Finite element methods of an operator splitting applied to population balance...

March 2, 2011 16:10

Finite element methods of an operator splitting applied to populationbalance equations

Naveed Ahmed

Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120,

D-39016 Magdeburg, Germany.

[email protected]

Gunar Matthies

Universität Kassel, Fachbereich 10 Mathematik und Naturwissenschaften, Institut für

Mathematik, Heinrich-Plett-Straße 40, 34132 Kassel, Germany.

[email protected]

Lutz Tobiska

Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120,D-39016 Magdeburg, Germany.

[email protected]

In population balance equations, the distribution of the entities depends not only onspace and time but also on their own properties referred to as internal coordinates.

The operator splitting method is used to transform the whole time-dependent problem

into two unsteady subproblems of a smaller complexity. The first subproblem is a time-dependent convection-diffusion problem while the second one is a transient transport

problem with pure advection. We use the backward Euler method to discretize the sub-

problems in time. Since the first problem is convection-dominated, the local projectionmethod is applied as stabilization in space. The transport problem in the one-dimensional

internal coordinate is solved by a discontinuous Galerkin method. The unconditionalstability of the method will be presented. Optimal error estimates are given. Numerical

results confirm the theoretical predictions.

Keywords: Operator splitting; discontinuous Galerkin; stabilized finite elements; popu-lation balance equations.

AMS Subject Classification: 65M12, 65M15, 65M60

1. Introduction

In this paper, we advocate the operator splitting method to approximate the so-lutions of population balance equations (PBE). This type of problems arises e.g.from models in the simulation of industrial crystallization process.20 In PBE, thedistribution of entities depends not only on space and time but also on their ownproperties referred to as internal coordinates and the source term usually involvesan integral operator. For efficient methods to handle integral operators we refer to

1

March 2, 2011 16:10

2 N. Ahmed, G. Matthies, L. Tobiska

Ref. 29. In this work we consider the source as a known function but still we have ahigh dimensional system of equations which is challenging from the computationalpoint of view.

In order to overcome the difficulty of higher dimensions, the operator splittingmethod is used to decompose the original problem into two unsteady subproblems ofsmaller complexity. The first subproblem is a time-dependent convection-diffusionproblem while the second one is a transport problem with pure advection. Operatorsplitting methods are widely used for time integration of unsteady problems. Thebasic theory of operator splitting for one-dimensional problems can be found inRef. 41, 45. The concept of operator splitting for time-dependent problems is todecompose the spatial operator into a sum of two or more operators. For examplein Ref. 33, the decomposition of convection-diffusion-reaction problem into pureconvection and diffusion-reaction problems was studied. For more details aboutoperator splitting methods for linear and non-linear convection-diffusion problems,see Ref. 22, 23, 24, 25, 30.

A detailed analysis of an alternating direction implicit (or operator-splitting)scheme is demonstrated in Ref. 26 for the Fokker-Planck equation. The basic idea inRef. 26 is to split the high dimensional problem into two low dimensional problemscorresponding to the configuration and the physical spaces. The solution of theconvection-diffusion type problem in configuration space is obtained by a Galerkinspectral method at each quadrature point corresponding to the physical domain.Furthermore, a type of L2 projection is used to update the right-hand side vectorat the second stage where the solution of advection equation in physical space isapproximated by a finite element method.

The main advantage of such splitting is that each subproblem can be discretizedand stabilized separately by the best suitable method independently of the othersubproblem(s). For example in Ref. 13 the Streamline-Upwind Petrov-Galerkinmethod (SUPG) has been combined with the standard Galerkin method. The maindisadvantage of SUPG scheme, in particular for unsteady problems, is that severalterms which include the time derivative, the source term, and second order deriva-tives have to be added into the stabilizing term in order to ensure the consistencyof the resulting method.

There are several other stabilization techniques as alternative to SUPG. We men-tion the local projection stabilization (LPS),3,4,34 the continuous interior penaltymethod (CIP),5,6,7 the subgrid scale modeling (SGS),15,32 and the orthogonal sub-scales method (OSS).10,11 The LPS method was originally proposed for the Stokesproblem in Ref. 2 as a two-level approach and extended to transport problems inRef. 3. An analysis of the local projection stabilization applied to Oseen problemscan be found in Ref. 4, 34 and for scalar convection-diffusion problems with mixedboundary condition in Ref. 35. The LPS in space combined with a discontinuousGalerkin (dG) method in time for transient convection-diffusion-reaction equationswas studied in Ref. 1. A comparison of one- and two-level approaches of local pro-jection stabilization for linear advection-diffusion-reaction problem is presented in

March 2, 2011 16:10

Operator splitting method for population balance equation 3

Ref. 27. For more details about local projection stabilization we refer to Ref. 43where an overview on recent development for this class of stabilization method ap-plied to scalar convection-diffusion, Stokes and linearized Navier-Stokes problemsis given. In this article, we will concentrate on the one-level local projection stabi-lization technique.

The second subproblem in our splitting method is a transport problem withpure advection, so one suitable choice is to approximate it by the discontinuousGalerkin (dG) method. The dG method was first introduced for the neutron trans-port problem in Ref. 39 and then analyzed in Ref. 31. The theoretical analysis ofthe dG method for scalar hyperbolic equations can be found in Ref. 21 and for thespace-time dG method in Ref. 12. For an introduction to discontinuous Galerkinmethod we refer to Ref. 9. The application of dG method makes the mass matrixcorresponding to the internal coordinate diagonal which leads to the feasibility ofparallel implementation without any projection steps between the two sub-stepsduring the computing process.

The aim of the present paper is to combine the local projection stabilizationmethod in space with discontinuous Galerkin method in internal coordinate. Wewill give the stability and convergence estimates for fully discrete two-step schemebased on an operator splitting method.

The format of the paper is as follows: Section 2 introduces the model problemunder consideration and defines basic notations. In Section 3, the operator split-ting technique is applied to decompose the problem into two simpler ones. We shallformulate the backward Euler discretization and derive the weak form of the twosubproblems. Further, we derive the unconditional stability of the two-step method.We then discretize the subproblems in space and internal coordinate using local pro-jection stabilization and discontinuous Galerkin methods, respectively, in Section 4.We show the unconditional stability of the fully discrete two-step method. Section 5presents the error analysis of the fully discrete scheme. Some implementation issuesof the method are discussed in Section 6. Finally, we present in Section 7 somecomputational results supporting our theoretical results.

2. Model problem

Let Ωx be a domain in Rd (d = 2 or 3) with boundary ∂Ωx, Ω` = [`min, `max] ⊂ R andT > 0. The state of individual particle in population balance equation may consistsof external coordinate x, referring to its position in the physical space, and internalcoordinate `, representing the properties of particles, such as size, temperature,volume etc. A population balance for a solid process such as crystallization with oneinternal coordinate can be described by the following partial differential equation:

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Find z : (0, T )× Ω` × Ωx → R such that

∂z

∂t+∂(Gz)∂`

− ε∆xz + b(x) · ∇xz = f in (0, T ]× Ω` × Ωx,

z(0, ·) = z0 in Ω` × Ωx,z|`min = zmin on (0, T ]× Ωx,z = 0 on (0, T ]× Ω` × ∂Ωx,

(2.1) {ss1_e1}

where the diffusion coefficient ε > 0 is a given constant, ∆x and ∇x represent theLaplacian and gradient with respect to x, respectively, b is a given velocity fieldsatisfying ∇x ·b = 0, and f is a source function. Here G > 0 represents the growthrate of the particles that depends on ` but is independent of x and t, we also assumethat ∂`G ≥ 0, see Ref. 36, 37. Furthermore, let us consider the data of the problemG, b, f , z0 and zmin to be sufficiently smooth functions.

Let us introduce some standard notations. Let Hm(Ω) denote the Sobolev spaceof functions with derivatives up to order m in L2(Ω). We denote by (·, ·) the innerproduct in L2(Ω` × Ωx) and by ‖ · ‖0 the corresponding L2-norm defined by

(v, w) =∫

Ω`×Ωxvw d`dx and ‖v‖20 = (v, v).

