Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
Contents lists available at SciVerse ScienceDirect
Journal of Computational and AppliedMathematics
journal homepage: www.elsevier.com/locate/cam
Finite element methods of an operator splitting applied to populationbalance equationsNaveed Ahmed a,∗, Gunar Matthies b, Lutz Tobiska a
a Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germanyb Universität Kassel, Fachbereich 10 Mathematik und Naturwissenschaften, Institut für Mathematik, Heinrich-Plett-Straße 40, 34132 Kassel, Germany
a r t i c l e i n f o
Article history:Received 15 April 2011Received in revised form 8 August 2011
In population balance equations, the distribution of the entities depends not only on spaceand time but also on their own properties referred to as internal coordinates. The operatorsplitting method is used to transform the whole time-dependent problem into twounsteady subproblems of a smaller complexity. The first subproblem is a time-dependentconvection–diffusion problem while the second one is a transient transport problem withpure advection. We use the backward Euler method to discretize the subproblems in time.Since the first problem is convection-dominated, the local projection method is applied asstabilization in space. The transport problem in the one-dimensional internal coordinate issolved by a discontinuous Galerkin method. The unconditional stability of the method willbe presented. Optimal error estimates are given. Numerical tests confirm the theoreticalresults.
Population balance equations (PBEs) have many applications in various branches of engineering and science [1]. Forexample, in [2] a precipitation process involving chemical reaction in a flow field has been modeled by a populationbalance system. It consists of equations describing the flow field by the Navier–Stokes equations, the chemical reactionsby convection–diffusion–reaction equations, and the particle size distribution by transport equations. These equationsare strongly coupled such that inaccuracies in concentration of one species directly effect the concentration of all otherspecies. In addition to the coupling of the equations in the system, the other difficulty in the simulation is that the particlesize distribution depends not only on space and time but also on its own properties refereed to as internal or propertycoordinates. Consequently, the dimension of the population balance equation is higher than those of the other equationsin the system. Due to the higher dimension of the population balance equation, the numerical simulation of the coupledsystem with standard numerical schemes is a challenge from the computational point of view.
In recent years, several numerical methods have been introduced for population balance equations, for example, themethod of moments and its variants [3,4], characteristics method of lines [5], finite differences [6], etc. The method ofmoments and the characteristics method of lines reduce the PBE to a set of simpler differential equations. Instead ofcomputing density functions, themethod of moments computes their moments. However, the reconstruction of the densityfunction from itsmoments is a difficult task [7] since the inverse problem is ill-posed.Moreover, the size dependent functions
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1605
violate the closure condition in the methods of moments. The characteristics method transforms the population balanceequation into a system of ODEs that is then solved along the path lines of the particles. The method is highly efficientwhen the physics is simple. However, the approach is restricted to variables without discontinuities. Furthermore, it is arelatively complicated procedure which cannot be easily implemented since time and property coordinates are coupled.Most of these and other methods are restricted to only internal coordinates. Hence, it is motivated to find a computationallyefficient numerical scheme for solving multidimensional population balance equations.
In order to overcome the curse of dimensionality associated with the population balance equations, we advocate theoperator splitting method to approximate the solution of the population balance equation [8]. Operator splitting methodshave been used in different contexts, i.e., splitting with respect to coordinates or directions, physics etc., see [9–12].A detailed analysis of an alternating direction implicit (or operator-splitting) scheme is demonstrated in [13] for thesix-dimensional Fokker–Planck equation. The basic idea in [13] is to split the high dimensional problem into two lowdimensional problems corresponding to the configuration and the physical spaces. The solution of the convection–diffusiontype problem in configuration space is obtained by a Galerkin spectral method at each quadrature point corresponding tothe physical domain. Furthermore, a type of L2 projection is used to update the right-hand side vector at the second stagewhere the solution of the advection equation in physical space is approximated by a finite element method.
In our context, the idea is to split the original problem with respect to internal and external directions, resultingin a time-dependent convection–diffusion problem and a transient transport problem with pure advection. The mainadvantage of such splitting is that each subproblem can be discretized and stabilized separately by the best suitable methodindependently of the other subproblem(s). For example, in [8] the Streamline-Upwind Petrov–Galerkin method (SUPG) ininternal coordinates has been combinedwith the standard Galerkinmethod in space. The SUPGmethod, originally proposedfor steady state problems in [14] has been recently extended to transient problems [15,16]. Standard energy argumentsapplied to the fully discrete problem yields error estimates under the conditions which couple the choice of stabilizationparameters to the length of the time step. In particular, the SUPG stabilization vanishes in the time-continuous limit. Thisbehavior is caused by the time derivative appearing in the stabilization terms of the SUPG to guarantee consistency andproduces a non-symmetric term difficult to handle. Nevertheless, in [16], using a different analysis for the case of time-independent coefficients on uniform grids optimal error estimates with the standard choice of the stabilization parameterindependent of the time step have been proven and confirmed by numerical experiments.
An alternative to SUPG is the local projection stabilization method. The local projection method provides additionalcontrol over the fluctuations of the gradient or parts of it. Although, the method is weakly consistent only, the consistencyerror can be bounded such that the optimal order of convergence is maintained. Furthermore, neither time derivativesnor second order derivatives have to be assembled for the stabilization term of LPS. Further, no coupling conditions ofstabilization parameters to the length of the time step arise. Originally the local projection stabilization method wasintroduced for the Stokes problem [17]. Over the last decade, a significant amount of research has been published inthe literature, e.g., see [18–22] and references therein. The LPS in space combined with a discontinuous Galerkin (dG)method in time for transient convection–diffusion–reaction equations has been studied in [23]. A comprehensive analysisfor symmetric stabilizations in space, such as the continuous interior penalty method (CIP), the local projection stabilizationmethod and the quasi-static orthogonal subscales method (OSS), combined with the θ-scheme and the second orderbackward difference formula (BDF2) can be found in [24].
The second subproblem in our splitting method is a transient transport problem with pure advection, thus one suitablechoice is to approximate it by the discontinuous Galerkin (dG) method. The dGmethod was first introduced for the neutrontransport problem in [25] and then analyzed in [26]. The theoretical analysis of the dGmethod for scalar hyperbolic equationscan be found in [27] and for the space–time dG method in [28]. For an introduction to discontinuous Galerkin methods werefer to [29].
The aim of the present paper is to combine the local projection stabilization method in space with a discontinuousGalerkin method in internal coordinates. We will give stability and convergence estimates for the fully discrete two-stepscheme based on an operator splitting method.
The format of the paper is as follows: Section 2 introduces the model problem under consideration and defines basicnotations. In Section 3, the operator splitting technique is applied to decompose the problem into two simpler ones. Weshall formulate the backward Euler discretization and derive the weak form of the two subproblems. Further, we derive theunconditional stability of the two-stepmethod.We then discretize the subproblems in space and internal coordinates usinglocal projection stabilization and discontinuous Galerkin methods, respectively, in Section 4. We show the unconditionalstability of the fully discrete two-step method. Section 5 presents the error analysis of the fully discrete scheme. Someimplementation issues of themethod are discussed in Section 6. Finally, we present in Section 7 some computational resultssupporting our theoretical predictions.
2. Model problem
Let Ωx be a domain in Rd (d = 2 or 3) with boundary ∂Ωx,Ωℓ = [ℓmin, ℓmax] ⊂ R and T > 0. The state ofindividual particles in population balance equation may consist of the external coordinate x, referring to its position in thephysical space, and the internal coordinate ℓ, representing the properties of particles, such as size, temperature, volume etc.
1606 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
Apopulation balance for a solid process such as crystallizationwith one internal coordinate can be described by the followingpartial differential equation:
Find z : (0, T )×Ωℓ ×Ωx → R such that∂z∂t
+∂(Gz)∂ℓ
− ε∆xz + b(x) · ∇xz = f in (0, T ] ×Ωℓ ×Ωx,
z(0, ·) = z0 inΩℓ ×Ωx,
z|ℓmin = zmin on (0, T ] ×Ωx,
z = 0 on (0, T ] ×Ωℓ × ∂Ωx,
(1)
where the diffusion coefficient ε > 0 is a given constant,∆x and∇x represent the Laplacian and the gradient with respect tox, respectively, b is a given velocity field satisfying ∇x · b = 0, and f is a source function. Here G > 0 represents the growthrate of the particles that depends on ℓ but is independent of x and t , we also assume that ∂ℓG ≥ 0, see [30,31]. Furthermore,let us consider the data of the problem G, b, f , z0 and zmin to be sufficiently smooth functions.
