Numerical solutions of systems with ( p , δ ...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2014; 76:855–874Published online 24 September 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/f ld.3955

Numerical solutions of systems with .p; ı/-structure using localdiscontinuous Galerkin finite element methods

Dietmar Kröner1, Michael Ružicka1 and Ioannis Toulopoulos2,*,†

1Abteilung für Angewandte Mathematik, Universität Freiburg, Freiburg, Germany2Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Vienna, Austria

SUMMARY

In this paper, we present LDG methods for systems with .p; ı/-structure. The unknown gradient and thenonlinear diffusivity function are introduced as auxiliary variables and the original (p; ı/ system is decom-posed into a first-order system. Every equation of the produced first-order system is discretized in thediscontinuous Galerkin framework, where two different nonlinear viscous numerical fluxes are implemented.An a priori bound for a simplified problem is derived. The ODE system resulting from the LDG discretiza-tion is solved by diagonal implicit Runge–Kutta methods. The nonlinear system of algebraic equations withunknowns the intermediate solutions of the Runge–Kutta cycle is solved using Newton and Picard iterativemethodology. The performance of the two nonlinear solvers is compared with simple test problems. Numer-ical tests concerning problems with exact solutions are performed in order to validate the theoretical spatialaccuracy of the proposed method. Further, more realistic numerical examples are solved in domains withnon-smooth boundary to test the efficiency of the method. Copyright © 2014 John Wiley & Sons, Ltd.

Received 24 February 2014; Revised 21 July 2014; Accepted 22 August 2014

KEY WORDS: local discontinuous Galerkin methods; .p; ı/-structure system of equations; .p; ı/-structurepenalty jump terms; Newton–Picard iterative methods; numerical solutions in domains withnon-smooth boundary

1. INTRODUCTION

In this paper, we present some computational issues of approximating solutions of systems with.p; ı/-structure. This type of equations appear as a mathematical model describing several physicalproblems such as non-Newtonian flows, plasticity, and glaciology [1, 2]. The most common p-typeproblem of the present general .p; ı/ model is the p-Laplace problem, for which ı D 0. There areseveral contributions analyzing the discretization of p-Laplace equations, for example, [3] usingfinite differences, [4, 5] using mixed finite element volume methods. The first error analysis for theapproximation of the p-Laplace solutions by FEMs has been presented in [6] and Ciarlet in [7], chp.5, presented error estimates for the case of p > 2, by treating the p-Laplace operator in a generalclass of monotone operators. Barrett and Liu in several papers, for example, [8–10], introduce a so-called quasi norm for the error between the exact and the approximate solution and improved errorestimates. In [11], interpolation operators in Orlicz–Sobolev spaces were studied and were utilizedin FEMs for approximating solutions of .p; ı/-structure problems. Linear convergence rate of themethod has been shown in case of using linear base polynomials.

During the last two decades, there is an intense effort toward devising LDG methods for linearand nonlinear elliptic problems. The main feature of LDG methods is the introduction of the gradi-ent (flux) of the solution, say L D ru and the nonlinear diffusion term A D QA.L/L as new variables

*Correspondence to: Ioannis Toulopoulos, Johann Radon Institute for Computational and Applied Mathematics, AustrianAcademy of Sciences, Vienna, Austria.

†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

856 D. KRÖNER, M. RUŽICKA AND I. TOULOPOULOS

and to rewrite the original problem as a first-order system, (a technique similar to the mixed FEMs;however, LDG methods are different in the discretization procedure). The resulting system is thensolved in DG framework, where the variables L; A; u are approximated using the same orderlocal spaces.

Local discontinuous Galerkin methods were first proposed in [12] for convection–diffusion sys-tems, based on the numerical approach applied in [13] for the discretization of the viscous fluxes ofcompressible Navier–Stokes equations. We can say that this was the starting point of a systematicstudy of LDG methods for linear elliptic problems. Indicatively, we refer to the following papers,[14] where the first a priori analysis was presented, [15] where the performance of several LDGmethods for a model problem is shown, and for a detailed review [16], where LDG methods arepresented as a particular choice of DG methods. Recently, LDG methods have been proposed andanalyzed for applications to nonlinear elliptic problems. In [17–19], a theoretical study of LDGmethods is presented for problems with mixed boundary conditions, under the assumption that thediffusion operator is monotone and has p D 2 structure. Moreover, Santillana and Dawson studiedin [20] the applicability of LDG methods for nonlinear diffusion shallow water equations.