To distinguish the inner products and the corresponding norms with respect tothe internal coordinate and the space variable we need some more notations. Forthis, let us denote by (·, ·)` and ‖ · ‖L2(Ω`) the L2-inner product and the associatednorm in Ω`, respectively, and by (·, ·)x and ‖ · ‖L2(Ωx) the L2-inner product and theassociated norm in Ωx, i.e.,(

v, w)`

=∫

Ω`

vw d` and ‖v‖2L2(Ω`) = (v, v)`,(v, w

)x

=∫

Ωx

vw dx and ‖v‖2L2(Ωx) = (v, v)x.

The norm in the Sobolev space Hm(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 Bochnerspaces. For this, let X be a Banach space with norm ‖ · ‖X . Then we define

C(Ω`;X) ={v : Ω` → X, v continuous

},

L2(Ω`;X) ={v : Ω` → X,

∫Ω`

‖v(`)‖2Xd`

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where the derivatives ∂jv/∂`j are understood in the sense of distribution on Ω`.For spaces X and Y we use the short notation Y (X) := Y (Ω`;X). The norms inthe above defined spaces are given as follows

‖v‖C(X) = sup`∈Ω`‖v(`)‖X , ‖v‖L2(X) =

(∫Ω`

‖v(`)‖2Xd`)1/2

,

‖v‖Hm(X) =

∫Ω`

m∑j=0

∥∥∥∂jv∂`j

∥∥∥2Xd`

1/2 .3. Operator splitting method

The numerical method for solving (2.1) in d + 1 variable is based on an operatorsplitting with respect to (`, t) and (x, t) in Ω` and Ωx direction, respectively. Weconsider a uniform partition of the time interval τ = T/N , i.e. tn = τn, n =1, . . . , N . Then starting with u(t0) = z0, two subproblems are sequentially solvedon the sub-intervals (tn, tn+1], n = 0, 1, . . . , N − 1:

Given u(tn) find ũ : (tn, tn+1]× Ω` × Ωx → R such that∂ũ

∂t+ Lxũ = f in (tn, tn+1]× Ω` × Ωx

ũ = 0 on (tn, tn+1]× Ω` × ∂Ωxũ(tn+) = u(tn).

(3.1) {spl1}

Find u : (tn, tn+1]× Ω` × Ωx → R such that∂u

∂t+ L`u = 0 in (tn, tn+1]× Ω` × Ωxu|`min = zmin on (tn, tn+1]× Ωxu(tn+) = ũ(tn+1),

(3.2) {spl2}

where

L`z =∂(Gz)∂`

, Lxz = −ε∆xz + b · ∇xz. (3.3) {ss1_e2}

This two-steps operator splitting scheme defines u(tn), n = 1, . . . , N , as an approx-imation of z(tn).

In the framework of PBE, the first subproblem (3.1) is a time-dependentconvection-diffusion equation posed on Ωx parameterized by the variable ` in in-ternal coordinate and the second subproblem (3.2) is a one-dimensional transportproblem on Ω` parameterized by the space variable x.

Let us consider the spaces V = H10 (Ωx) and W = H1(Ω`). We introduce the

space

P ={v ∈ L2(Ω` × Ωx) : v ∈ L2(Ωx;W ) ∩ L2(Ω`;V )

}.

A variational form of (3.1) and (3.2) reads as follows:

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First step: Find ũ : (tn, tn+1]→ P with ũ(tn+) = u(tn) such that∫Ω`

(ũt, v

)xd`+

∫Ω`

a(ũ, v) d` =∫

Ω`

(f, v)xd` ∀v ∈ P, (3.4) {s2_e1}

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+) = ũ(tn+1) such that∫Ω`

(ut, v

)xd`+ b

(u, v)

=((Gz)min, v(`min)

)x

∀v ∈ P, (3.5){s2_e2}

where wmin = w(`min) and the bilinear form b is defined as

b(u, v) =∫

Ω`

(∂(Gu)∂`

, v)xd`+

((Gu)(`min), v(`min)

)x.

Note that we have imposed the boundary condition (u|`min = zmin) in `-direction inweak sense.

After discretizing in time by the backward Euler method, the first order accurateimplicit scheme is considered as two-step method:

First step: Given un ∈ P, find ũn+1 ∈ P such that∫Ω`

( ũn+1 − unτ

, v)xd`+

∫Ω`

a(ũn+1, v) d` =∫

Ω`

(fn+1, v)x d` (3.6){s2_e2a}

for all v ∈ P.

Second step: Update ũn+1 from the first step and find the solution un+1 ∈ Psuch that ∫

Ω`

(un+1 − ũn+1τ

, v)xd`+ b(un+1, v) =

(Gminz

n+1min , v(`min)

)x

(3.7){s2_e5}

for all v ∈ P, where zn+1min = zmin(tn+1, ·).The next paragraph gives the stability of the two-step method (3.6) and (3.7).

Lemma 3.1 (Stability). Assume that ũn, un , n = 1, 2 . . . , N , is the solutionobtained from the two-step algorithm (3.6) and (3.7). If ∂`G ≥ 0 and τ ≤ 14 , thenthe following estimate shows the stability

∥∥uN∥∥20

+ τN−1∑n=0

∫Ω`

{2ε∥∥ũn+1∥∥2

H1(Ωx)+ ∂`G

∥∥un+1∥∥2L2(Ωx)

}d`

≤ exp(3T/2)

{∥∥u0∥∥20

+ τN−1∑n=0

(2∥∥fn+1∥∥2

0+∥∥G1/2minzn+1min ∥∥2L2(Ωx))

}. (3.8){s2_e6}

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Proof. Setting v = ũn+1 in (3.6), yields∫Ω`

(ũn+1 − un, ũn+1)x d`+ τ∫

Ω`

a(ũn+1, ũn+1) d` = τ∫

Ω`

(fn+1, ũn+1)x d`.

Using the relation 2(a− b)a = a2 − b2 + (a− b)2, one can write∫Ω`

(ũn+1 − un, ũn+1)x d` =12‖ũn+1‖20 −

12‖un‖20 +

12‖ũn+1 − un‖20.

Integrating by parts with respect to x the second term in the bilinear form a(·, ·),one obtains ∫

Ω`

a(ũn+1, ũn+1) d` = ε∫

Ω`

‖ũn+1‖2H1(Ωx) d`

since ũn+1 vanishes on the boundary ∂Ωx and ∇x · b = 0. Hence by using Cauchy-Schwarz inequality for the right-hand side, we have for the first step

‖ũn+1‖20 − ‖un‖20 + ‖ũn+1 − un‖20 + 2τε∫

Ω`

‖ũn+1‖2H1(Ωx)d`

≤ τ‖fn+1‖20 + τ‖ũn+1‖20. (3.9) {s2_e7}

Substituting v = un+1 in the second step (3.7) gives∫Ω`

(un+1 − ũn+1, un+1)xd`+ τb(un+1, un+1) = τ(Gminz

n+1min , u

n+1(`min))x. (3.10) {s2_e7a}

Starting from

b(un+1, un+1) =∫

Ω`

(∂(Gun+1)∂`

, un+1)xd`+

(Gminu

n+1(`min), un+1(`min))x

an integration by parts twice with respect to ` gives

b(un+1, un+1) =12

∫Ω`

∂`G∥∥un+1∥∥2

L2(Ωx)d`+

12

∥∥G1/2maxun+1(`max)∥∥2L2(Ωx)+

12

∥∥G1/2minun+1(`min)∥∥2L2(Ωx).where Gmax = G(`max). Cauchy-Schwarz inequality gives for the right-hand side in(3.10)(

Gminzn+1min , u

n+1(`min))x≤ 1

2

∥∥G1/2minzn+1min ∥∥2L2(Ωx) + 12∥∥G1/2minun+1(`min)∥∥2L2(Ωx).Combining these two results in (3.10) and using the same relation 2(a − b)a =a2 − b2 + (a− b)2 for first term, we get for second step∥∥un+1∥∥2

0−∥∥ũn+1∥∥2

0+∥∥un+1 − ũn+1∥∥2

0+ τ

∫Ω`

∂`G∥∥un+1∥∥2

L2(Ωx)

≤ τ∥∥G1/2minzn+1min ∥∥2L2(Ωx). (3.11) {s2_e8}

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Adding (3.9) and (3.11), neglecting some contribution of positive terms, and sum-ming over n = 0, . . . , N − 1, we obtain

∥∥uN∥∥20

+ τN−1∑n=0

∫Ω`

{2ε∥∥ũn+1∥∥2

H1(Ωx)+ ∂`G

∥∥un+1∥∥2L2(Ωx)

}d`

≤∥∥u0∥∥2

0+ τ

N−1∑n=0

{∥∥fn+1∥∥20

+∥∥G1/2minzn+1min ∥∥2L2(Ωx)}+ τ N−1∑

n=0

∥∥ũn+1∥∥20.