Let us introduce some standard notations. Let Hm(Ω) denote the Sobolev space of functions with derivatives up to orderm in L2(Ω). We denote by (·, ·) the inner product in L2(Ωℓ × Ωx) and by ‖ · ‖0 the corresponding L2-norm. To distinguishthe inner products and the corresponding norms with respect to the internal coordinate and the space variable we needsome more notation. For this, let us denote by (·, ·)ℓ and ‖ · ‖L2(Ωℓ) the L2-inner product and the associated norm in Ωℓ,respectively, and by (·, ·)x and ‖ · ‖L2(Ωx) the L2-inner product and the associated norm inΩx. The norm in the Sobolev spaceHm(Ωx) is defined as
‖v‖m =
−|α|≤m
‖Dαv‖2L2(Ωx)
1/2
where α = (α1, α2, . . . , αd) is a multi-index.We also consider certain Bochner spaces. For this, let X be a Banach space withnorm ‖ · ‖X . For spaces X and Y we use the short notation Y (X) := Y (Ωℓ; X). Then we define
C(Ωℓ; X) =
v : Ωℓ → X : v continuous
,
L2(Ωℓ; X) =
v : Ωℓ → X :
∫Ωℓ
‖v(ℓ)‖2X < ∞
,
Hm(Ωℓ; X) =
v ∈ L2(Ωℓ; X) :
∂ jv
∂ℓj∈ L2(Ωℓ; X), 1 ≤ j ≤ m
,
where the derivatives ∂ jv/∂ℓj are understood in the sense of distribution onΩℓ. The norms in the above defined spaces aregiven as follows
‖v‖C(X) = supℓ∈Ωℓ
‖v(ℓ)‖X , ‖v‖L2(X) =
∫Ωℓ
‖v(ℓ)‖2X
1/2
, ‖v‖Hm(X) =
∫Ωℓ
m−j=0
∂ jv∂ℓj2X
1/2
.
3. Operator splitting method
The numerical method for solving (1) in d + 1 variables is based on an operator splitting with respect to (ℓ, t) and (x, t)in theΩℓ andΩx directions, respectively. We consider a uniform partition of the time interval, i.e., tn = τn, n = 1, . . . ,N ,with τ = T/N . Let w(tn+) = limt→tn+0w(t). Starting with u(t0) = z0, two subproblems are sequentially solved on thesub-intervals (tn, tn+1
], n = 0, 1, . . . ,N − 1:Given u(tn) find u : (tn, tn+1
] ×Ωℓ ×Ωx → R such that∂ u∂t
+ Lxu = f in (tn, tn+1] ×Ωℓ ×Ωx,
u = 0 on (tn, tn+1] ×Ωℓ × ∂Ωx,
u(tn+) = u(tn).
(2)
Given u(tn+1), find u : (tn, tn+1] ×Ωℓ ×Ωx → R such that
∂u∂t
+ Lℓu = 0 in (tn, tn+1] ×Ωℓ ×Ωx,
u|ℓmin = zmin on (tn, tn+1] ×Ωx,
u(tn+) = u(tn+1),
(3)
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1607
where
Lℓz =∂(Gz)∂ℓ
, Lxz = −ε∆xz + b · ∇xz. (4)
This two-step operator splitting scheme defines u(tn), n = 1, . . . ,N , as an approximation of z(tn).The critical issue of the operator splitting method is the overall accuracy of the two-step method. Using Taylor series
expansions first order accuracy of the two-stepmethod (2) and (3) can be shown. A detailed error analysis for the first orderLie operator splitting of the sum of two elliptic operators can be found in [32,33]. Unfortunately, we cannot use these resultsdue to the hyperbolic nature of the operator Lℓ.
In the framework of PBE, the first subproblem (2) is a time-dependent convection–diffusion equation posed on Ωxparametrized by the variable ℓ in the internal coordinate and the second subproblem (3) is a one-dimensional transportproblem onΩℓ parametrized by the space variable x.
Let us consider the spaces V = H10 (Ωx) and W = H1(Ωℓ). We introduce the space
P =
v ∈ L2(Ωℓ ×Ωx) : v ∈ L2(Ωx;W ) ∩ L2(Ωℓ; V )
.
A variational form of (2) and (3) reads as follows:First step. Find u : (tn, tn+1
] → P with u(tn+) = u(tn) such that∫Ωℓ
ut , v
x +
∫Ωℓ
a(u, v) =
∫Ωℓ
f , v
x ∀v ∈ P , (5)
where the bilinear form a is defined as
a(u, v) = ε(∇xu,∇xv)x + (b · ∇xu, v)x.
Second step. Find u : (tn, tn+1] → P with u(tn+) = u(tn+1) such that∫
Ωℓ
ut , v
x + b
u, v
=Gminzmin, v(ℓmin)
x ∀v ∈ P , (6)
where Gmin = G(ℓmin) and the bilinear form b is defined as
b(u, v) =
∫Ωℓ
∂(Gu)∂ℓ
, v
x
+
(Gu)(ℓmin), v(ℓmin)
x.
Note that we have imposed the boundary condition (u|ℓmin = zmin) in the ℓ-direction in a weak sense.After discretizing in time by the backward Euler method, the first order accurate implicit scheme is considered as a two-
step method:First step. Given un
∈ P , find un+1∈ P such that∫
Ωℓ
un+1
− un
τ, v
x
+
∫Ωℓ
a(un+1, v) =
∫Ωℓ
(f n+1, v)x v ∈ P . (7)
Second step. Update un+1 from the first step and find the solution un+1∈ P such that∫
Ωℓ
un+1
− un+1
τ, v
x
+ b(un+1, v) =
Gminzn+1
min , v(ℓmin)x
(8)
for all v ∈ P , where zn+1min = zmin(tn+1, ·).
The next paragraph gives the stability of the two-step method (7) and (8).
Lemma 1. Assume that un, un, n = 1, 2 . . . ,N, are the solutions obtained from the two-step algorithm (7) and (8). If ∂ℓG ≥ 0and τ ≤
14 , then the estimate
‖uN‖20 + τ
N−1−n=0
∫Ωℓ
2ε‖un+1
‖2H1(Ωx)
+ ∂ℓG‖un+1‖2L2(Ωx)
≤ exp(2T )
‖u0
‖20 + τ
N−1−n=0
43‖f n+1
‖20 + ‖G1/2
minzn+1min ‖
2L2(Ωx)
(9)
holds.
1608 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
Proof. Setting v = un+1 in (7) yields∫Ωℓ
(un+1− un, un+1)x + τ
∫Ωℓ
a(un+1, un+1) = τ
∫Ωℓ
(f n+1, un+1)x.
Using the relation 2(a − b)a = a2 − b2 + (a − b)2, one can write∫Ωℓ
(un+1− un, un+1)x =
12‖un+1
‖20 −
12‖un
‖20 +
12‖un+1
− un‖20.
Integrating by parts with respect to x the second term in the bilinear form a(·, ·), one obtains∫Ωℓ
a(un+1, un+1) = ε
∫Ωℓ
‖un+1‖2H1(Ωx)
since un+1 vanishes on the boundary ∂Ωx and ∇x · b = 0. Hence by using the Cauchy–Schwarz inequality for the right-handside, we have for the first step
‖un+1‖20 − ‖un
‖20 + ‖un+1
− un‖20 + 2τε
∫Ωℓ
‖un+1‖2H1(Ωx)
≤ τ‖f n+1‖20 + τ‖un+1
‖20. (10)
Substituting v = un+1 in the second step (8) gives∫Ωℓ
(un+1− un+1, un+1)x + τb(un+1, un+1) = τ
Gminzn+1
min , un+1(ℓmin)
x. (11)
Starting from
b(un+1, un+1) =
∫Ωℓ
∂(Gun+1)
∂ℓ, un+1
x
+
Gminun+1(ℓmin), un+1(ℓmin)
x
an integration by parts twice with respect to ℓ gives
b(un+1, un+1) =12
∫Ωℓ
∂ℓG‖un+1‖2L2(Ωx)
+12‖G1/2
maxun+1(ℓmax)‖
2L2(Ωx)
+12‖G1/2
minun+1(ℓmin)‖
2L2(Ωx)
where Gmax = G(ℓmax). The Cauchy–Schwarz inequality gives for the right-hand side in (11)Gminzn+1
min , un+1(ℓmin)
x≤
12‖G1/2
minzn+1min ‖
2L2(Ωx)
+12‖G1/2
minun+1(ℓmin)‖
2L2(Ωx)
.