The objective of this study is to present a complete framework for the discretization of .p; ı/-structure systems using LDG methods. We generalize the nonlinear penalty jump terms of thenumerical flux proposed in [21] for the case of ı D 0 to ı > 0 (see (11b)) to be compatible withthe .p; ı/-structure of the problem. Furthermore, we apply new penalty jump terms (see (13))which, instead of the jump of uh, use the trace of Lh in order to compute the ‘diffusivity’ onthe inter-element boundaries. Comparisons of the numerical results computed by (11b) and thenew numerical flux (13) are shown in the last section of the paper. The problem is discretized intime using diagonal implicit Runge–Kutta methods. Initially, the nonlinear algebraic system withunknowns of the intermediate solutions of one Runge–Kutta cycle is solved using a Newton methodcombined with matrix-free GMRES linear iterative solver. Despite the fact that this nonlineariterative solution approach achieved fast convergence in many test problems, the performance speedof the method appeared to be dependent on the values of the problem parameters. In particular, whenwe solve the problem on fine meshes, the CPU time of p < 2 test cases is significantly increasedin comparison with the CPU time of p > 2 test cases. This has motivated us to develop and applya Picard-type method which is computationally less expensive and non-parameter dependent in thesense that the convergence speed performance is similar for all p test cases. In order to increase theconverge speed of the Picard method, a new idea of applying nested Jacobi-type iterations is uti-lized. Comparisons concerning the performance of the two different nonlinear iterative processesare shown in the numerical examples. We point out that many real fluids are covered by the range ofvalues for p considered in the numerical experiments, [22]. Moreover, the .p; ı/-structure ensuresthat many popular models in the engineering literature are contained in our investigation.

The outline of the paper is as follows. We begin by presenting the model problem in Section 2.We continue in the Section 3 with the definition of LDG spaces, the numerical fluxes, and finally,the semi-discrete analog of the problem. In the last paragraph, a stability bound is given for a scalarmodel problem. In Section 4, we present the time marching scheme based on diagonal implicitRunge–Kutta methods. A detailed description of the implementation of the Newton and the Picardnonlinear iterative processes is also included. Lastly, in Section 5, we present the experimentalerror convergence rates for different test problems. For first-order polynomial space, we confirmthe theoretical results of the error analysis presented in [15, 17, 18, 23]. Also, we give comparisonsfor the two nonlinear iterative solvers. In the last part of the section, we present numerical testsin domains with corner singular boundary points aimed to investigate the efficiency of the LDGmethod on realistic diffusive problems.

2. .P; ı/-STRUCTURE SYSTEM: THE MODEL PROBLEM

Let � be a bounded domain in Rd ; d D 2; 3 with polygonal (polyhedral) Lipschitz boundary @�which is decomposed into @� D �N

S�D with �N

T�D D Ø; j�Dj ¤ Ø and let .0; T � be a

time interval. We consider the following system

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2014; 76:855–874DOI: 10.1002/fld

LDG METHOD FOR .p; ı/-STRUCTURE PROBLEMS 857

ut � divA.ru/ D f in � � .0; T �; (1a)

u.:; 0/ D u0 in �; (1b)

u D uD on �D � .0; T �; (1c)

A.ru/ � n D aN on �N � .0; T �; (1d)

where

A.ru/ D QA.ru/ru D .ı C jruj/p�2ru: (2)

Here, f W � � .0; T � ! R;uD W �D � .0; T � ! R; aN W �N � .0; T � ! R;u0 W � ! R aregiven appropriately smooth functions, u D .u1; : : : ; udu/