From (3.9) we have ∥∥ũn+1∥∥20≤ τ

1− τ∥∥fn+1∥∥2

0+

11− τ

∥∥un∥∥20. (3.12){s2_e8a}

Using this estimate in the last inequality, we get

∥∥uN∥∥20

+ τN−1∑n=0

∫Ω`

{2ε∥∥ũn+1∥∥2

H1(Ωx)+ ∂`G

∥∥un+1∥∥2L2(Ωx)

}d`

≤∥∥u0∥∥2

0+ τ

N−1∑n=0

{43

∥∥fn+1∥∥20

+∥∥G1/2minzn+1min ∥∥2L2(Ωx)

}+

4τ3

N−1∑n=0

∥∥un∥∥20,

where we have used 1/(1 − τ) ≤ 4/3 for τ ≤ 1/4. We conclude the statement byusing Gronwall’s lemma. This completes the proof.

The critical issue of the operator splitting method is the overall accuracy of thetwo-step method. Using Taylor series expansions first order accuracy of the two-step method (3.1) and (3.2) can be shown. A detail error analysis for the first orderLie operator splitting of the sum of two elliptic operators can be found in Ref. 16,17. Unfortunately, we can’t use these results due to the hyperbolic nature of theoperator L`.

4. Fully-discrete method

In view of different properties of operator L` and Lx, the operator splitting techniqueallows us to use different types of discretization methods to solve the problemsin Ω` and Ωx. Since the first subproblem (3.5) is convection-dominated, we usethe local projection method to stabilize the space discretization. While the secondsubproblem (3.7) is a transport problem with pure advection, one suitable choice isthe discontinuous Galerkin method for the discretization with respect to the internalcoordinate.

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 regular decompositions of Ωx into d-simplices, quadrilateralor hexahedra. The diameter of a cell K ∈ Th is denoted by hK and h describes

March 2, 2011 16:10


the maximum diameter of cells K. We will consider the one-level LPS in which theapproximation and projection space live on the same mesh. For other variants ofLPS we refer to Ref. 18, 28, 34, 40, 43, 44.

Let Vh ⊂ V denote the standard finite element space of continuous, piecewisepolynomials of degree r. The Galerkin discretization of the problem (3.5) is generallyunstable due to dominating advection when the diffusion coefficient is very smallε � 1. We will handle this difficulty by adding a stabilizing term based on localprojection. LetDh be the projection space of discontinuous, piecewise polynomials ofdegree r−1 with r ≥ 1. Let Dh(K) = {qh|K : qh ∈ Dh} be the local projection spaceand πK : L2(K) → Dh(K) the local L2-projection onto Dh(K). Define the globalprojection πh : L2(Ωx) → Dh by (πhv)|K := πK(v|K). The fluctuation operatorκh : L2(Ωx) → L2(Ωx) is given by κh := id − πh, where id : L2(Ωx) → L2(Ωx) isthe 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 overthe fluctuations of gradients. Note that one can also replace the gradient ∇xwh bythe derivative in streamline direction b · ∇xwh or (even better Ref. 27, 28) bybK · ∇xwh where bK is a piecewise constant approximation of b, which leads tosimilar results.

The stabilized bilinear form is then defined as

ah(uh, vh) = a(uh, vh) + Sh(uh, vh). (4.1) {ss3_1e3}

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

, (4.2) {ss3_1e4}

that is ah(vh, vh) ≥ |||vh|||2 for all vh ∈ Vh. The stability and convergence propertiesof the LPS method are based on the following assumptions with respect to the pair(Vh, Dh), see Ref. 34, 40.

Assumption A1 : There is an interpolation operator jh : H2(Ω)→ Vh such thatthe approximation properties,

‖v − jhv‖0,K + hK |v − jhv|1,K ≤ Chlk‖v‖l,K ∀v ∈ H l(Ωx), 2 ≤ l ≤ r + 1, (4.3) {ss3_1e6}

for all K ∈ Th and the orthogonality

(v − jhv, qh) = 0 ∀qh ∈ Dh,∀v ∈ H2(Ω) (4.4) {ss3_1e7}

hold true.

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Assumption A2 : The fluctuation operator κh satisfies the following approxima-tion property

‖κhq‖0,K ≤ ChlK |q|l,K ∀K ∈ Th, ∀q ∈ H l(K), 0 ≤ l ≤ r. (4.5) {ss3_1e8}

In numerical computations, we use mapped finite element spaces, see Ref. 8, whereon the reference cell K̂ the enriched spaces are given by

P bubbler (K̂) = Pr(K̂) + b̂4Pr−1(K̂),

Qbubbler (K̂) = Qr(K̂) + span{b̂�x̂

r−1i , i = 1, 2

}.

Here, b̂4 and b̂� are the cubic bubble on the reference triangle and the biquadraticbubble on the reference square, respectively. The pairs (P bubbler , P

discr−1), r ≥ 1, on

triangles and the pairs (Qbubbler , Pdiscr−1), r ≥ 1, on quadrilaterals fulfill assumptions

A1 and A2. Further examples of spaces (Vh, Dh) satisfying A1 and A2 are given inRef. 34, 40.

4.2. Discontinuous Galerkin method in internal coordinate

To discretize (3.5) and (3.7) in internal coordinate `, we apply a discontinuousGalerkin method. Let M > 0 be a given positive integer and `min = `0 < `1 < · · · <`M = `max is a partition of Ω` with Ii = (ì−1, ì], ki = ì − ì−1, and k = max

iki.

Let us introduce the function space of discontinuous piecewise polynomials of degreeq ≥ 1 as

Sqk ={v : Ω` → R : v|Ii(`) =

q∑j=0

vj`j with vj ∈ R, j = 0, . . . , q

}.

Then we give the fully discrete space Sr,qh,k as follows

Sr,qh,k = Vh ⊗ Sqk

={v : Ω` × Ωx → R : v|Ii(`) =

q∑j=0

vj`j with vj ∈ Vh, j = 0, . . . , q

}. (4.6){ss3_1e10}

The functions in these spaces are allowed to be discontinuous at the nodes ì,i = 1, . . . ,M − 1. The jumps across the nodes are defined by [φ]i = φ(`+i )− φ(`

−i ),

where

ϕ±m = ϕ(`±m) = lim

`→`m±0ϕ(`).

In the next paragraph, we define the fully discrete scheme based on two-step method.

First step : For given unh,k ∈ Sr,qh,k, find ũ

n+1h,k ∈ S

r,qh,k such that∫

Ω`

( ũn+1h,k − unh,kτ

,X)xd`+

∫Ω`

ah(ũn+1h,k , X) d` =∫

Ω`

(fn+1, X)x d` (4.7){ss3_1e11}

for all X ∈ Sr,qh,k where u0h,k is a suitable approximation of z0 in Sr,qh,k.

March 2, 2011 16:10


Second step : Update the solution ũn+1h,k from (4.7) and find un+1h,k ∈ S

r,qh,k such

that ∫Ω`

(un+1h,k − ũn+1h,kτ

,X)xd`+B(un+1h,k , X) =

(Gminz

n+1min,h, X(`

+0 ))x

(4.8){ss3_1e12}

for all X ∈ Sr,qh,k, where zn+1min,h ∈ S

r,qh,k is an approximation of z

n+1min and the bilinear

form B is defined as

B(u, v) =M∑i=1

∫Ii

(∂(Gu)∂`

, v)xd`+

M−1∑i=1

([(Gu)]i, v(`+i )

)x

+(Gminu(`+0 ), v(`

+0 ))x.