Combining these two results in (11) and using the same relation 2(a − b)a = a2 − b2 + (a − b)2 for first term, we get forthe second step
‖un+1‖20 − ‖un+1
‖20 + ‖un+1
− un+1‖20 + τ
∫Ωℓ
∂ℓG‖un+1‖2L2(Ωx)
≤ τ‖G1/2minz
n+1min ‖
2L2(Ωx)
. (12)
Adding (10) and (12), neglecting some contribution of non-negative terms, and summing over n = 0, . . . ,N − 1, we obtain
‖uN‖20 + τ
N−1−n=0
∫Ωℓ
2ε‖un+1
‖2H1(Ωx)
+ ∂ℓG‖un+1‖2L2(Ωx)
≤ ‖u0
‖20 + τ
N−1−n=0
‖f n+1
‖20 + ‖G1/2
minzn+1min ‖
2L2(Ωx)
+ τ
N−1−n=0
‖un+1‖20.
From (10) we have
‖un+1‖20 ≤
τ
1 − τ‖f n+1
‖20 +
11 − τ
‖un‖20. (13)
Using this estimate in the last inequality, we get
‖uN‖20 + τ
N−1−n=0
∫Ωℓ
2ε‖un+1
‖2H1(Ωx)
+ ∂ℓG‖un+1‖2L2(Ωx)
≤ ‖u0
‖20 + τ
N−1−n=0
43‖f n+1
‖20 + ‖G1/2
minzn+1min ‖
2L2(Ωx)
+
4τ3
N−1−n=0
‖un‖20,
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1609
where we have used 1/(1 − τ) ≤ 4/3 for τ ≤ 1/4. We conclude the statement by using Gronwall’s lemma. This completesthe proof.
4. Fully-discrete method
In view of different properties of the operators Lℓ and Lx, the operator splitting technique allows us to use different typesof discretization methods to solve the problems in Ωℓ and Ωx. Since the first subproblem (7) is convection-dominated,we use the local projection method to stabilize the space discretization. While the second subproblem (8) is a transportproblem with pure advection, one suitable choice is the discontinuous Galerkin method for the discretization with respectto the internal coordinate.
4.1. Local projection stabilization in space
In this subsection, we discretize the subproblem in space. For this, let us denote by Th a family of shape regulardecompositions of Ωx into d-simplices, quadrilateral or hexahedra. The diameter of a cell K ∈ Th is denoted by hK andh describes the maximum diameter of cells K .
Let Vh ⊂ V denote the standard finite element space of continuous, piecewise polynomials of degree r . The Galerkindiscretization of problem (7) is generally unstable due to dominating advection when the diffusion coefficient is very smallε ≪ 1. We will consider the one-level LPS in which the approximation and projection spaces live on the same mesh.For other variants of LPS we refer to [34,35,20,22]. We will handle this difficulty by adding a stabilizing term based onlocal projection. Let Dh be the projection space of discontinuous, piecewise polynomials of degree r − 1 with r ≥ 1. LetDh(K) = qh|K : qh ∈ Dh be the local projection space andπK : L2(K) → Dh(K) the local L2-projection ontoDh(K). Definethe global projection πh : L2(Ωx) → Dh by (πhv)|K := πK (v|K ). The fluctuation operator κh : L2(Ωx) → L2(Ωx) is given byκh := id − πh, where id : L2(Ωx) → L2(Ωx) is the identity mapping.
We define the stabilizing term Sh
Sh(uh, vh) =
−K∈Th
µK
κh(∇xuh), κh(∇xvh)
K
with user chosen non-negative constant µK , K ∈ Th. It gives additional control over the fluctuations of gradients. Notethat one can also replace the gradient ∇xwh by the derivative in the streamline direction b · ∇xwh or (even better [35]) bybK · ∇xwh where bK is a piecewise constant approximation of b, which leads to similar results.
The stabilized bilinear form is then defined asah(uh, vh) = a(uh, vh)+ Sh(uh, vh). (14)
The bilinear form ah is coercive on Vh with respect to the mesh dependent norm
|||v||| :=
ε|v|2H1(Ωx)
+
−K∈Th
µK‖κh(∇xv)‖2L2(K)
1/2
, (15)
that is ah(vh, vh) ≥ |||vh|||2 for all vh ∈ Vh. The stability and convergence properties of the LPS method are based on the
following assumptions with respect to the pair (Vh,Dh), see [20,22].
Assumption A1. There is an interpolation operator jh : H2(Ω) → Vh such that the approximation properties
‖v − jhv‖0,K + hK |v − jhv|1,K ≤ ChlK‖v‖l,K ∀v ∈ H l(Ωx), 2 ≤ l ≤ r + 1, (16)
for all K ∈ Th and the orthogonality
(v − jhv, qh) = 0 ∀qh ∈ Dh, ∀v ∈ H2(Ω) (17)
hold true.
Assumption A2. The fluctuation operator κh satisfies the following approximation property
‖κhq‖0,K ≤ ChlK |q|l,K ∀K ∈ Th, ∀q ∈ H l(K), 0 ≤ l ≤ r. (18)
In numerical computations,we usemapped finite element spaces, see [36],where on the reference cellK the enriched spacesare given by
Pbubbler (K) = Pr(K)+ bPr−1(K),
Q bubbler (K) = Qr(K)+ span
bxr−1
i , i = 1, 2.
Here, b and b are the cubic bubble on the reference triangle and the biquadratic bubble on the reference square,respectively. The pairs (Pbubble
r , Pdiscr−1), r ≥ 1, on triangles and the pairs (Q bubble
r , Pdiscr−1), r ≥ 1, on quadrilaterals fulfill
Assumptions A1 and A2. Further examples of spaces (Vh,Dh) satisfying A1 and A2 are given in [20,22].
1610 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
4.2. Discontinuous Galerkin method in internal coordinates
To discretize (7) and (8) in internal coordinate ℓ, we apply a discontinuousGalerkinmethod. LetM > 0 be a given positiveinteger and ℓmin = ℓ0 < ℓ1 < · · · < ℓM = ℓmax is a partition ofΩℓ with Ii = (ℓi−1, ℓi], ki = ℓi − ℓi−1, and k = maxi ki. Letus introduce the function space of discontinuous piecewise polynomials of degree q ≥ 1 as
Sqk =
v : Ωℓ → R : v|Ii(ℓ) =
q−j=0
vjℓj with vj ∈ R, j = 0, . . . , q
.
Then we give the fully discrete space Sr,qh,k as follows
Sr,qh,k = Vh ⊗ Sqk =
v : Ωℓ ×Ωx → R : v|Ii(ℓ) =
q−j=0
vjℓj with vj ∈ Vh, j = 0, . . . , q
.
The functions in these spaces are allowed to be discontinuous at the nodes ℓi, i = 1, . . . ,M−1. The jumps across the nodesare defined by [φ]i = φ(ℓ+
i )− φ(ℓ−
i ), where
ϕ±
m = ϕ(ℓ±
m) = limℓ→ℓm±0
ϕ(ℓ).
In the next paragraph, we define the fully discrete scheme based on the two-step method.First step. For given un
h,k ∈ Sr,qh,k , find un+1h,k ∈ Sr,qh,k such that∫
Ωℓ
un+1h,k − un
h,k
τ, X
x
+
∫Ωℓ
ah(un+1h,k , X) =
∫Ωℓ
(f n+1, X)x (19)
for all X ∈ Sr,qh,k where u0h,k is a suitable approximation of z0 in Sr,qh,k .