> is the unknown vector, where throughthis paper, we consider the case of d D du, ru is the tensor gradient of u, ı > 0 is the parameter,and QA W Rd�d ! RC is the diffusivity function given by QA.ru/ D .ı C jruj/p�2. Our motivationfor developing LDG methods for the model problem (1) arises from our interest to develop efficientmethods for non-Newtonian incompressible fluids modeled by p-power low systems, [22, 24]. Forı D 0 and p < 2, in case of jruj ! 0, model (1) leads to a singular system. For non-Newtonianfluids, this possesses an infinite zero shear limit, whereas for ı > 0, the zero shear limit is finite.Thus, choosing a relative small value for the parameter ı in (2), we approximate the solution ofthe singular model problem. Results concerning existence and regularity of the solution of (1) andthe interpretation of the time derivative can be found in [25–27]. A detailed discussion about theseresults is out of the context of the present work. The main goal here is to show how problem (1) canbe discretized by the LDG method.

In order to obtain the LDG formulation, we introduce auxiliary variables L D ru and A DQA.L/L and we rewrite system (1) as a first-order system

L D ru in � � .0; T �; (3a)

A D QA.L/L in � � .0; T �; (3b)

ut � div.A/ D f in � � .0; T �: (3c)

In that way, problem (1) is reformulated as (3), and consequently, we have to approximate L;A; uin an appropriate way.

Remark 1We note that operator A falls in the general class of .p; ı/-structure operators [28, 29], this meansthat there exist p 2 .1;1/; ı 2 Œ0;1/ and constants C0; C1 such that

dXj;lD1

dXi;kD1

@klAij .P/QijQkl >C0.ı C jPj/p�2jQj2; (4a)

j@klAij .P/j 6C1.ı C jPj/p�2; (4b)

are satisfied for all P; Q 2 Rd�d with P ¤ 0 and all i; k; j; l D 1; : : : ; d .

3. THE LDG METHOD

3.1. Preliminaries—DG notation

For u; v 2 Rd and A WD .Aij /;T WD .Tij / 2 Rd�d , we use standard notation, u˝ v D uivj ; 2

Rd�d , A W T DPdi;jD1AijTij , u � .Av/ D

Pdi;jD1 uiAij vj D A W .u˝ v/.

Let Th D ¹EiºNEiD1 be a regular subdivision of � in triangular (or tetrahedral) elements withdiameter hEi and h D maxEi2ThhEi . We denote the collection of the edges (or faces) of the



elements by Eh D ¹eºNeiD1. We group the edges into three sets. The set EI of the interior edges,EI WD ¹e W e 2 Eh � @�º. For e 2 EI , there are two adjacent elements E1; E2 that share e. Theset of the Dirichlet boundary edges ED WD ¹e 2 Eh W e \ �D ¤ Øº and the set of the Neumannboundary edges EN WD ¹e 2 Eh W e \ �N ¤ Øº.

We denote by Pk; k 2 N0 the space of scalar polynomials of degree less than or equal to k andby p0 the conjugate exponent 1

pC 1p0D 1. On Th, we define the discontinuous finite element spaces

�V kh

�dW D

²vh 2 .L

p.�//dW vhjE 2

�Pk.E/

�d;8E 2 Th

³; (5)

�Y kh

�d�dW D

°Lh 2 .L

p.�//d�dW LhjE 2 .P

k.E//d�d ;8E 2 Th±; (6)

�Xkh

�d�dW D

²Ah 2

�Lp0

.�/�d�d

W AhjE 2�Pk.E/

�d�d;8E 2 Th

³: (7)

3.1.1. Interface jumps and averages. For a function v 2�V kh

�d, we do not impose any continu-

ity requirements on the interfaces of the elements. Thus, for two elements E1; E2 which share acommon edge e D @E1

T@E2, we assume the outward normal vector ne;12 to be oriented from E1

toward the interior of E2 and conversely ne;21 from E2 toward the interior of E1. We define theaverage and the jump, respectively, of v on e by

