-splitting finite element method

The operators Δ_x,∇x and ∇ in the population balance equation (1.1) are the decomposition of unmixed partial derivatives of the Cartesian coordinates x and , respectively. Thus, we can take advantage of the decomposition, and discretize equation (1.1) in space with d- and s-dimensional finite elements instead of (d+s)-dimensional finite elements. Using the Lie’s operator-splitting method, see for e.g., [9], in the time interval (tⁿ⁻¹, tⁿ), the operator-split equations of (1.1) read:

Step 1(L-direction).

For given ˆuⁿ⁻¹=u(tⁿ⁻¹, x, ), find ˆuⁿ in (tⁿ⁻¹, tⁿ) such that for allx∈Ω_X,

∂uˆ

∂t +g· ∇uˆ=f inΩ_L, ˆ

u(t, x, ) = 0 in∂Ω_L,

(3.1)

by consideringxas a parameter. In this step the solution is updated in theL-direction. Then, this solution ˆuⁿ is taken as the initial solution for theX-direction update.

Step 2(X-direction).

For given ˜uⁿ⁻¹= ˆuⁿ, find ˜uin (tⁿ⁻¹, tⁿ) such that for all for all ∈Ω_L,

∂u˜

∂t −Δ_xu˜+b· ∇xu˜= 0 inΩ_X,

˜

u(t, x, ) = 0 in∂Ω_X,

(3.2)

by considering as a parameter. Here the solutionuⁿ in the time step (tⁿ⁻¹, tⁿ) is obtained by first updating L-direction operators (Eq. (3.1)) and then updatingX-direction operators (Eq. (3.2)). Note that theL-direction equation (3.1) has to be solved only for inner pointx∈Ω_X due to the Dirichlet boundary condition. However, if we consider non-Dirichlet boundary conditions, then equation (3.1) has to be solved also for all boundary pointsx∈∂Ω_X. Similar arguments hold for theX-direction equation (3.2).

Next, to derive the discrete forms of the operator-split equations (3.1) and (3.2), we define ˆ

uⁿ_h(x_j, ) :=

N l=1

ˆ

uⁿ_j,lψ_l(), u˜ⁿ_h(x, _l) :=

M j=1

˜ uⁿ_j,lφ_j(x)

as the finite element functions for the equations (3.1) and (3.2), respectively. Here,x_j∈Ω_X,j= 1, . . . , N XP, and _l ∈ Ω_L, l = 1, . . . , N LP are the Cartesian coordinates which are necessary to evaluate the nodal functionals of the finite element spaces V_h and Q_h, respectively. As discussed before, the X-direction equation (3.2) is a standard convection-diffusion equation, and with sufficiently large in comparison to

|b|, it can be solved with the standard Galerkin method. However, the L-direction equation (3.1) is a pure advection equation and a stabilization method has to be used since the standard Galerkin method induce spurious oscillations in the solution. After applying the SUPG to (3.1) and the standard Galerkin to (3.2), the discrete form of the operator-split equations (3.1) and (3.2) in the time interval (tⁿ⁻¹, tⁿ) withu⁰_h=u₀read:

Step 1(L-direction).

For a givenfⁿ and ˆuⁿ⁻¹_h =uⁿ⁻¹_h , find ˆuⁿ_h∈Q_h such that ∂¯uˆⁿ_h, ψ_h

+a_S(ˆuⁿ_h, ψ_h)+

K∈Sh

δ_K∂¯uˆⁿ_h, g_h· ∇ψ_h

K = (fⁿ, ψ_h)+F_S^L(fⁿ, ψ_h), ∀ψ_h∈Q_h ∀x∈Ω_X, (3.3)

by consideringxas a parameter. Here, a_S(u_h, ψ_h) :=

ΩL

g_h· ∇u_h ψ_h+

K∈Sh

δ_K(g_h· ∇u_h, g_h· ∇ψ_h)_K, F_S^L(f, ψ_h) :=

K∈Sh

δ_K(f, g_h· ∇ψ_h)_K.

Step 2(X-direction).

For the given ˜uⁿ⁻¹_h = ˆuⁿ_h, find ˜uⁿ_h∈V_h,such that ∂¯u˜ⁿ_h, φ_h

x+a_X(˜uⁿ_h, φ_h) = 0, ∀ φ_h∈V_h, ∀ ∈Ω_L, (3.4) by consideringas a parameter. Here,

a_X(˜u_h, φ_h) :=

ΩX

∇xu˜_h· ∇xφ_h +

ΩX

b_h· ∇xu˜_hφ_h.