Second step. Update the solution un+1h,k from (19) and find un+1
h,k ∈ Sr,qh,k such that∫Ωℓ
un+1h,k − un+1
h,k
τ, X
x
+ B(un+1h,k , X) =
Gminzn+1
min,h, X(ℓ+
0 )x
(20)
for all X ∈ Sr,qh,k where zn+1min,h ∈ Sr,qh,k is an approximation of zn+1
min and the bilinear form B is defined as
B(u, v) =
M−i=1
∫Ii
∂(Gu)∂ℓ
, v
x
+
M−1−i=1
Gu
i, v(ℓ+
i )x+
Gminu(ℓ+
0 ), v(ℓ+
0 )x. (21)
Integrating by parts∫Ii
∂(Gu)∂ℓ
, v
x
=
Gu(ℓ−
i ), v(ℓ−
i )x−
Gu(ℓ+
i−1), v(ℓ+
i−1)x−
∫Ii
Gu,
∂v
∂ℓ
x
leads to the representation
B(u, v) = −
M−i=1
∫Ii
Gu,
∂v
∂ℓ
x
−
M−1−i=1
u(ℓ−
i ),Gv
i
x+
Gmaxu(ℓ−
M), v(ℓ−
M)x. (22)
For L2 functions v : Ωℓ×Ωx → R having traces v(ℓ+
i−1, ·) and v(ℓ−
i , ·) in L2(Ωx), i = 1, . . . ,M , we introduce the seminorm
‖v‖dG =
M−i=1
∫Ii∂ℓG‖v‖2
L2(Ωx)+ ‖G1/2
minv(ℓ+
0 )‖2L2(Ωx)
+
M−1−i=1
‖[(G1/2v)]i‖2L2(Ωx)
+ ‖G1/2maxv(ℓ
−
M)‖2L2(Ωx)
1/2
(23)
which becomes a norm when ∂ℓG > 0.
Lemma 2. The bilinear form B is coercive with respect to the mesh dependent norm ‖ · ‖dG, i.e.,
B(v, v) ≥12‖v‖2
dG (24)
holds for all v ∈ Sr,qh,k .
Proof. Setting u = v in (21) and (22), then adding them together we conclude the statement of the lemma.
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1611
The next lemma provides a stability result of the fully discrete two-step method (19) and (20).
Lemma 3. Let ∂ℓG ≥ 0 and τ ≤ 1/2, then the solution unh,k and un
h,k, n = 1, 2, . . . ,N, of (19) and (20), respectively, satisfies
‖uNh,k‖
20 + 2τ
N−1−n=0
∫Ωℓ
|||un+1h,k |||
2+ τ
N−1−n=0
‖un+1h,k ‖
2dG ≤ exp(2T )
‖u0
h,k‖20 + τ
N−1−n=0
43‖f n+1
‖20 + ‖(G1/2
minzn+1min,h)‖
2L2(Ωx)
.
Proof. Following similar derivation steps as in Lemma 1, we get the proof of the lemma.
5. Error analysis
In this section, we derive the error estimates of the fully discrete two-step scheme (19) and (20). First we define a specialinterpolantΠkw(t, ·, x) ∈ Sqk of a functionw(t, ℓ, x) by
Πkw(ℓ−
i ) = w(ℓ−
i ), i = 1, . . . ,M − 1, (25)∫Ii(Πkw − w)ℓs = 0, s ≤ q − 1, i ≥ 1, (26)
i.e., Πkw interpolates at the nodal points and the interpolation error is orthogonal to the space of polynomials of degreeq− 1 on Ii. For this type of interpolant we have the following error estimates (with | · |0 = ‖ · ‖L2(Ωx) and | · |1 = |∇x · |L2(Ωx))
sup0≤ℓ≤ℓM
|Πkw(ℓ)− w(ℓ)|j ≤ Ckq+1 sup0≤ℓ≤ℓM
|w(q+1)(ℓ)|j, j = 0, 1. (27)
In addition, we have the stability property of interpolantΠk given by∫Ωℓ
‖Πku‖2Hr+1(Ωx)
≤ C∫Ωℓ
‖u‖2Hr+1(Ωx)
(28)
sinceΠk acts in ℓ-direction and the norms are with respect to the space direction. Furthermore, it holds∫Ii|Πkw
see [37]. In order to obtain the error estimate for the splitting method in space and internal coordinates, we define aprojection operator Rh which maps onto the tensor product space Sr,qh,k . It is defined as follows
Rhw = jhΠkw = Πkjhw ∀w ∈ P , (30)where jh is the special interpolant in space satisfying Assumption A1. Let us consider ξ n := u(tn) − Rhu(tn) and ηn :=
Rhu(tn)−unh,k. We also denote ξ n := u(tn)−Rhu(tn) and ηn := Rhu(tn)− un
h,k, then the error u(tn)−unh,k can be decomposed
as followsen = u(tn)− un
h,k = ξ n + ηn
where unh,k is the solution for fully discrete scheme (19) and (20) and u(tn) is the solution of (2) and (3). Furthermore, to
obtain the separate estimates in space and internal coordinate we use the following decomposition of errors
ξ = Rhw − w =Rhw −Πkw
+Πkw − w
= ϑ + ϕ. (31)
Assumption A3. Let u, ut , utt , u, ut , utt , zmin and z0 satisfy the following regularity conditions
u, u ∈ H10, T ; L2Ωℓ;Hr+1(Ωx)
∩ H10, T ;Hq+1Ωℓ;H1(Ωx)
,
ut , ut ∈ L20, T ; L2(Ωℓ;Hr+1(Ωx))
∩ L2
0, T ;Hq+1(Ωℓ; L2(Ωx))
,
utt , utt ∈ L20, T ; L2
Ωℓ; L2(Ωx)
, z0 ∈ L2
Ωℓ;Hr+1(Ωx)
∩ Hq+1Ωℓ; L2(Ωx)
,
zmin ∈ H10, T ;Hr+1(Ωx).
Lemma 4. Let the Assumptions A1–A3 be fulfilled. Then for all t ∈ (0, T ], we have the following estimates for the interpolationerror
‖ϑ(t)‖dG ≤ Chr+1‖u(t)‖L2(Hr+1) + ‖u(t)‖C(Hr+1)
,
‖ϕ(t)‖dG ≤ C kq+1/2‖u(t)‖Hq+1(L2).
Proof. For simplicitywe skip the dependency of t within the proof. Since for the interpolation error the jumps [jhu−u]i, i =
1, . . . ,M − 1, vanish due to the continuity of jhu in the internal coordinate, we have from (23), the interpolation error
1612 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
estimates (16) and condition (28)
‖ϑ‖2dG ≤
M−1−i=1
∫Ii∂ℓG‖ϑ‖
2L2(Ωx)
+ ‖G1/2minϑ(ℓ
+
0 )‖2L2(Ωx)
+ ‖G1/2maxϑ(ℓ
−
M)‖2L2(Ωx)
≤ Ch2r+2‖u‖2
L2(Hr+1)+ ‖u‖2
C(Hr+1)
.
For the second estimate with respect to the internal coordinate, we use the definition of interpolant Πku, i.e., theinterpolation Πku satisfies Πku(ℓ−
i ) = u(ℓ−
i ), i = 1, . . . ,M , thus from the second representation (22) of the bilinearform B and interpolation estimates (27), (29), we have
‖ϕ‖2dG ≤ B(ϕ, ϕ) =
M−i=1
∫Ii−
Gϕ,
∂ϕ
∂ℓ
x
≤
M−i=1
∫Ii‖Gϕ‖L2(Ωx)‖∂ℓϕ‖L2(Ωx)
≤ C k2q+1M−i=1
∫Ii‖u(q+1)
‖2L2(Ωx)
≤ C k2q+1‖u‖2
Hq+1(L2)
which completes the proof of the lemma.
Lemma 5. Let Assumptions A1–A3 be fulfilled and µK ∼ hK . Then for all t ∈ (0, T ], the following estimates hold∫Ωℓ
ahϑ(t), η(t)
≤ C(ε1/2 + h1/2)hr
‖u(t)‖L2(Hr+1)
∫Ωℓ
|||η(t)|||21/2
+ Chr+1‖u(t)‖L2(Hr+1)‖η(t)‖0, (32)
∫Ωℓ
ahϕ(t), η(t)
≤ C(ε1/2 + h1/2)kq+1
‖u(t)‖Hq+1(H1)
∫Ωℓ
|||η(t)|||21/2
+ C kq+1‖u(t)‖Hq+1(H1)‖η(t)‖0, (33)
Bϑ(t), η(t)
≤ Chr+1
[‖u(t)‖H1(Hr+1)‖η(t)‖0 + ‖u(t)‖C(Hr+1)‖η(t)‖dG
], (34)
Bϕ(t), η(t)
≤ C kq+1
‖u(t)‖Hq+1(L2)‖η(t)‖0. (35)
Proof. For simplicity of the presentation we again skip the dependency on the time within the proof. From the definition ofthe stabilized bilinear form ah, we have∫
Ωℓ
ahϑ, η
= ε
∫Ωℓ
∇xϑ,∇xη
x +
∫Ωℓ
b · ∇xϑ, η
x +
∫Ωℓ
Shϑ, η
= I1 + I2 + I3. (36)
We start by estimating the first term on the right-hand side. Using the Cauchy–Schwarz inequality, the interpolationestimates (16) of jh and condition (28), it follows that
|I1| ≤ ε
∫Ωℓ
||ϑ ||H1(Ωx)||η||H1(Ωx) ≤ Cε1/2hr∫Ωℓ
‖Πku‖Hr+1 |||η|||
≤ Cε1/2hr∫
Ωℓ
‖u‖2Hr+1
1/2 ∫Ωℓ
|||η|||21/2
≤ Cε1/2hr‖u‖L2(Hr+1)
∫Ωℓ
|||η|||21/2
.