¹vº WD1

2.vjE1 C vjE2/; �v˝ n� WD vjE1 ˝ ne;12 C vjE2 ˝ ne;21; (8)

¹Aº WD1

2.AjE1 C AjE2/; �An� WD AjE1ne;12 C AjE2ne;21: (9)

3.2. The LDG discretization

We multiply each equation in (3) by a test function .Xh;Yh; zh/ 2��Xkh

�d�d��Y kh

�d�d��V kh

�d�,

respectively, integrate over one element E 2 Th and use partial integration to obtain the discreteformulation: we look for the LDG approximations Lh 2

�Y kh

�d�d; Ah 2

�Xkh

�d�d; uh 2

�V kh

�dof .L; A; u/ of (3) such that for E 2 Th and t 2 .0; T �, the following equations are satisfiedZ

E

Lh W Xh dx D �ZE

uh � divXh dx CZ@E

buh � .Xhn/ ds; (10a)ZE

Ah W Yh dx DZE

QA.Lh/Lh W Yh dx; (10b)ZE

uht � zh dx D �ZE

Ah W rzh dx CZ@E

bAh W .zh ˝ n/ ds CZE

f � zh dx; (10c)

where the fluxes O.:/ on @E in (10) are the numerical fluxes, which must be defined suitably in orderto ensure stability and convergence of the method (e.g., see [30] and [31] for comprehensive analysisof the numerical fluxes for a linear elliptic model). As in [23], we choose

buhe WD8<:

uh on e 2 EN ;uD on e 2 ED;¹uhº on e 2 EI ;

(11a)

cAhe WD8<ˆ:

aN on e 2 EN ;Ah � � QA

��uh˝n�h

��uh˝n�h

on e 2 ED;

¹Ahº � � QA�

�uh˝n�h

��uh˝n�h

on e 2 EI ;(11b)



where � > 0 is a constant (which will be specified in the numerical examples) and for e 2 ED , thejump is defined as �uh ˝ n� D .uh � uD/˝ n. The term

� QA

��uh ˝ n�

h

��uh ˝ n�

h(12)

is called penalty jump term and complies with the .p; ı/-structure of A, see (2). Here, with thenumerical flux (11b), we have generalized the numerical flux from [21], which was proposed for thecase ı D 0 to the general case of ı > 0.

Another type of numerical flux. In addition to the numerical flux defined in (11b), we proposea new numerical flux

bAeh;L WD8<:

aN on e 2 EN ;Ah � � QA.Lh/

�uh˝n�h

on e 2 ED;¹Ahº � �¹ QA.Lh/º

�uh˝n�h

on e 2 EI ;(13)

where � > 0 is a constant and the diffusivity function QA on the edges depends on Lh. The flux (13)increases the performance of the computational procedure. This will be explained in the numericaltests, where we perform comparisons between the results computed by the numerical fluxes (11b)and (13). Here is the first time where the numerical fluxes (11b), (13) are applied in LDG methodsfor solving problems with .p; ı/-structure. Notice that for the linear case p D 2, the two differentpenalty jump terms are the same.

3.3. Stability bounds for the scalar case

In the scalar case where u W � � .0; T / ! R, the model problem (1) with ı D 0 (for ı > 0, werefer to [23] for steady problems) has the form

ut � div . Qa.ru/ru/ D f in � � .0; T �; (14a)

u0.x/ D u.x; 0/ in �; (14b)

. Qa.ru/ru/ � n D aN on �N � .0; T �; (14c)

u D uD; on �D � .0; T �; (14d)

where Qa.v/ D jvjp�2; v 2 Rd . Following the same procedure as in the previous section, wecan easily obtain the corresponding LDG formulation (10) for problem (14). Next, using expres-sion (11) for the numerical fluxes and summing up over all E 2 Th, we obtain after some simplemanipulations, see details in [16], the discrete variational formulation: find the LDG approxima-tions .lh; ah; uh/ 2 .