(4.9) {b_uv_1}

Integrating by parts∫Ii

(∂(Gu)∂`

, v)xd` =

((Gu)(`−i ), v(`

−i ))x−((Gu)(`+i−1), v(`

+i−1)

)x−∫Ii

(Gu,

∂v

∂`

)xd`

leads to the representation

B(u, v) = −M∑i=1

∫Ii

(Gu,

∂v

∂`

)xd`−

M−1∑i=1

(u(`−i ),

[(Gv)]i

)x

+(Gmaxu(`−M ), v(`

−M ))x.

(4.10) {b_uv_2}We introduce the mesh dependent norm

‖v‖2dG =M∑i=1

∫Ii

∂`G‖v‖2L2(Ωx) d`+∥∥G1/2minv(`+0 )∥∥2L2(Ωx) + M−1∑

i=1

∥∥[(G1/2v)]i

∥∥2L2(Ωx)

+∥∥G1/2maxv(`−M )∥∥2L2(Ωx). (4.11) {dg_norm}

Lemma 4.1. The bilinear form B is coercive with respect to the mesh dependentnorm ‖ · ‖dG, i.e.,

B(v, v) ≥ 12‖v‖2dG. (4.12) {B_coer}

holds for all v ∈ Sr,qh,k.

Proof. Setting u = v in (4.9) and (4.10), then adding them together we concludethe statement of the lemma.

The next lemma provides a stability result of the fully discrete two-step method(4.7) and (4.8).

March 2, 2011 16:10


Lemma 4.2 (Stability). Let ∂`G ≥ 0 and τ ≤ 1/2, then the solution ũnh,k andunh,k, n = 1, 2, . . . , N , of (4.7) and (4.8),respectively, satisfies∥∥uNh,k∥∥20 + 2τ N−1∑

n=0

∫Ω`

∣∣∣∣∣∣ũn+1h,k ∣∣∣∣∣∣2 d`+ τ N−1∑n=0

∥∥un+1h,k ∥∥2dG≤ exp(3T/2)

{∥∥u0h,k∥∥20 + τ N−1∑n=0

(43

∥∥fn+1∥∥20

+∥∥(G1/2minzn+1min,h)∥∥2L2(Ωx)

)}.

(4.13) {ss3_1e19}

Proof. Following the similar derivation steps as in Lemma 3.1, we get the proof oflemma.

5. Error analysis

In this section, we derive the error estimates of the fully discrete two-step scheme(4.7) and (4.8). First we define a special interpolant Πkw(t, ·, x) ∈ Sqk of a functionw(t, `, x) by

Πkw(`−i ) = w(`−i ), i = 1, . . . ,M − 1, (5.1){int_pi1} ∫

Ii

(Πkw − w)`s d` = 0, s ≤ q − 1, i ≥ 1, (5.2){int_pi2}

i.e., Πkw interpolates at the nodal points and the interpolation error is orthogonalto the space of polynomials of degree q − 1 on Ii. For this type of interpolant wehave the following error estimates

sup0≤`≤`M

|Πkw(`)− w(`)|j ≤ Ckq+1 sup0≤`≤`M

|w(q+1)(`)|j , j = 0, 1, (5.3){int_pi3} ∫Ii

|Πkw(i)(`)− w(i)(`)|2j d` ≤ Ck2(q+1−i)∫Ii

|w(q+1)(`)|j d`, i, j = 0, 1, (5.4){int_pi4}

see Ref. 38, 42. In order to obtain the error estimate for the splitting method inspace and internal coordinate, we define a projection operator Rh which maps ontothe tensor product space Sr,qh,k. It is defined as follows

Rhw = jhΠkw = Πkjhw ∀w ∈ P, (5.5){R}

where jh is the special interpolant in space satisfying Assumption A1. In addition,we have the stability property of interpolant Πk given by∫

Ω`

∥∥Πku∥∥2Hr+1(Ωx) d` ≤ C ∫Ω`

‖u‖2Hr+1(Ωx) d` (5.6){stb_jpi}

since Πk acts in `-direction and the norms are with respect to the space direction.Let us consider ξn := u(tn) − Rhu(tn) and ηn := Rhu(tn) − unh,k. We also denoteξ̃n := ũ(tn)−Rhũ(tn) and η̃n := Rhũ(tn)− ũnh,k, then the error u(tn)−unh,k can bedecomposed as follows

en = u(tn)− unh,k = ξn + ηn

March 2, 2011 16:10


where unh,k is the solution for fully discrete scheme (4.7) and (4.8) and u(tn) is the

solution of (3.1) and (3.2). Furthermore, to obtain the separate estimates in spaceand internal coordinate we use the following decomposition of errors

Rhw − w =(Rhw −Πkw

)+(Πkw − w

)= ϑ+ ϕ. (5.7) {sp_sp_int}

Assumption A3 : Let u, ut, utt, ũ, ũt, ũtt, zmin and z0 satisfy the followingregularity conditions

u, ũ ∈ H1(L2(Hr+1)

)∩H1

(Hq+1(H1)

), ut, ũt ∈ L2

(L2(Hr+1)

)∩ L2

(Hq+1(L2)

),

utt, ũtt ∈ L2(L2(L2)

), z0 ∈ L2

(Ω`;Hr+1(Ωx)

)∩Hq+1

(Ω`;L2(Ωx)

),

zmin ∈ H1(0, T ;Hr+1(Ωx)

).

Lemma 5.1. Let the assumptions A1-A3 be fulfilled. Then for all t ∈ (0, T ], wehave the following estimates for the interpolation error

‖ϑ(t)‖dG ≤ C hr+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 simplicity we skip the dependency of t within the proof. Since for theinterpolation error the jumps [jhu − u]i, i = 1, . . . ,M − 1, vanishes due to thecontinuity of jhu in internal coordinate, we have from (4.11), the interpolationerror estimates (4.3) and condition (5.6)

‖ϑ‖2dG ≤M−1∑i=1

∫Ii

∂`G‖ϑ‖2L2(Ωx) d`+ ‖G1/2minϑ(`

+0 )‖2L2(Ωx) + ‖G

1/2maxϑ(`

−M )‖

2L2(Ωx)

≤ C h2r+2{‖u‖2L2(Hr+1) + ‖u‖

2C(Hr+1)

}.

For the second estimate with respect to the internal coordinate, we use thedefinition of interpolant Πku, i.e., the interpolation Πku satisfies Πku(`−i ) = u(`i),i = 1, . . . ,M , thus from the second representation (4.10) of the bilinear form B andinterpolation estimates (5.3), (5.4), we have

‖ϕ‖2dG ≤ B(ϕ,ϕ) =M∑i=1

∫Ii

−(Gϕ,

∂ϕ

∂`

)xd`

≤M∑i=1

∫Ii

‖Gϕ‖L2(Ωx)‖∂`ϕ‖L2(Ωx) d`

≤ C k2q+1M∑i=1

∫Ii

‖uq+1‖2L2(Ωx) d` ≤ C k2q+1‖u‖2Hq+1(L2)

which completes the proof of the lemma.

March 2, 2011 16:10


Lemma 5.2. Let the assumptions A1-A3 be fulfilled and τK ∼ hK . Then for allt ∈ (0, T ], the following estimates hold∫

Ω`

ah(ϑ(t), η(t)

)d` ≤ C (ε1/2 + h1/2)hr‖u(t)‖L2(Hr+1)

(∫Ω`

|||η(t)|||2 d`)1/2

+ C hr+1‖u(t)‖L2(Hr+1) ‖η(t)‖0, (5.8) {c4:lem7:atheta}∫Ω`

ah(ϕ(t), η(t)

)d` ≤ C (ε1/2 + h1/2) kq+1‖u(t)‖Hq+1(H1)

(∫Ω`

|||η(t)|||2 d`)1/2

+ C kq+1‖u(t)‖Hq+1(H1)‖η(t)‖0, (5.9) {c4:lem7:aphi}

B(ϑ(t), η(t)

)≤ C hr+1

{‖u(t)‖H1(Hr+1)‖η(t)‖0 + ‖u(t)‖C(Hr+1)‖η(t)‖dG

}, (5.10) {c4:lem7:btheta}

B(ϕ(t), η(t)

)≤ C kq+1‖u(t)‖Hq+1(L2)‖η(t)‖0. (5.11) {c4:lem7:bphi}

Proof. For simplicity of the presentation we again skip the dependency of the timewithin the proof. From the definition of the stabilized bilinear form ah, we have∫

Ω`

ah(ϑ, η)d` = ε

∫Ω`

(∇xϑ,∇xη

)x

+∫

Ω`

(b · ∇xϑ, η

)xd`+

∫Ω`

Sh(ϑ, η)d`

= I1 + I2 + I3. (5.12){c4:lem7:e1}

We start by estimating the first term on the right-hand side. Using Cauchy-Schwarzinequality, the interpolation estimates (4.3) of jh and the condition (5.6), it followsthat

|I1| ≤ ε∫

Ω`

||ϑ||H1(Ωx)||η||H1(Ωx) d` ≤ C ε1/2hr

∫Ω`

‖Πku‖Hr+1 |||η||| d`

≤ Cε1/2hr(∫

Ω`

‖u‖2Hr+1d`)1/2(∫

Ω`

|||η|||2 d`)1/2

≤ C ε1/2hr‖u‖L2(Hr+1)(∫

Ω`

|||η|||2 d`)1/2

.