Integrating I2 by parts with respect to the space variable x, using the orthogonality property of interpolant jh and theCauchy–Schwarz inequality to get
|I2| =
∫Ωℓ
b · ∇xϑ, η
x
=
∫Ωℓ
ϑ, b · ∇xη
x
≤
∫Ωℓ
ϑ, κh(b · ∇xη)
x
≤
∫Ωℓ
−K∈Th
‖ϑ‖L2(K)‖κh(b · ∇xη)‖L2(K).
Let b be the L2-projection of b in the space of piecewise constant functions with respect to Th. Using the L2-stability of thefluctuation operator κh, an inverse inequality and κh(b · ∇x)η = b · κh(∇xη), we get in the same fashion as in [20] thefollowing estimate
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1613
Thus inserting this in the previous estimate, using (16), µK ∼ hK , and (28) we get
|I2| ≤ C∫Ωℓ
−K∈Th
‖ϑ‖L2(K)
[|b|1,∞,K‖η‖L2(K) + |b|0,∞,K‖κh(∇xη)‖L2(K)
]≤ Chr+1
∫Ωℓ
‖u‖Hr+1(Ωx)‖η‖L2(Ωx) + Chr+1/2∫Ωℓ
‖u‖Hr+1(Ωx)|||η|||
≤ Chr+1/2
h1/2
‖η‖0 +
∫Ωℓ
|||η|||2
1/2‖u‖L2(Hr+1).
For I3, the Cauchy–Schwarz inequality and interpolation error estimates give
|I3| =
∫Ωℓ
Shϑ, η
≤
∫Ωℓ
Shϑ, ϑ
1/2Shη, η
1/2≤ Chr+1/2
∫Ωℓ
‖u‖Hr+1(Ωx)|||η||| ≤ Chr+1/2‖u‖L2(Hr+1)
∫Ωℓ
|||η|||21/2
.
Inserting these estimates into (36), we get the desired estimate∫Ωℓ
ahϑ, η
≤ C(ε1/2 + h1/2)hr
‖u‖L2(Hr+1)
∫Ωℓ
|||η|||21/2
+ Chr+1‖u‖L2(Hr+1)‖η‖0.
Next, we find the estimates in the internal coordinate. From the definition, we have∫Ωℓ
ah(ϕ, η) = ε
∫Ωℓ
∇xϕ,∇xη
x +
∫Ωℓ
b · ∇xϕ, η
x +
∫Ωℓ
Shϕ, η
.
Then by using the Cauchy–Schwarz inequality, the stability property of the fluctuation operator κh, the approximationproperties (27) of interpolantΠk and the parameter choice µK ∼ hK , we get∫
To obtain the last two estimates, we use the two different representations (21) and (22) of B. Note that the jump terms[jhu − u]i, i = 1, . . . ,M − 1, vanish due to the continuity of the interpolant jhu in the ℓ-direction. We have from (16), (21)and (28)
B(ϑ, η) =
M−i=1
∫Ii
∂(Gϑ)∂ℓ
, η
x
+
Gminϑ(ℓ
+
0 ), η(ℓ+
0 )x
≤
M−i=1
∫Ii‖∂ℓ(Gϑ)‖L2(Ωx)‖η‖L2(Ωx) + ‖G1/2
minϑ(ℓ+
0 )‖L2(Ωx)‖G1/2minη(ℓ
+
0 )‖L2(Ωx)
≤ Chr+1‖u‖H1(Hr+1)‖η‖0 + ‖u‖C(Hr+1)‖η‖dG
.
The interpolationΠku satisfiesΠku(ℓ−
i ) = u(ℓ−
i ), i = 1, . . . ,M . Hence, we get from (22) the relation
B(ϕ, η) =
M−i=1
∫Ii−
Gϕ,
∂η
∂ℓ
x
.
LetΠ0G be the L2-projection of G in a space of piecewise constant functions in the ℓ-direction. Using the orthogonality (26)of the interpolantΠk, we get
B(ϕ, η) = −
M−i=1
∫Ii
ϕ, (G −Π0G)
∂η
∂ℓ
x
≤
M−i=1
∫Ii‖ϕ‖L2(Ωx)‖(G −Π0G)∂ℓη‖L2(Ωx)
≤ Ckq+1‖u‖Hq+1(L2)‖η‖0.
1614 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
Here, we used the Cauchy–Schwarz inequality, the inverse inequality and the interpolation error estimates (27). Thiscompletes the proof.
Theorem 6. Let u(tn), u(tn) and unh,k, u
nh,k, be the solutions of two-step methods (2), (3) and (19), (20), respectively. Under the
Assumptions A1–A3 and µK ∼ hK there holds for ηn = Rhu(tn)− unh,k and η
n= Rhu(tn)− un
h,k
‖ηN‖20 + τ
N−1−n=0
∫Ωℓ
|||ηn+1|||2+
12‖ηn+1
‖2dG
≤ Cue9T/2
[‖Rhz0 − u0
h,k‖20 + τ 2 + (ε + h)h2r
+ k2q+2]
(37)
and for en = u(tn)− unh,k and en = u(tn)− un
h,k
‖eN‖20 + τ
N−1−n=0
∫Ωℓ
|||en+1|||2+
12‖en+1
‖2dG
≤ Cue9T/2
[‖Rhz0 − u0
h,k‖20 + τ 2 + (ε + h)h2r
+ k2q+1]
(38)
where Cu depends on u, ut , utt , u, ut , utt and zmin.
Note that the error to the interpolant Rhu is superclosewith respect to the internal coordinate (order q+1 instead of q+1/2).
Proof. From the result of Lemma 3, we can write for ηn = Rhu(tn)− unh,k
12‖ηN‖
20 −
12‖η0‖2
0 + τ
N−1−n=0
∫Ωℓ
|||ηn+1|||2+τ
2
N−1−n=0
‖ηn+1‖2dG ≤ T1 + T2 (39)
where
T1 = τ
N−1−n=0
∫Ωℓ
ηn+1
− ηn
τ, ηn+1
x
+ ah(ηn+1, ηn+1)
, (40)
T2 = τ
N−1−n=0
∫Ωℓ
ηn+1
− ηn+1
τ, ηn+1
x
+ B(ηn+1, ηn+1)
. (41)
We first consider T1. Using (19), we obtain
T1 = τ
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn)
τ, ηn+1
x
+ ahRhu(tn+1), ηn+1
−
un+1h,k − un
h,k
τ, ηn+1
x
ah(un+1h,k , η
n+1)
= τ
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn)
τ, ηn+1
x
+ ahRhu(tn+1), ηn+1
−f n+1, ηn+1
x
.
For the last term on the right-hand side of the above equation, using (5) at t = tn+1, we get for the first term
T1 = τ
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn)
τ− ut(tn+1), ηn+1
x
+ τ
N−1−n=0
∫Ωℓ
aRhu(tn+1)− u(tn+1), ηn+1
+ τ
N−1−n=0
∫Ωℓ
ShRhu(tn+1), ηn+1
= τ
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn)
τ− ut(tn+1), ηn+1
x
+ τ
N−1−n=0
∫Ωℓ
ahRhu(tn+1)− u(tn+1), ηn+1
+ τ
N−1−n=0
∫Ωℓ
Shu(tn+1), ηn+1
= T1,1 + T1,2 + T1,3. (42)
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1615
We treat the contribution of the terms on the right-hand side of (42) separately. For the first term, using the Cauchy–Schwarzinequality, Young’s inequality and the initial condition u(tn) = u(tn) for first step
|T1,1| ≤ τ
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn)τ
− ut(tn+1)
L2(Ωx)
‖ηn+1‖L2(Ωx)
≤τ
2
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn)τ
− ut(tn+1)
2L2(Ωx)
+τ
2
N−1−n=0
∫Ωℓ
‖ηn+1‖2L2(Ωx)
≤ τ
N−1−n=0
Rhu(tn+1)− Rhu(tn)τ
− Rhut(tn+1)
20+ τ
N−1−n=0
‖Rhut(tn+1)− ut(tn+1)‖20 +
τ
2
N−1−n=0
‖ηn+1‖20.