�Y kh

�d�d��Xkh

�d�d��V kh

�d/ such that the following discrete equations are

satisfied Z�

lh � xh dx DZ�

ruh � xh dx �ZEI

�uhn� � ¹xhº ds

�

ZEDuhxh � n ds C

ZEDuDxh � n ds; (15a)Z

�

ah � yh dx DZ�

Qa.lh/lh � yh dx; (15b)Z�

uht´h dx C

Z�

ah � r´h dx

D

ZEI¹ahº � �´hn� ds �

ZEI� Qa

��uhn�

h

��uhn�

h� �´hn� ds

C

ZED

ah � n´h ds �ZED� Qa

��uhn�

h

��uhn�

h� ´hn ds

C

ZENaN´h ds C

Z�

f ´h dx (15c)



for xh 2�Xkh

�d; yh 2

�Y kh

�d; ´h 2 V

kh

. Next, we derive an a priori bound for the LDG scheme(15) in a special case, namely, we assume �N D Ø and uD D 0 in problem (14). Choosing in (15)´h D uh; xh D ah; yh D lh and adding all equations, we obtain after some manipulations

Z�

uhtuh dx C

Z�

Qa.lh/jlhj2 dx C �

ZEIQa

��uhn�

h

��uhn�2

hds

C �

ZEDQa

��uhn�

h

��uhn�2

hds D

Z�

f uh dx:

(16)

By virtue of the form of Qa and using that Qa�

�uhn�h

��uhn�2

hD 1

hp�1j�uhn�jp on e 2 EI [ ED , we

have that Z�

uhtuh dx C�

hp�1

ZEI[ED

j�uhn�jp ds C

Z�

jlhjp dx D

Z�

f uh dx: (17)

Next, applying Cauchy–Schwartz inequality on the right-hand side of (17) and using uhtuh D12@@t.uh.t//

2, we have

1

2

d

dtkuh.t/k

2L2.�/

C klhkp

Lp.�/C

�

hp�1k�uhn�k

p

Lp.EI[ED/

6 12kf k2

L2.�/C1

2kuhk

2L2.�/

:

(18)

Integrating from 0 to t yields

kuh.t/k2L2.�/

C 2

Z t

0

klh.�/kp

Lp.�/C

�

hp�1k�uh.�/n�k

p

Lp.EI[ED/ d�

6Z t

0

kf .�/k2L2.�/

C kuh.�/k2L2.�/

d� C kuh.0/k2L2.�/

:

(19)

Applying Gronwall’s inequality in (19), we can obtain the following a priori bound for the LDGsolutions of (15),

kuh.t/k2L2.�/

C 2

Z t

0

klh.�/kp

Lp.�/C

�

hp�1k�uh.�/n�k

p

Lp.EI[ED/ d�

6 et�Z t

0

kf .�/k2L2.�/

d� C kuh.0/k2L2.�/

�:

(20)

4. TIME MARCHING SCHEME

We denote by S.t/ D ŒL.t/;A.t/;U.t/�> the vector with the global DOFs of the expressions of(Lh;Ah;uh/ in

��Xkh

�d�d��Y kh

�d�d��V kh

�d�. By the discrete formulation (10), we obtain the

following nonlinear ODE system,0@ 0

0

M dU.t/dt

1A D

[email protected];S.t//

R2.t;S.t//R3.t;S.t//

1A ; (21)

whereM is the mass matrix of uh and the components of R1;R2;R3 are derived from the variationalformulation (10). We discretize the system (21) by applying s-stage diagonal implicit Runge–Kutta(sDIRK) methods [32, 33]. In sDIRK methods, the coefficient matrix Œaij � 2 Rs�s of Butcher’s table



is lower triangular with equal diagonal elements, offering a significant computational advantage,because for every time step s, decoupled ODEs systems must be solved.