Here, (·,·) and (·,·)_x in equations (3.3) and (3.4) denote the L²-inner products in Ω_L and Ω_X, respectively.

Finally, we obtain the global discrete solution uⁿ_h(x, ) =

M j=1

N l=1

u_j,lφ_j(x)ψ_l()

by settingu_j,l= ˜u_j,l. In the next section, we address the consistency error, the stability and the convergence of the operator-splitting Galerkin/SUPG finite element scheme for equations (3.3) and (3.4).

4. Analysis of the operator-splitting finite element method

To obtain the stability anda priori error estimates for the operator-split equations (3.3) and (3.4), we first derive the equivalent one-step operator-split discrete form of the operator-split equations (3.3) and (3.4).

Lemma 4.1 (consistency of the operator-splitting method). The equivalent one-step discrete form of the operator-split equations (3.3)and (3.4)is

∂u¯ ⁿ_h, v_h

+a_LS(uⁿ_h, v_h) +a_OS(uⁿ_h, v_h) +

ΩX

K∈Sh

δ_K∂u¯ ⁿ_h, g_h· ∇v_h

K = (fⁿ, v_h) +F_S(fⁿ, v_h) (4.1) where the consistency error induced by the operator-splitting is

a_OS(uⁿ_h, v_h) =τ

Ω

g_h· ∇(∇xuⁿ_h)∇xv_h+τ

Ω

g_h· ∇(b_h· ∇xuⁿ_h)v_h

+τ

ΩX

K∈Sh

δ_K(g_h· ∇(∇xuⁿ_h), g_h· ∇(∇xv_h))_K

+τ

ΩX

K∈Sh

δ_K(g_h· ∇(b_h· ∇xuⁿ_h), g_h· ∇v_h)_K

+

ΩX

K∈Sh

δ_K(−Δxuⁿ_h, g_h· ∇v_h)_K

+

ΩX

K∈Sh

δ_K(b_h· ∇xuⁿ_h, g_h· ∇v_h)_K.

(4.2)

Proof. The algebraic form of theL-direction equation (3.3) can be written as

(M+S+τ(A+G)) ˆUⁿ=τ(F_Lⁿ+F_L^S,n) + (M+S) ˆUⁿ⁻¹, (4.3) where

F_Lⁿ:=

ΩL

fⁿψ_h, F_L^S,n =

K∈Sh

δ_K(fⁿ,g_h· ∇ψh)_K.

Here, the matrices S, G and F_L^S,n belong to the SUPG stabilization terms. Next, the algebraic form of the X-direction equation (3.4) can be written as

(M_x+τ A_x) ˜Uⁿ =M_xU˜ⁿ⁻¹. (4.4)

In the above equations (4.3) and (4.4),

Uˆⁿ= (Uⁿ)^T, U˜ⁿ=Uⁿ, Fⁿ :=

ΩL

fⁿψ_h. Now, multiply (4.3) by M_x⊗I, and (4.4) byI⊗(M+S+τ A+τ G), we get

((M_x⊗M) + (M_x⊗S) +τ(M_x⊗A) +τ(M_x⊗G)) ˆUⁿ =τ(M_x⊗F_Lⁿ) +τ(M_x⊗F_L^S,n) + ((M_x⊗M) + (M_x⊗S)) ˆUⁿ⁻¹ and

((M_x⊗M) + (M_x⊗S) +τ{(Ax⊗M) + (A_x⊗S) + (M_x⊗A) + (M_x⊗G)}

+τ²{(A_x⊗A) + (A_x⊗G)}

U˜ⁿ= ((M_x⊗M) + (M_x⊗S) +τ(M_x⊗A) +τ(M_x⊗G)) ˜Uⁿ⁻¹, respectively. Equating, the above equations, we get

((M_x⊗M) + (M_x⊗S) +τ{(Ax⊗M) + (A_x⊗S) + (M_x⊗A) + (M_x⊗G)}

+τ²{(Ax⊗A) + (A_x⊗G)}U¯ⁿ =τ(M_x⊗F_Lⁿ) +τ(M_x⊗F_L^S,n) + ((M_x⊗M) + (M_x⊗S)) ¯Uⁿ⁻¹. Using the definitions (2.10) in the above equation, we get

M +M^S+τ A+τ A^S+τ²(A_x⊗A) +τ²(A_x⊗G) +τ(A_x⊗S) U¯ⁿ

=τ(M_x⊗F_Lⁿ) +τ(M_x⊗F_L^S,n) +

M +M^SU¯ⁿ⁻¹. (4.5) For theA_x⊗A term, we have

(A_x⊗A) ¯Uⁿ =

Ω

uⁿ_j,l∇xφ_j· ∇xφ_i g_h· ∇ψ_lψ_k + uⁿ_j,lb_h· ∇xφ_jφ_i g_h· ∇ψ_lψ_k

=

Ω

g_h· ∇(uⁿ_j,l∇xφ_jψ_l)∇xv_h + g_h· ∇(b_h·(uⁿ_j,l∇xφ_j)ψ_l)v_h

=

Ω

g_h· ∇(∇xuⁿ_h)∇xv_h + g_h· ∇(b_h· ∇xuⁿ_h)v_h.