Integrating I2 by parts with respect to the space variable x, using the orthogonalityproperty of interpolant jh and Cauchy-Schwarz inequality to get

|I2| =∣∣∣ ∫

Ω`

(b · ∇xϑ, η

)xd`∣∣∣ = ∣∣∣ ∫

Ω`

(ϑ,b · ∇xη

)xd`∣∣∣

≤∣∣∣ ∫

Ω`

(ϑ, κh(b · ∇xη)

)xd`∣∣∣

≤∫

Ω`

∑K∈Th

‖ϑ‖L2(K)‖κh(b · ∇xη)‖L2(K) d`.

Let b be the L2-projection of b in the space of piecewise constant functions with re-spect to Th. Using the L2-stability of the fluctuation operator κh, inverse inequality

March 2, 2011 16:10


and κh(b · ∇x)η = b · κh(∇xη), we get in a same fashion as in Ref. 34 the followingestimate∥∥κh(b · ∇x)η∥∥L2(K) ≤ C|b|1,∞,K‖η‖L2(K) + ‖b‖0,∞,K∥∥κh(∇xη)∥∥L2(K). (5.13){C4_th1_14a}Thus inserting this in the previous estimate, using (4.3), µK ∼ hK , and (5.6) to get

|I2| ≤ C∫

Ω`

∑K∈Th

|b|1,∞,K‖ϑ‖L2(K)‖η‖L2(K) d`

+ C∫

Ω`

∑K∈Th

|b|0,∞,K‖ϑ‖L2(K)∥∥κh(∇xη)∥∥L2(K) d`

≤ C hr+1∫

Ω`

‖u‖Hr+1(Ωx)‖η‖L2(Ωx) d`+ C hr+1/2

∫Ω`

‖u‖Hr+1(Ωx) |||η||| d`

≤ C hr+1/2{h1/2‖η‖0 +

(∫Ω`

|||η|||2 d`)1/2}

‖u‖L2(Hr+1).

For I3, the Cauchy-Schwarz inequality and interpolation error estimates give

|I3| =∣∣∣ ∫

Ω`

Sh(ϑ, η)d`∣∣∣ ≤ ∫

Ω`

Sh(ϑ, ϑ

)1/2Sh(η, η)1/2

d`

≤ Chr+1/2∫

Ω`

‖u‖Hr+1(Ωx) |||η||| d` ≤ Chr+1/2‖u‖L2(Hr+1)

(∫Ω`

|||η|||2 d`)1/2

.

Combining I1, I2 and I3, we get the desired estimate∫Ω`

ah(ϑ, η)d` ≤ C (ε1/2 + h1/2)‖u‖L2(Hr+1)

(∫Ω`

|||η|||2 d`)1/2

+ C hr+1‖u‖L2(Hr+1) ‖η‖0.

Next, we find the estimates in internal coordinate. From the definition, we have∫Ω`

ah(ϕ, η) d` = ε∫

Ω`

(∇xϕ,∇xη

)xd`+

∫Ω`

(b · ∇xϕ, η

)xd`+

∫Ω`

Sh(ϕ, η

)d`

= I4 + I5 + I6.

Then by using the Cauchy-Schwarz inequality, the stability property of the fluc-tuation operator κh, the approximation properties (5.3) of interpolant Πk and theparameter choice µK ∼ hK , we get for I4, I5, and I6 the following estimates

|I4| ≤ ε∫

Ω`

‖Πku− u‖H1(Ωx)‖η‖H1(Ωx) d`

≤ ε1/2∫

Ω`

‖Πku− u‖H1(Ωx)|||η||| d`

≤ ε1/2(∫

Ω`

‖Πku− u‖2H1(Ωx) d`)1/2(∫

Ω`

|||η|||2 d`)1/2

≤ C ε1/2kq+1‖u‖Hq+1(H1)(∫

Ω`

|||η|||2 d`)1/2

.

|I5| ≤ C kq+1‖u‖Hq+1(H1)‖η‖0.

March 2, 2011 16:10


|I6| ≤∫

Ω`

∑K∈Th

µK∥∥κh(∇x(Πku− u))∥∥L2(K)∥∥κh(∇xη)∥∥L2(K) d`

≤ C h1/2∫

Ω`

∥∥∇x(Πku− u)∥∥L2(Ωx)|||η||| d`≤ C h1/2kq+1‖u‖Hq+1(H1)

(∫Ω`

|||η|||2 d`)1/2

.

Hence, combining these estimates we get the second statement of the lemma∫Ω`

ah(ϕ, η

)d` ≤ C (ε1/2 + h1/2) kq+1‖u‖Hq+1(H1)

(∫Ω`

|||η|||2 d`)1/2

+ C kq+1‖u‖Hq+1(H1)‖η‖0.

To obtain the last two estimates, we use the two different representations (4.9) and(4.10) of B. Note that the jump terms [jhu− u]i, i = 1, . . . ,M − 1, vanishes due tothe continuity of the interpolant jhu in `-direction. We have from (4.9), (4.3), and(5.6)

B(ϑ, η) =M∑i=1

∫Ii

(∂(Gϑ)∂`

, η)xd`+

(Gminϑ(`+0 ), η(`

+0 ))x

≤M∑i=1

∫Ii

∥∥∂`(Gϑ)∥∥L2(Ωx)‖η‖L2(Ωx) d`+ ∥∥G1/2minϑ(`+0 )∥∥L2(Ωx)∥∥G1/2minη(`+0 )∥∥L2(Ωx)≤ C hr+1

{‖u‖H1(Hr+1)‖η‖0 + ‖u‖C(Hr+1)‖η‖dG

}.

The interpolation Πku satisfies Πku(`−i ) = u(`i), i = 1, . . . ,M . Hence, we get from(4.10) the relation

B(ϕ, η) =M∑i=1

∫Ii

−(Gϕ,

∂η

∂`

)xd`.

Let Π0G be the L2-projection of G in a space of piecewise constant functions in`-direction. Using the orthogonality (5.2) of the interpolant Πk, we get

B(ϕ, η) =M∑i=1

∫Ii

(ϕ, (G−Π0G)

∂η

∂`

)xd`

≤M∑i=1

∫Ii

‖ϕ‖L2(Ωx)∥∥(G−Π0G)∂`η∥∥L2(Ωx) dx

≤ C kq+1‖u‖Hq+1(L2)‖η‖0.

Here, we used the Cauchy-Schwarz inequality, the inverse inequality and the inter-polation error estimates (5.3). This complete the proof.

March 2, 2011 16:10


Theorem 5.1. Let ũ(tn), u(tn) and ũnh,k, unh,k, be the solutions of two-step method

(3.1), (3.2) and (4.7), (4.8), respectively. Under the assumptions A1-A3 and µK ∼hK there holds for ηn = Rhu(tn)− unh,k and η̃n = Rhu(tn)− unh,k

∥∥ηN∥∥20

+ τN−1∑n=0

∫Ω`

∣∣∣∣∣∣η̃n+1∣∣∣∣∣∣2 d`+ τ2

N−1∑n=0

∥∥ηn+1∥∥2dG

≤ Cue9T/2[∥∥Rhz0 − u0h,k∥∥20 + τ2 + (ε+ h)h2r + k2q+2] (5.14){err_e23}

and for en = u(tn)− unh,k and ẽn = ũ(tn)− ũnh,k

∥∥eN∥∥20

+ τN−1∑n=0

∫Ω`

∣∣∣∣∣∣ẽn+1∣∣∣∣∣∣2 d`+ τ2

N−1∑n=0

∥∥en+1∥∥2dG

≤ Cue9T/2[∥∥Rhz0 − u0h,k∥∥20 + τ2 + (ε+ h)h2r + k2q+1] (5.15) {err_e23a}

where Cu depends on u, ut, utt, ũ, ũt, ũtt and zmin.