If �1; : : : ; �s are the quadrature points and b1; : : : ; bs are their weights, supposing that we havecomputed the solution Sn at time step tn, then the solution SnC1 at time step tnC1 D tn C �t iscomputed by the following formula

0 DR1.tnC1;SnC1/; (22a)

0 DR2.tnC1;SnC1/; (22b)

UnC1 DUn C�tsXiD1

biM�1R3.tn;i ;Sn;i /; (22c)

where Sn;i are intermediate solutions at tn;i D tn C �i�t given by

0 DR1.tn;i ;Sn;i /; (23a)

0 DR2.tn;i ;Sn;i /; (23b)

Un;i DUn C�tiX

jD1

aijM�1R3.tn;j ;Sn;i /: (23c)

The computation of the intermediate solutions requires the solution of the nonlinear system (in the

general case of p ¤ 2), which is achieved by an iterative process. Next, we develop and apply twowell-known iterative processes, the Newton method (the most widespread second order iterativemethod) and the Picard method (first-order convergence rate), see details in [34].

Newton iterative process. The Newton method for the system (23) takes the form0B@

@R1@S@R2@S

M ��tai i@R3@S

1CA�Sn;i;kC1 D

0@ R1;S

R2;S�tR3;S

1A ; k D 0; 1; 2; : : : (24a)

�Sn;i;kC1 DSn;i;kC1 � Sn;i;k; (24b)

where @Rj@S , Rj;S ; j D 1; 2; 3 are the associated Jacobian matrices and the resulting residual

vectors, which are calculated at Sn;i;k . In the materialization of (24), the Jacobian matrices arenot computed explicitly but are replaced by numerical approximations (matrix-free implementa-tion). Consequently, a Krylov subspace projection method for non-symmetric systems (GMRES) isapplied, see [35], in order to approximate the solution of the derived linear system. Further detailsfor implementing sDIRK methods for nonlinear equations are presented in [36]. In the numericalcomputations, .k C 1/ order sDIRK method is applied.

We mention that several multigrid or preconditioned techniques have been proposed in the litera-ture for solving nonlinear systems similar to (24) produced by the discretization of p-type problemsas (1), [37]. In our numerical examples, we do not use any particular of these techniques.

The computational effort of the whole method depends strongly on the number of iterations ofthe Newton process due to the fact that in every iteration k, the Jacobian matrices in (24) mustbe computed. Also, it is important for the efficiency of the iterative process, the solution of thelinear system (23) to be obtained in few GMRES iterations. It is known that, in many nonlinearproblems, the condition number of the resulting Jacobian matrices is very large, and as a result,the convergence speed of the iterative solver is very slow (or even the iterative solver can fail),[34]. During the performance of the numerical examples, we saw that the performance speed of theiterative process (Newton and GMRES) is affected by the value of � and p. Specifically, we sawthat choosing � > 2 for the p < 2 test cases, the CPU time is increased at a higher rate comparedwith the p > 2 test cases. For this reason, we develop and apply a Picard-type method, which is less



costly (but still first order) and more stable in the sense that the performance speed is not stronglyaffected by the parameter � and is similar for all p test cases.

Picard method. The Picard iteration scheme for (23) is expressed as follows

A�

Sn;i;k�

Sn;i;kC1 D F; k D 0; 1; 2; : : : (25a)

The coefficient matrix A in block-form can be written as

24 QM 0 BQA.Ln;i;k/ QM 0

0 �tai iB> C.Ln;i;k/

35Sn;i;kC1 D

0@ F10

F3

1A ; (25b)

where QM is the mass matrix of Lh, the matrices B; QA.Ln;i�1/ are defined by the discrete variationalformulation (10), the components F1;F2 are derived by the formula (23), and C.Ln;i;k/ D M ��tai i QA.¹Ln;i;kº/.

The convergence of the iteration scheme (25) can further speed up when a block Jacobi approachis utilized. In this approach, the whole structure of the system (25) is retained, but the iterative matrixA.Sn;i;k/ is split into a block diagonal matrix D.Sn;i;k/, which is related to the volume integralsof (10) and into off-block diagonal O.Sn;i;k/, which arises by the computation of the numericalfluxes on the interfaces. The matrix O.Sn;i;k/ is transferred to the right-hand side of (25a) and aJacobi interior iterative process withD as iterative matrix applied to obtain an intermediate solutionSn;i;kC