Similarly, for the cross termA_x⊗G, we have (A_x⊗G) ¯Uⁿ

=

ΩX

K∈Sh

K

δ_Kuⁿ_j,l∇xφ_j· ∇xφ_i g_h· ∇ψ_lg_h· ∇ψ_k + δ_Kuⁿ_j,lb_h· ∇xφ_jφ_ig_h· ∇ψ_lg_h· ∇ψ_k

=

ΩX

K∈Sh

K

δ_Kuⁿ_j,lg_h· ∇(∇xφ_jψ_l)g_h· ∇(∇xφ_iψ_k) + δ_Kuⁿ_j,lg_h· ∇(b_h· ∇xφ_jψ_l)g_h· ∇(φ_iψ_k)

=

ΩX

K∈Sh

K

δ_Kg_h· ∇(∇xuⁿ_h)g_h· ∇(∇xv_h) + δ_Kg_h· ∇(b_h· ∇xuⁿ_h)g_h· ∇v_h. Also for the (A_x⊗S) term, we have

(A_x⊗S) ¯Uⁿ =

ΩX

K∈Sh

K

δ_Kuⁿ_j,l∇xφ_j· ∇xφ_i ψ_lg_h· ∇ψ_k + δ_Kuⁿ_j,lb_h· ∇xφ_jφ_i ψ_lg_h· ∇ψ_k

=

ΩX

K∈Sh

K

−φi∇x·

uⁿ_j,l∇xφ_jδ_Kψ_l g_h· ∇ψ_k

+ δ_Kuⁿ_j,lb_h· ∇x(φ_jψ_l)g_h· ∇(φ_iψ_k)

=

ΩX

K∈Sh

K

−uⁿ_j,lδ_KΔ_x(φ_jψ_l)g_h· ∇(φ_iψ_k) + δ_Kb_h· ∇xuⁿ_h g_h· ∇v_h

=

ΩX

K∈Sh

K

−δKΔ_xuⁿ_h g_h· ∇v_h + δ_Kb_h· ∇xuⁿ_h g_h· ∇v_h.

Next, we show that the source termM_x⊗F_Lⁿ =Fⁿ. Each equation in the algebraic system (4.5) is obtained by applying summation to the ansatz indices j andk on both sides of the system. Therefore, the source term in the algebraic system (4.5) becomes

M_x⊗F_Lⁿ =

ΩX×ΩL

M i=1

M j=1

N k=1

fⁿφ_iφ_jψ_k.

Thus, the right hand side vector,rhs_i,k, i= 1, . . . ,M, k= 1, . . . ,N, can be written as rhs_i,k=

ΩX×ΩL

M j=1

fⁿφ_iφ_jψ_k =

ΩX×ΩL

fⁿφ_iψ_k M j=1

φ_j =

ΩX×ΩL

fⁿφ_iψ_k,

which is the source term in the algebraic system (2.9). Thus, we haveM_x⊗F_Lⁿ=Fⁿ. Similar argument holds

forM_x⊗F_L^S,n=F^S,n. Hence, the statement of the lemma.

Note that the discrete bilinear form (4.1) of the two-step operator-split equations is not same as the original discrete form (2.8). The difference is the consistency error due to the operator-splitting method.

Lemma 4.2. Let the discrete form of the assumptions (1.2)and (1.3)be satisfied. Then, for all v_h ∈W_h, we

have

Ω

g_h· ∇(b_h· ∇xv_h)v_h= 0,

ΩX

K∈Sh

δ_K(g_h· ∇(b_h· ∇xvⁿ_h), g_h· ∇v_h)_K= 0,

ΩX

K∈Sh

δ_K(Δ_xuⁿ_h, g_h· ∇v_h)_K= 0,

ΩX

K∈Sh

δ_K(b_h· ∇xv_h, g_h· ∇v_h)_K = 0.

Proof. We use the definitions of the finite element spaces (2.2) to show this property. In particular, we repeat- edly use the properties that the basis functions ofV_handQ_hare independent of∈Ω_Landx∈Ω_X, respectively.