For first term, applying Taylor’s theoremwith an integral remainder term and for second term the approximation propertiesof interpolants jh andΠk with the stability condition (28) yields
|T1,1| ≤ τ 2N−1−n=0
∫ tn+1
tn‖utt‖
20 +
τ
2
N−1−n=0
‖ηn+1‖20 + Cτ
N−1−n=0
[h2r+2
‖ut(tn+1)‖2L2(Hr+1)
+ k2q+2‖ut(tn+1)‖2
Hq+1(L2)
].
To find the estimates for T1,2, we use the decomposition (31) of errors into space and internal coordinates and get
T1,2 = τ
N−1−n=0
∫Ωℓ
[ahϑn+1, ηn+1
+ ahϕn+1, ηn+1].
Then from the results (32) and (33) of Lemma 5, we obtain
|T1,2| ≤ C(ε + h)τN−1−n=0
[h2r
‖u(tn+1)‖2L2(Hr+1)
+ k2q+2‖u(tn+1)‖2
Hq+1(H1)
]+τ
2
N−1−n=0
‖ηn+1‖20
+ CτN−1−n=0
[h2r+2
‖u(tn+1)‖2L2(Hr+1)
+ k2q+2‖u(tn+1)‖2
Hq+1(H1)
]+τ
4
N−1−n=0
∫Ωℓ
|||ηn+1|||2.
The estimate for T1,3 follows from the approximation properties of the fluctuation operator κh and the choice of thestabilizing parameter µK ∼ hK . We have
|T1,3| ≤ τ
N−1−n=0
∫Ωℓ
Shu(tn+1), u(tn+1)
+τ
4
N−1−n=0
∫Ωℓ
Shηn+1, ηn+1
≤ Ch2r+1τ
N−1−n=0
‖u(tn+1)‖2L2(Hr+1)
+τ
4
N−1−n=0
∫Ωℓ
|||ηn+1|||2.
Finally, by inserting the estimates T1,1, T1,2, and T1,3 into (42), we obtain
|T1| ≤ τ 2N−1−n=0
∫ tn+1
tn‖utt‖
20 +
τ
2
N−1−n=0
∫Ωℓ
|||ηn+1|||2+ τ
N−1−n=0
‖ηn+1‖20
+ Ch2rτ
N−1−n=0
[(ε + h)‖u(tn+1)‖2
L2(Hr+2)+ h2
‖ut(tn+1)‖2L2(Hr+1)
]
+ Ck2q+2τ
N−1−n=0
[(ε + h + 1)‖u(tn+1)‖2
Hq+1(H1)+ ‖ut(tn+1)‖2
Hq+1(L2)
]. (43)
Now we estimate the second term T2. Using (20) and (6) we obtain the following error equation for the second step
T2 = τ
N−1−n=0
∫Ωℓ
Rhu(tn+1)− Rhu(tn+1)
τ− ut(tn+1), ηn+1
x
+ τ
N−1−n=0
BRhu(tn+1)− u(tn+1), ηn+1
− τ
N−1−n=0
Gminzn+1
min − Gminzn+1min,h, η
n+1(ℓ+
0 )x
= T2,1 + T2,2 + T2,3. (44)
1616 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
The estimates for the first term can be obtained by using the same procedure as for T1,1. We get
|T2,1| ≤ τ
N−1−n=0
Rhu(tn+1)− Rhu(tn+1)
τ− Rhut(tn+1)
20+ τ
N−1−n=0
‖Rhut(tn+1)− ut(tn+1)‖20 +
τ
2
N−1−n=0
‖ηn+1‖20
≤ CτN−1−n=0
[h2r+2
‖ut(tn+1)‖2L2(Hr+1)
+ k2q+2‖ut(tn+1)‖2
Hq+1(L2)
]+ τ 2
N−1−n=0
∫ tn+1
tn‖utt‖
20 +
τ
2
N−1−n=0
‖ηn+1‖20.
Note that in the above estimates we have used the initial condition u(tn) = u(tn+1) from (3). The bounds on the secondterm T2,2 are obtained by using the error decomposition (31) and the estimates (34) and (35)
|T2,2| =
τ N−1−n=0
Bϑn+1, ηn+1
+ Bϕn+1, ηn+1
≤ Ch2r+2τ
N−1−n=0
[‖u(tn+1)‖2
H1(Hr+1)+ ‖u(tn+1)‖2
C(Hr+1)
]+ Ck2q+2τ
N−1−n=0
‖u(tn+1)‖2Hq+1(L2)
+τ
2
N−1−n=0
‖ηn+1‖20 +
τ
8
N−1−n=0
‖ηn+1‖2dG.
The Cauchy–Schwarz inequality and Young’s inequality give for T2,3
|T2,3| ≤ τ
N−1−n=0
‖G1/2minzmin(tn+1)− G1/2
minzn+1min,h‖L2(Ωx)‖G
1/2minη
n+1(ℓ+
0 )‖L2(Ωx)
≤ Ch2r+2τ
N−1−n=0
‖zmin(tn+1)‖2Hr+1(Ωx)
+τ
8
N−1−n=0
‖ηn+1‖2dG.
Finally using these estimates in (44) we get for T2
|T2| ≤ τ 2N−1−n=0
∫ tn+1
tn‖utt‖
20 + τ
N−1−n=0
‖ηn+1‖20 +
τ
4
N−1−n=0
‖ηn+1‖2dG
+ Cτh2r+2N−1−n=0
[‖u(tn+1)‖2
H1(Hr+1)+ ‖zmin(tn+1)‖2
Hr+1(Ωx)+ ‖ut(tn+1)‖2
L2(Hr+1)+ ‖u(tn+1)‖2
C(Hr+1)
]
+ Cτk2q+2N−1−n=0
[‖u(tn+1)‖2
Hq+1(L2) + ‖ut(tn+1)‖2Hq+1(L2)
].
Inserting T1 and T2 in (39), absorbing the triple norm and the dG norm contributions in the left-hand side and using (13),we get
12‖ηN‖
20 −
12‖η0‖2
0 + τ
N−1−n=0
∫Ωℓ
|||ηn+1|||2dℓ+
τ
2
N−1−n=0
‖ηn+1‖2dG
≤ τ 2N−1−n=0
∫ tn+1
tn‖utt‖
20 + τ
N−1−n=0
γn‖ηn‖20 + 2τ
N−1−n=0
‖f n+1‖20
+ Ch2rτ
N−1−n=0
[(ε + h)‖u(tn+1)‖2
H1(Hr+1)+ h2
‖ut(tn+1)‖2L2(Hr+1)
+ h2‖zmin(tn+1)‖2
Hr+1(Ωx)
+ ‖u(tn+1)‖2C(Hr+1)
]+ Ck2q+2τ
N−1−n=0
[(ε + h + 1)‖u(tn+1)‖2
Hq+1(H1)+ ‖ut(tn+1)‖2
Hq+1(L2)
]where γ0 = 2, γN = 1 and γn = 3, n = 1, . . . ,N − 1. We conclude by applying Gronwall’s lemma in the same fashion asin Lemma 1.
In the diffusion-dominated case ε = 1, no stabilization is needed, i.e., µK = 0, and the standard Galerkin discretizationworks fine. The statement of Theorem 6 simplifies as follows.
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1617
Theorem 7. Let u(tn), u(tn) and unh,k, u
nh,k, be the solutions of two-step methods. Under Assumption A3, there holds for ηn =
Rhu(tn)− unh,k and η
n= Rhu(tn)− un
h,k
‖ηN‖20 + τ
N−1−n=0
∫Ωℓ
|||ηn+1|||2+
12‖ηn+1
‖2dG
≤ Cue9T/2
[‖Rhz0 − u0
h,k‖20 + τ 2 + h2r
+ k2q+2]
(45)
and for en = u(tn)− unh,k and en = u(tn)− un
h,k
‖eN‖20 + τ
N−1−n=0
∫Ωℓ
|||en+1|||2+
12‖en+1
‖2dG
≤ Cue9T/2
[‖Rhz0 − u0
h,k‖20 + τ 2 + h2r
+ k2q+1]
(46)
where Cu depends on u, ut , utt , u, ut , utt and zmin.