Note that the error to the interpolant Rhu is superclose with respect to theinternal coordinate (order k + 1 instead of k + 1/2).

Proof. From the result of the Lemma 4.2, we can write for ηn = Rhu(tn)− unh,k

12

∥∥ηN∥∥20− 1

2

∥∥η0∥∥20

+ τN−1∑n=0

∫Ω`

∣∣∣∣∣∣η̃n+1∣∣∣∣∣∣2 d`+ τ2

N−1∑n=0

∥∥ηn+1∥∥2dG≤ T1 + T2 (5.16) {err_e24}

where

T1 = τN−1∑n=0

∫Ω`

{( η̃n+1 − ηnτ

, η̃n+1)x

+ ah(η̃n+1, η̃n+1)}d`, (5.17)

T2 = τN−1∑n=0

{∫Ω`

(ηn+1 − η̃n+1τ

, ηn+1)xd`+B(ηn+1, ηn+1)

}. (5.18)

We first consider T1. Using (4.7), we obtain

T1 = τN−1∑n=0

∫Ω`

{(Rhũ(tn+1)−Rhu(tn)τ

, η̃n+1)x

+ ah(Rhũ(tn+1), η̃n+1

)−( ũn+1h,k − unh,k

τ, η̃n+1

)x−∫

Ω`

ah(ũn+1h,k , η̃n+1)

}d`

= τN−1∑n=0

∫Ω`

{(Rhũ(tn+1)−Rhu(tn)τ

, η̃n+1)x

+ ah(Rhũ(tn+1), η̃n+1

)−(fn+1, η̃n+1

)x

}d`.

March 2, 2011 16:10


For the last term on the right-hand side of the above equation, using (3.4) att = tn+1, we get for the first term

T1 = τN−1∑n=0

∫Ω`

(Rhũ(tn+1)−Rhu(tn)τ

− ũt(tn+1), η̃n+1)xd`

+ τN−1∑n=0

∫Ω`

a(Rhũ(tn+1)− ũ(tn+1), η̃n+1

)d`+ τ

N−1∑n=0

∫Ω`

Sh(Rhũ(tn+1), η̃n+1

)d`

= τN−1∑n=0

∫Ω`

(Rhũ(tn+1)−Rhu(tn)τ

− ũt(tn+1), η̃n+1)xd`

+ τN−1∑n=0

∫Ω`

ah(Rhũ(tn+1)− ũ(tn+1), η̃n+1

)d`+ τ

N−1∑n=0

∫Ω`

Sh(ũ(tn+1), η̃n+1

)d`

= T1,1 + T1,2 + T1,3. (5.19) {T_1}

We treat the contribution of the terms on the right-hand side of (5.19) separately.For the first term, using Cauchy-Schwarz inequality, the Young’s inequality and theinitial condition ũ(tn) = u(tn) for first step

|T1,1| ≤ τN−1∑n=0

∫Ω`

∥∥∥∥Rhũ(tn+1)−Rhu(tn)τ − ũt(tn+1)∥∥∥∥L2(Ωx)

∥∥η̃n+1‖L2(Ωx) d`≤ τ

2

N−1∑n=0

∫Ω`

∥∥∥∥Rhũ(tn+1)−Rhu(tn)τ − ũt(tn+1)∥∥∥∥2L2(Ωx)

d`

+τ

2

N−1∑n=0

∫Ω`

‖η̃n+1‖2L2(Ωx) d`

≤ τN−1∑n=0

∥∥∥∥Rhũ(tn+1)−Rhũ(tn)τ −Rhũt(tn+1)∥∥∥∥2

0

+ τN−1∑n=0

∥∥Rhũt(tn+1)− ũt(tn+1)∥∥20 + τ2N−1∑n=0

∥∥η̃n+1∥∥20.

For first term, applying Taylor’s theorem with integral remainder term and forsecond term the approximation properties of interpolant jh and Πk with the stabilityproperty (5.6) yields

|T1,1| ≤ τ2N−1∑n=0

∫ tn+1tn

∥∥ũtt∥∥20 dt+ τ2N−1∑n=0

∥∥η̃n+1∥∥20

+ CτN−1∑n=0

[h2r+2

∥∥ũt(tn+1)∥∥2L2(Hr+1) + k2q+2∥∥ũt(tn+1)∥∥2Hq+1(L2)].To find the estimates for T1,2, we use the decomposition (5.7) of errors into space

March 2, 2011 16:10


and internal coordinate and get

T1,2 = τN−1∑n=0

{ah(ϑ̃n+1, η̃n+1

)+ ah

(ϕ̃n+1, η̃n+1

)}.

Then from the results (5.8) and (5.9) of Lemma 5.2, we obtain

|T1,2| ≤ C (ε+ h) τN−1∑n=0

[h2r∥∥ũ(tn+1)∥∥2

L2(Hr+1)+ k2q+2

∥∥ũ(tn+1)∥∥2Hq+1(H1)

]

+ C τN−1∑n=0

[h2r+2

∥∥ũ(tn+1)∥∥2L2(Hr+1)

+ k2q+2∥∥ũ(tn+1)∥∥2

Hq+1(H1)

]

+τ

4

N−1∑n=0

∫Ω`

∣∣∣∣∣∣η̃n+1∣∣∣∣∣∣2 d`+ τ2

N−1∑n=0

∥∥η̃n+1∥∥20.

The estimate for T1,3 follows from the approximation properties of the fluctuationoperator κh and the choice of the stabilizing parameter µK ∼ hK . We have

|T1,3| ≤ τN−1∑n=0

∫Ω`

Sh(ũ(tn+1), ũ(tn+1)

)d`+

τ

4

N−1∑n=0

∫Ω`

Sh(η̃n+1, η̃n+1

)d`

≤ C h2r+1τN−1∑n=0

∥∥ũ(tn+1)∥∥2L2(Hr+1)

+τ

4

N−1∑n=0

∫Ω`

∣∣∣∣∣∣η̃n+1∣∣∣∣∣∣2 d`.Finally, by inserting the estimates T1,1, T1,2, and T1,3 into (5.19), we obtain

|T1| ≤ τ2N−1∑n=0

∫ tn+1tn

∥∥ũtt∥∥20 dt+ τ2N−1∑n=0

∫Ω`

∣∣∣∣∣∣η̃n+1∣∣∣∣∣∣2 d`+ τ N−1∑n=0

∥∥η̃n+1∥∥20

+ C h2rτN−1∑n=0

[(ε+ h)

∥∥ũ(tn+1)∥∥2L2(Hr+2)

+ h2∥∥ũt(tn+1)∥∥2L2(Hr+1)]

+ C k2q+2 τN−1∑n=0

[(ε+ h+ 1)

∥∥ũ(tn+1)∥∥2Hq+1(H1)

+∥∥ũt(tn+1)∥∥2Hq+1(L2)].