The convergence order O(hr) is optimal since in the diffusion-dominant case the triple norm in space is dominated by theH1 seminorm.
6. Implementation of numerical method
This section indicates the implementation of the operator splitting method in the context of finite element methods. Formore details, we refer to [38].
Using the bases
Vh = spanφi, 1 ≤ i ≤ Nx, Sqk = spanψk, 1 ≤ k ≤ Nℓ,
the tensor product space Sr,qh,k is defined as follows
Sr,qh,k =
v =
Nx−i=1
Nℓ−k=1
αikφi(x)ψk(ℓ), αik ∈ R, 1 ≤ i ≤ Nx, 1 ≤ k ≤ Nℓ
.
The finite element functions are represented as
unh,k =
Nx−i=1
Nℓ−k=1
ξ nikφi(x)ψk(ℓ), X =
Nx−j=1
Nℓ−l=1
xjlφj(x)ψl(ℓ).
We define the matricesMx, Tx,Dx, Sx ∈ RNx×Nx by
(Mx)ij =φi(x), φj(x)
x, (Dx)ij = ε
∇xφi(x),∇xφj(x)
x
(Tx)ij =b · ∇xφi(x), φj(x)
x, (Sx)ij = Sh
φi(x), φj(x)
.
Similarly we define the matricesMℓ, Tℓ ∈ RNℓ×Nℓ as
(Mℓ)kl =
ψk(ℓ), ψl(ℓ)
ℓ,
(Tℓ)kl =
Nℓ−i=1
∂ℓ(Gψk(ℓ)), ψl(ℓ)
Ii+
Nℓ−1−i=1
[Gψk(ℓ)]iψl(ℓ+
i )+ Gψk(ℓ+
0 )ψl(ℓ+
0 ).
For the ease of presentation let us consider (1)with source term f = 0. Then the computing scheme for the operator splittingmethod described in (19) and (20) is as follows:
Within each time interval (tn, tn+1], we begin with the x-direction step where we are looking for the solution of the
time-dependent convection–diffusion equation (19). We compute ηn+1j ∈ RNx , j = 1, . . . ,Nℓ, by solving the linear systems
(Mx + τDx + τTx + τSx)ηn+1j = Mxη
nj , j = 1, . . . ,Nℓ.
With obtaining the solutions ηn+1j , j = 1, . . . ,Nℓ, the x-direction step is completed. Then, we continue with the ℓ-direction
step where we update the solution from the first step and compute the solution of the one-dimensional transport problem(20) by a discontinuous Galerkin method. In this step we solve the linear systems
(Mℓ + τTℓ)ηn+1j = Mℓη
n+1j , j = 1, . . . ,Nx,
and the obtained solutions ηn+1j , j = 1, . . . ,Nℓ, are used as input for the time interval (tn+1, tn+2
].
1618 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
Fig. 1. Meshes forΩx on level 0.
Table 1Errors and order of convergence in space for Q1 and dG(1), k = 1/64 and τ = 10−3 .
We report in this section the numerical computations illustrating the theoretical results obtained in the previous section.The two-step method (19) and (20) in the context of finite element method in space and discontinuous Galerkin method inthe internal coordinate is implemented in the finite element package MooNMD [39].
The tests are made in two plus one dimensions, i.e., we consider Ωx = (0, 1) × (0, 1) as a two-dimensional domainin space and Ωℓ = (0, 1) as a one-dimensional domain in the internal coordinate. We consider the velocity field b asb1 = b2 = 0.1, the growth rate G(ℓ) = 1 and two different choices for the diffusion coefficient ε, ε = 1 and ε ≪ 1. Thesource term f and the boundary and initial conditions are chosen such that the analytical solution of the problem (1) is
z(t, ℓ, x, y) = e−0.1t sin(πℓ) cos(πx) cos(πy).
Let en := z(tn) − unh,k, where z is the exact solution of (1) and the numerical solution un
h,k is obtained by two-step method(19) and (20). We use the following notations
‖e‖0 =
τ
N−n=1
‖en‖2L2(L2) + τ
N−n=1
‖en‖2dG
1/2
, ‖e‖1 =
τ
N−n=1
‖en‖2L2(H1)
+ τ
N−n=1
‖en‖2dG
1/2
,
‖e‖DG =
τ
N−n=1
∫Ωℓ
|||en|||2dℓ+ τ
N−n=1
‖en‖2dG
1/2
.
In order to illustrate the convergence order in time, internal coordinate and space, we use the well known strategy, i.e., theconvergence order in time can be obtained by assuming that the mesh sizes k and h are small enough compared to thetime-step size τ .
In the numerical computations, we have used triangular and quadrilateral meshes which are generated by successiverefinements starting from the coarsest meshes (level 0) as in Fig. 1 for the two-dimensional domain Ωx and a line dividedinto two cells for the one-dimensional domainΩℓ.Case ε = 1. In this case, the Galerkin finite element method in space is combined with a discontinuous Galerkin method inthe internal coordinate. For time discretization, the backward Euler time stepping scheme is used with final time T = 1.One can expect a convergence for ‖ · ‖0-norm of order O(hr+1) and for ‖ · ‖1-norms of order O(hr) using Qr and Pr finiteelements in space with sufficiently small time step length τ and mesh size k. The results are presented in Tables 1–4.
Tables 1 and 2 show the second order convergence in the ‖ · ‖0-norm and first order convergence in the ‖ · ‖1-normfor both Q1 and P1 finite elements in space with dG(1) in the internal coordinate. The length of the time step was set to beτ = 10−3 and mesh size to k = 1/64. For Q2 and P2 finite elements in space with dG(2) in the internal coordinate, the timestep length was set to τ = 10−4 and mesh size k = 1/64. The results of Tables 3 and 4 show third order convergence forthe ‖ · ‖0-norm and second order for the ‖ · ‖1-norm.
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1619
Table 2Errors and order of convergence in space for P1 and dG(1), k = 1/64 and τ = 10−3 .
In Tables 5 and 6, the errors and convergence orders for internal coordinate and time are listed.We expect a convergenceof order O(kq+1/2) in the internal coordinate and a convergence of O(τ ) in time. The errors for dG(1) in the internalcoordinate with Q1 on level 7 and time step length τ = 2.5 · 10−4 are presented in Table 5. We see that the expectedorders of convergence are achieved. The numerical errors and convergence orders in time are listed in Table 6 for dG(1)with k = 1/32 and Q1 on level 6. The theoretically predicted convergence order is achieved.Case ε = 10−9. In the case of dominating convection, we apply local projection stabilization in space. Discontinuous Galerkinmethods of the first and second order are used for the discretization in the internal coordinate. For time discretization, thebackward Euler time stepping scheme is used.
The numerical tests are performed using for (Vh,Dh) the pairs (Pbubble1 , Pdisc
0 ), (Pbubble2 , Pdisc
1 ), (Q bubble1 , Pdisc
0 ), and(Q bubble
2 , Pdisc1 ). The stabilization parameters µK have been chosen as
µK := µ0hK ∀K ∈ Th
where µ0 denotes a constant which will be given for each of the test calculations.
1620 N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621
Table 7Errors and order of convergence in space for (Q bubble
1 , Pdisc0 ) and (Pbubble
1 , Pdisc0 ) and dG(1), k = 1/64, τ = 10−3 and µK = 5hK .
In Tables 7 and 8 we show the convergence results for space in norm ‖ · ‖DG. Table 7 shows the error in space withstabilizing parameter µ0 = 5, time step length τ = 10−3 and mesh size k = 1/64 for (Q bubble
1 , Pdisc0 ) and (Pbubble
1 , Pdisc0 )
with dG(1) in the internal coordinate. In Table 8, the convergence results for (Q bubble2 , Pdisc
1 ) and (Pbubble2 , Pdisc
1 )with dG(2) inthe internal coordinate with µ0 = 5, k = 1/64 and τ = 10−4 are listed. We see that the expected orders of convergenceO(hr+1/2) are almost achieved. For smallermesh size h, the relative influence of the error in the internal coordinate increases.
The numerical errors and convergence orders in the internal coordinate are listed in Table 9 for dG(1) and (Q bubble1 , Pdisc
0 )
with µ0 = 5 on level 7 and τ = 2.5 · 10−4. The convergence order starts to decrease for small mesh size k since the errorsin space have increasing influence.