(5.20) {T_1_n}

Now we estimate the second term T2. Using (4.8) and (3.5) we obtain the followingerror equation for second step

T2 = τN−1∑n=0

∫Ω`

(Rhu(tn+1)−Rhũ(tn+1)τ

− ut(tn+1), ηn+1)xd`

+ τN−1∑n=0

B(Rhu(tn+1)− u(tn+1), ηn+1

)− τ

N−1∑n=0

(Gminz

n+1min −Gminz

n+1min,h, η

n+1(`+0 ))x

= T2,1 + T2,2 + T2,3. (5.21) {T_2}

March 2, 2011 16:10


The estimates for the first term can be obtained by using the same procedure asfor T1,1 and get

|T2,1| ≤ τN−1∑n=0

∥∥∥∥Rhu(tn+1)−Rhũ(tn+1)τ −Rhut(tn+1)∥∥∥∥2

0

+ τN−1∑n=0

∥∥Rhut(tn+1)− ut(tn+1)∥∥20 + τ2N−1∑n=0

∥∥ηn+1∥∥20

≤ τ2N−1∑n=0

∫ tn+1tn

∥∥utt∥∥20 dt+ τ2N−1∑n=0

∥∥ηn+1∥∥20

+ CτN−1∑n=0

[h2r+2

∥∥ut(tn+1)∥∥2L2(Hr+1) + k2q+2∥∥ut(tn+1)∥∥2Hq+1(L2)].Note that in above estimates we have used the initial condition u(tn) = ũ(tn+1)from (3.2). The bounds on the second term T2,2 are obtained by using the errordecomposition (5.7) and the estimates (5.10) and (5.11)

|T2,2| =

∣∣∣∣∣τN−1∑n=0

{B(ϑn+1, ηn+1

)+B

(ϕn+1, ηn+1

)}∣∣∣∣∣≤ C h2r+2 τ

N−1∑n=0

[∥∥u(tn+1)∥∥2H1(Hr+1)

+∥∥u(tn+1)∥∥2

C(Hr+1)

]

+ C k2q+2 τN−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.

Cauchy-Schwarz inequality and Young’s inequality give for T2,3

|T2,3| ≤ τN−1∑n=0

∥∥G1/2minzmin(tn+1)−G1/2minzn+1min,h∥∥L2(Ωx) ∥∥G1/2minηn+1(`+0 )∥∥L2(Ωx)≤ C h2r+2 τ

N−1∑n=0

∥∥zmin(tn+1)∥∥2Hr+1(Ωx) + τ8N−1∑n=0

∥∥ηn+1∥∥2dG.

Finally using these estimates in (5.21) we get for T2

|T2| ≤ τ2N−1∑n=0

∫ tn+1tn

∥∥utt∥∥20 dt+ 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)∥∥2H1(Hr+1)

+∥∥zmin(tn+1)∥∥2Hr+1(Ωx) + ∥∥ut(tn+1)∥∥2L2(Hr+1)

+∥∥u(tn+1)∥∥2

C(Hr+1)

]+ Cτk2q+2

N−1∑n=0

[∥∥u(tn+1)∥∥2Hq+1(L2)

+∥∥ut(tn+1)∥∥2Hq+1(L2)].

March 2, 2011 16:10


Inserting T1 and T2 in (5.16), absorbing the triple norm and the dG norm contri-butions in the left-hand side and using (3.12), we get

12

∥∥ηN∥∥20− 1

2

∥∥η0∥∥20

+ τN−1∑n=0

∫Ω`

∣∣∣∣∣∣η̃n+1∣∣∣∣∣∣2 d`+ τ2

N−1∑n=0

∥∥ηn+1∥∥2dG

≤ τ2N−1∑n=0

∫ tn+1tn

∥∥utt∥∥20 dt+ τ N−1∑n=0

γn∥∥ηn∥∥2

0+ 2τ

N−1∑n=0

∥∥fn+1∥∥20

+ C h2r τN−1∑n=0

[(ε+ h)

∥∥u(tn+1)∥∥2H1(Hr+1)

+ h2∥∥ut(tn+1)∥∥2L2(Hr+1)

+ h2∥∥zmin(tn+1)∥∥2Hr+1(Ωx) + ∥∥u(tn+1)∥∥2C(Hr+1)

]+ C k2q+2 τ

N−1∑n=0

[(ε+ h+ 1)

∥∥u(tn+1)∥∥2Hq+1(H1)

+∥∥ut(tn+1)∥∥2Hq+1(L2)]

where γ0 = 2, γN = 1 and γn = 3, n = 1, . . . , N − 1. We conclude by applying theGronwall’s Lemma in the same fashion as in Lemma 3.1.

6. Implementation of numerical method

This section indicates the implementation of the operator splitting method in thecontext of finite element methods. For more details, we refer to Ref. 14.

Using the bases

Vh = span{φi}, 1 ≤ i ≤ Nx, Sqk = span{ψk}, 1 ≤ k ≤ N`,

the tensor product space Sr,qh,k is defined as follows

Sr,qh,k =

{v =

Nx∑i=1

N∑̀k=1

αikφi(x)ψk(`), αik ∈ R, 1 ≤ i ≤ Nx, 1 ≤ k ≤ N`

}.

The finite element functions are represented as

unh,k =Nx∑i=1

N∑̀k=1

ξnikφi(x)ψk(`), X =Nx∑j=1

N∑̀l=1

xjlφj(x)ψl(`).

We define the matrices Mx, 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 matrices M`, T` ∈ RN`×N` as

(M`)kl =(ψk(`), ψl(`)

)`,

(T`)kl =N∑̀i=1

(∂`(Gψk(`)), ψl(`)

)Ii

+N`−1∑i=1

[Gψk(`)]i ψl(`+i ) +Gψk(`+0 )ψl(`

+0 ).

March 2, 2011 16:10


For the ease of presentation let us consider (2.1) with source term f = 0. Then thecomputing scheme for the operator splitting method described in (4.7) and (4.8) isas follows:

Within each time interval (tn, tn+1], we begin with the x-direction step wherewe are looking for the solution of the time-dependent convection-diffusion equa-tion (4.7). We compute η̃n+1j ∈ RNx , j = 1, . . . , N`, by solving the linear systems

(Mx + τDx + τTx + τSx)η̃n+1j = Mxηnj , j = 1, . . . , N`.

With obtaining the solutions η̃n+1j , j = 1, . . . , N`, the x-direction step is completed.Then, we continue with the `-direction step where we update the solution from thefirst step and compute the solution of the one-dimensional transport problem (4.8)by a discontinuous Galerkin method. In this step we solve the linear systems

(M` + τT`)ηn+1j = M`η̃n+1j , j = 1, . . . , Nx,

and the obtained solutions ηn+1j , j = 1, . . . , N`, are used as input for the timeinterval (tn+1, tn+1].

7. Numerical tests

We report in this section the numerical computations illustrating the theoreticalresults obtained in the previous section. The two-step method (4.7) and (4.8) inthe context of finite element method in space and discontinuous Galerkin methodin internal coordinate is implemented in the finite element package MooNMD.19

The tests are made in two plus one dimensions, i.e, we consider Ωx = (0, 1)×(0, 1)as two-dimensional domain in space and Ω` = (0, 1) as one-dimensional domain inthe internal coordinate. We consider the velocity field b as b1 = b2 = 0.1, the growthrate G(`) = 1 and two different choices for the diffusion coefficient ε, ε = 1 andε� 1. The source term f and the boundary and initial conditions are chosen suchthat the analytical solution of the problem (2.1) is

z(t, `, x, y) = e−0.1t sin(π`) cos(πx) cos(πy).

Let en := z(tn) − unh,k, where z is the exact solution of (2.1) and the numericalsolution unh,k is obtained by two-step method (4.7) and (4.8). We use the followingnotations

‖e‖0 =

(τ

N∑n=1

‖en‖2L2(L2) + τN∑n=1

‖en‖2dG

)1/2,

‖e‖1 =

(τ

N∑n=1

‖en‖2L2(H1) + τN∑n=1

‖en‖2dG

)1/2,

‖e‖DG =

(τ

N∑n=1

∫Ω`

∣∣∣∣∣∣en∣∣∣∣∣∣2 d`+ τ N∑n=1

‖en‖2dG

)1/2.

March 2, 2011 16:10


In order to illustrate the convergence order in time, internal coordinate and space,we use the well known strategy, i.e., the convergence order in time can be obtainedby assuming that the mesh sizes k and h are small enough compared to the time-stepsize τ .

In the numerical computations, we have used triangular and quadrilateralmeshes which are generated by successive refinement starting from coarsest meshes(level 0) as in Fig. 1 for two-dimensional domain Ωx and a line divided into twocells for one-dimensional domain Ω`.

��

��

��

��

��

�

��

��

��

��

��

Fig. 1: Meshes for Ωx on level 0.

Case ε = 1: In this case, the Galerkin finite element method in space is combinedwith a discontinuous Galerkin method in internal coordinate. For time discretiza-tion, the backward Euler time stepping scheme is used with final time T = 1. Onecan expect a convergence for ‖ · ‖0-norm of order O(hr+1) and for ‖ · ‖1-norms oforder O(hr) using Qr and Pr finite elements in space with sufficiently small timestep 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 firstorder convergence in the ‖·‖1-norm for both Q1 and P1 finite elements in space withdG(1) in internal coordinate. The length of the time step was set to be τ = 10−3 andmesh size to k = 1/64. For Q2 and P2 finite elements in space with dG(2) in internalcoordinate, the time step length was set to τ = 10−4 and mesh size k = 1/64. Theresults of Tables 3 and 4 show third order convergence for the ‖·‖0-norm and secondorder for the ‖ · ‖1-norm.

In Tables 5 and 6, the errors and convergence orders for internal coordinateand time are listed. We expect a convergence of order O(kq+1/2) in the internalcoordinate and a convergence of O(τ) in time. The errors for dG(1) in internalcoordinate with Q1 on level 7 and time step length τ = 2.5 · 10−4 are presented inTable 5. We see that the expected orders of convergence are achieved. The numericalerrors and convergence orders in time are listed in Table 6 for dG(1) with k = 1/32and Q1 on level 6. The theoretically predicted convergence order is achieved.

Case ε = 10−9: In the case of convection-dominated convection-diffusion, we

March 2, 2011 16:10


Table 1: Errors and rate of convergence in space for Q1 and dG(1), k = 1/64 andτ = 10−3.

Level ‖e‖0 ‖e‖1error order error order

0 1.719554e-01 —— 1.006185 ——1 4.746460e-02 1.8571 4.892384e-01 1.0403

2 1.206219e-02 1.9764 2.412003e-01 1.02033 3.167958e-03 1.9289 1.201483e-01 1.0054

Table 2: Errors and rate of convergence in space for P1 and dG(1), k = 1/64 andτ = 10−3.


0 2.353104e-01 —— 1.432599 ——

1 7.412177e-02 1.6666 7.996426e-01 0.84132 1.981996e-02 1.9029 4.113880e-01 0.9589

3 5.144843e-03 1.9458 2.072235e-01 0.9893

Table 3: Errors and rate of convergence in space for Q2 and dG(2), k = 1/64 andτ = 10−4.


0 1.916287e-02 —— 2.396151e-01 ——1 2.599528e-03 2.8820 6.137457e-02 1.9650

2 3.354662e-04 2.9540 1.561139e-02 1.9750

Table 4: Errors and rate of convergence in space for P2 and dG(2), k = 1/64 andτ = 10−3.


0 3.511498e-02 —— 5.583590e-01 ——1 4.796648e-03 2.8720 1.526520e-01 1.87102 6.138514e-04 2.9661 3.929766e-02 1.9577

consider local projection as stabilization in space. Discontinuous Galerkin methodsof first and second order are used for the discretization in internal coordinate. Fortime discretization, the backward Euler time stepping scheme is used.

The numerical tests are performed using for (Vh, Dh) the pairs (P bubble1 , Pdisc0 ),

March 2, 2011 16:10


Table 5: Errors and rate of convergence in internal coordinate for dG(1), Q1 onlevel 6 and τ = 2.5 · 10−4.

k ‖e‖01/2 6.696513e-02 ——

1/4 1.829413e-02 1.73981/8 6.521805e-03 1.4880

Table 6: Errors and rate of convergence in time for Q1 and dG(1) on level = 6 andk = 1/32.

τ ‖e‖0 ‖e‖1error order error order

1/10 1.815303e-01 —— 4.027364 ——

1/20 9.577853e-02 0.9224 2.170105 0.89211/40 4.983170e-02 0.9427 1.141479 0.9269

1/80 2.567753e-02 0.9566 5.869174e-01 0.9597

(P bubble2 , Pdisc1 ), (Q

bubble1 , P

disc0 ), and (Q

bubble2 , P

disc1 ). 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.In Tables 7 and 8 we show the convergence results for space in norm ‖ · ‖DG.

Table 7 shows the error in space with stabilizing parameter µ0 = 5, time steplength τ = 10−3 and mesh size k = 1/64 for (Qbubble1 , P0) and (P

bubble1 , P0) with

dG(1) in internal coordinate. In Table 8, the convergence results for (Qbubble2 , Pdisc1 )

and (P bubble2 , Pdisc1 ) with dG(2) in internal coordinate with µ0 = 5, k = 1/64 and

τ = 10−4 are listed. We see that the expected orders of convergence O(hr+1/2) areachieved for quadrangles. For smaller mesh size h, the convergence order starts todecrease for triangles. This is because the influence of the error in internal coordinateincreases, i.e., the mesh size k is not small enough that one can see the correspondingconvergence rate in space for higher order elements.

The numerical errors and convergence orders in internal coordinate are listed inTable 9 for dG(1) and (Qbubble1 , P0) with µ0 = 5 on level 7 and τ = 2.5 · 10−4. Theconvergence order starts to decrease for small mesh size k since the errors in spacehave increasing influence. Finally, Table 10 shows the errors and convergence ordersin time for (Qbubble1 , P0) on level 6 with µ0 = 2.5 and dG(1) with k = 1/32. We seethat the time stepping scheme is of first order convergent.

March 2, 2011 16:10


Table 7: Errors and rate of convergence in space for (Qbubble1 , P0) and (Pbubble1 , P0)

and dG(1), k = 1/64, τ = 10−3 and µK = 5hK .

(Qbubble1 , P0) (Pbubble1 , P0)

Level ‖en‖DG ‖en‖DG0 1.756772 —— 1.93314 ——

1 6.394630e-01 1.4580 7.247844e-01 1.41532 2.280495e-01 1.4875 2.661525e-01 1.4453

3 8.245890e-02 1.4678 1.086554e-01 1.2925

Table 8: Errors and rate of convergence in space for (Qbubble2 , Pdisc1 ) and

(P bubble2 , Pdisc1 ) and dG(2), k = 1/64, τ = 10

−4 and µK = 5hK .

(Qbubble2 , Pdisc1 ) (P

bubble2 , P

disc1 )

Level ‖en‖DG ‖en‖DG0 1.272972 —— 1.234504 ——1 2.558153e-01 2.3151 2.352103e-01 2.3919

2 4.700162e-02 2.4443 5.094834e-02 2.20693 8.010563e-03 2.5527 1.222369e-02 2.0593

Table 9: Errors and rate of convergence in internal coordinate for dG(1) and(Qbubble1 , P0) on level 7 with µK = 5hK and τ = 2.5 · 10−4.

k ‖en‖DG1/2 2.493607e-01 ——

1/4 9.283060e-02 1.42561/8 3.425394e-02 1.4383

1/16 1.446166e-02 1.2441

Table 10: Errors and rate of convergence in time for dG(1) and (Qbubble1 , P0) onlevel with µK = 2.5hK and k = 1/32.

τ ‖en‖DG1/10 8.017623e-01 ——

1/20 4.318566e-01 0.8926

1/40 2.270064e-01 0.92781/80 1.166372e-01 0.9607

8. Conclusion

In this paper we have been concerned with the numerical solution of the populationbalance equation with one internal coordinate posed on the domain Ω` × Ωx ind+ 1 dimension. We proposed an operator splitting method to reduce the original

March 2, 2011 16:10


problem into two subproblems. The method combines the continuous finite elementmethod (and local projection stabilization) in space with discontinuous Galerkinmethod in internal coordinate. We have considered first order backward Euler timestepping scheme. Under certain regularity of exact solution, we have derived errorestimates for the two-step method, i.e., using polynomials of degree r in space andof degree q in internal coordinate the error is of order O(τ +hr+1/2 + kq+1/2) whenε� 1 and O(τ + hr + kq+1/2) when ε = 1.

The application of discontinuous Galerkin method makes the mass matrix cor-responding to the internal coordinate diagonal, which leads to the feasibility ofthe implementation without any projection between the two-steps in the compu-tation process. Computational results shown in Section 7 confirms the theoreticalprediction of error estimates.

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IntroductionModel problemOperator splitting methodFully-discrete methodLocal projection stabilization in space Discontinuous Galerkin method in internal coordinate

Error analysisImplementation of numerical methodNumerical testsConclusion

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