Finally, Table 10 shows the errors and convergence orders in time for (Q bubble1 , Pdisc
0 ) on level 6 with µ0 = 2.5 and dG(1)with k = 1/32. We see that the time stepping scheme is of first order convergent.
8. Conclusion
In this paper we have been concerned with the numerical solution of the population balance equation with one internalcoordinate posed on the domainΩℓ ×Ωx in the d + 1 dimension. We proposed an operator splitting method to reduce theoriginal problem into two subproblems. The method combines the continuous finite element method (with local projectionstabilization) in space with a discontinuous Galerkin method in the internal coordinate. We have considered the first orderbackward Euler time stepping scheme. Under certain regularity assumptions on the exact solution, we have derived errorestimates for the two-stepmethod, i.e., using polynomials of degree r in space and of degree q in the internal coordinate theerror is of order O(τ + hr+1/2
N. Ahmed et al. / Journal of Computational and Applied Mathematics 236 (2011) 1604–1621 1621
The application of discontinuous Galerkin methods makes the mass matrix corresponding to the internal coordinatediagonal which leads to the feasibility of the implementation without any projection between the two-steps in thecomputation process. Computational results shown in Section 7 confirms the theoretical prediction of error estimates.
Acknowledgments
The authors acknowledge the financial support from the Federal Ministry of Education and Research (BMBF) under grant03TOPAA1, Germany, and Kohat University of Science and Technology (KUST-HEC), Pakistan.
References
[1] D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego, CA, 2000.[2] V. John, M. Roland, T. Mitkova, K. Sundmacher, L. Tobiska, A. Voigt, Simulations of population balance systems with one internal coordinate using
finite element methods, Chem. Eng. Sci. 64 (2009) 733–741.[3] M. Strumendo, H. Arastoopour, Solution of population balance equations by the finite size domain complete set of trial functions method of moments
(FCMOM) for inhomogeneous systems, Ind. Eng. Chem. Res. 49 (11) (2010) 5222–5230.[4] D. Marchisio, R. Fox, Solution of population balance equations using the direct quadrature method of moments, J. Aerosol Sci. 36 (2005) 43–73.[5] K. Sanjeev, D. Ramkrishna, On the solution of population balance equations by discretization-III. Nucleation, growth and aggregation of particles,
Chem. Eng. Sci. 52 (1997) 4659–4679.[6] R. Davida, J. Villermauxa, P. Marchalb, J.-P. Kleinc, Crystallization and precipitation engineering-IV. Kinetic model of adipic acid crystallization, Chem.
Eng. Sci. 46 (4) (1991) 1129–1136.[7] V. John, I. Angelov, A.A. Öncül, D. Thévenin, Techniques for the reconstruction of a distribution from a finite number of its moments, Chem. Eng. Sci.
62 (11) (2007) 2890–2904.[8] S. Ganesan, Population balance equations, Streamline-Upwind Petrov–Galerkin finite element methods, operator-splitting method, backward Euler
scheme, error analysis, Preprint 1531, WIAS, Berlin, 2010.[9] J. Kačur, B. Malengier, M. Remešíková, Convergence of an operator splittingmethod on a bounded domain for a convection–diffusion–reaction system,
J. Math. Anal. Appl. 348 (2) (2008) 894–914.[10] K.H. Karlsen, K. Brusdal, H.K. Dahle, S. Evje, K.-A. Lie, The corrected operator splitting approach applied to a nonlinear advection–diffusion problem,
Comput. Methods Appl. Mech. Engrg. 167 (3–4) (1998) 239–260.[11] K.H. Karlsen, K.-A. Lie, An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA J. Numer. Anal. 19 (4) (1999)
609–635.[12] K.H. Karlsen, N.H. Risebro, An operator splitting method for nonlinear convection–diffusion equations, Numer. Math. 77 (3) (1997) 365–382.[13] D.J. Knezevic, E. Süli, A heterogeneous alternating-direction method for a micro–macro dilute polymeric fluid model, M2AN Math. Model. Numer.
Anal. 43 (6) (2009) 1117–1156.[14] T.J.R. Hughes, A.N. Brooks, A multidimensional upwind scheme with no crosswind diffusion, in: Finite Element Methods for Convection Dominated
Flows, Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979, in: AMD, vol. 34, Amer. Soc. Mech. Engrs. (ASME), New York, 1979,pp. 19–35.
[15] E. Burman, Consistent SUPG-method for transient transport problems: stability and convergence, Comput. Methods Appl. Mech. Engrg. 199 (17–20)(2010) 1114–1123.
[16] V. John, J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection–diffusion–reaction equations, SIAM J. Numer. Anal.49 (3) (2011) 1149–1176.
[17] R. Becker,M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38 (4) (2001) 173–199.[18] R. Becker, M. Braack, A two-level stabilization scheme for the Navier–Stokes equations, in: Numerical Mathematics and Advanced Applications,
Springer, Berlin, 2004, pp. 123–130.[19] M. Braack, E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM J. Numer.
Anal. 43 (6) (2006) 2544–2566.[20] G. Matthies, P. Skrzypacz, L. Tobiska, A unified convergence analysis for local projection stabilisations applied to the Oseen problem, M2AN Math.
Model. Numer. Anal. 41 (4) (2007) 713–742.[21] G. Matthies, P. Skrzypacz, L. Tobiska, Stabilization of local projection type applied to convection–diffusion problems with mixed boundary conditions,
Electron. Trans. Numer. Anal. 32 (2008) 90–105.[22] H.-G. Roos,M. Stynes, L. Tobiska, Robust NumericalMethods for Singularly PerturbedDifferential Equations: Convection–Diffusion–Reaction and Flow
Problems, 2nd ed., in: Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 2008.[23] N. Ahmed, G. Matthies, L. Tobiska, H. Xie, Discontinuous Galerkin time stepping with local projection stabilization for transient convec-
tion–diffusion–reaction problems, Comput. Methods Appl. Mech. Engrg. 200 (2011) 1747–1756.[24] E. Burman, M.A. Fernández, Finite element methods with symmetric stabilization for the transient convection–diffusion–reaction equation, Comput.
Methods Appl. Mech. Engrg. 198 (33–36) (2009) 2508–2519.[25] W.H. Reed, T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory.[26] P. Lasaint, P.-A. Raviart, On a finite element method for solving the neutron transport equation, in: Mathematical Aspects of Finite Elements in
Partial Differential Equations, Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974, in: Publication Math. Res. Center, Univ. ofWisconsin–Madison, vol. 33, Academic Press, New York, 1974, pp. 89–123.
[27] C. Johnson, J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (173) (1986) 1–26.[28] M. Feistauer, J. Hájek, K. Svadlenka, Space–time discontinuous Galerkin method for solving nonstationary convection–diffusion–reaction problems,
Appl. Math. 52 (3) (2007) 197–233.[29] B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, in: Advanced Numerical Approximation of
Nonlinear Hyperbolic Equations, Cetraro, 1997, in: Lecture Notes in Math., vol. 1697, Springer, Berlin, 1998, pp. 151–268.[30] A. Mersmann, Batch precipitation of barium carbonate, Chem. Eng. Process. 38 (1993) 6177–6184.[31] A. Mersmann, Crystallization and precipitation, Chem. Eng. Process. 38 (1999) 345–353.[32] E. Hansen, A. Ostermann, Dimension splitting for evolution equations, Numer. Math. 108 (4) (2008) 557–570.[33] E. Hansen, A. Ostermann, Dimension splitting for time dependent operators, in: Dynamical Systems, Differential Equations and Applications. 7th AIMS
Conference Suppl., Discrete Contin. Dyn. Syst. (2009) 322–332.[34] L. He, L. Tobiska, The two-level local projection stabilization as an enriched one-level approach, Adv. Comput. Math., in press (doi:10.1007/s10444-
011-9188-1.[35] P. Knobloch, A generalization of the local projection stabilization for convection–diffusion–reaction equations, SIAM J. Numer. Anal. 48 (2) (2010)
659–680.[36] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, in: Studies in Mathematics and its Applications, vol. 4, North-Holland Publishing Co.,
Amsterdam, 1978.[37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., in: Springer Series in Computational Mathematics, vol. 25, Springer-
Verlag, Berlin, 2006.[38] S. Ganesan, L. Tobiska, Implementation of an operator splitting finite element method for high-dimensional parabolic problems, Preprint 11-04,
Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, 2011.[39] V. John, G. Matthies, MooNMD—a program package based on mapped finite element methods, Comput. Vis. Sci. 6 (2–3) (2004) 163–169.