12 . In the next Picard iterative step, we use the solution Sn;i;kC

12 for updating the matrix A in

(25a). The iterative process (25a) including the inner Jacobi iterations can be written as

A�

Sn;i;kC12

�Sn;i;kC1 D F; k D 0; 1; 2; : : : ; (26a)

where the solution Sn;i;kC12 is obtained by

D�

Sn;i;kj

�Sn;i;kjC1 DF �O

�Sn;i;kj

�Sn;i;kj ; j D 0; 1; 2; : : : ; JM : (26b)

We set Sn;i;kC12 WDSn;i;kjDJM

: (26c)

The advantage of this approach is the following: the matrix D is block diagonal where every block,say DE , is associated with an element E 2 T〈. The first and last rows of DE are formed by linearterms. The second row of DE includes nonlinear entries which are formed by the integral termsof (10b). Having found the solution Sn;i;kj (by the previous Jacobi iteration step), the nonlinearentries ofDE are updated easily by applying simple integration rules. Thus, the linear system (26b)with unknowns Sn;i;kjC1 can be solved block by block using an LU (stored) factorization procedure.Therefore, after the last Jacobi iteration, we can update the entries of the coefficient matrix (26a)by the solution Sn;i;kC

12 which is ‘closer’ to Sn;i;kC1. In fact, Sn;i;kC

12 is ‘closer to’ Sn;i;kC1 than

the solution that we obtain in the case where we apply explicit time stepping from tn to tnC1. Moreprecisely, in this case, the expression (26b) takes the form of an explicit treatment of (21) (or (25b)),that is,

ML;A;USn;i;kjC1 D F � QO�

Sn;i;kj

�; j D 0; 1; 2; : : : ; JM ; (27)

where ML;A;U is the block diagonal mass matrix of L;A;U, and QO.Sn;i;kj / includes all the rest fluxterms that appear in (21). The interior method (27) inherits the small size of �t that we have toapply for solving the original problem (21) by an explicit method. Thus, the intermediate solutionSn;i;kC

12 obtained by (27) does not differ much by the initial starting solution Sn;i;k , (specially for

small number JM ), and as a result, we do not have a remarkable improvement in the performance ofthe original Picard method (25). In order to have more clear picture of this, we make comparisonsin the numerical examples, see also comments in the next paragraph implementation remarks.



We point out that the nested approach (26b) is expected to have similar convergence rate perRunge–Kutta cycle as the original scheme (25a) but improved performance in terms of CPU time.This is shown in the numerical examples. Henceforth, we will call the iterative scheme (26) asJacobi–Picard and denote as Jc–Picard and the iterative scheme with the explicit nested method(27) explicit Jacobi–Picard and denote as ExplJc–Picard.

4.1. Implementation remarks

In practice, the iteration processes are stopped when k�Sn;i;kC1kl2 6 � for a prescribed tolerance�, and we set Sn;i WD Sn;i;kC1 at the last iteration. We set � D 1:E � 06.

In both Picard’s iteration schemes, we increased the efficiency of the iterative linear solver byapplying a preconditioned GMRES iterative solver. The use of a preconditioned GMRES solver wasnot possible in the previous implementation of the matrix-free Newton method.

The codes materialized for all iterative processes have the ability of adjusting (increasing ordecreasing) the size of �t at the next time step, according to a criterion which is derived by usingthe number of the nonlinear iterations and the number of the iterations of the linear solver (GMRES)of the current time step. The solution of ODE problem (21) starts by using an initial�t0 D .

hi16/kC1

and an initial guess given either by the initial conditions of (1) or by previous time steps solutions.The code, which materializes (27), keeps fixed time step �tExplicit D

�t200

. Both Jc–Picard andExplJc–Picard iterative schemes utilize JM D 2 inner Jacobi iterations.

5. NUMERICAL TESTS

In this section, we present several numerical results in order to illustrate the performance of theproposed LDG method to problem (1) with (p; ı)-structure. In the first paragraph, we considerthe problem (1) with known exact solutions for verifying experimentally the spatial convergencerate of the method. Then, we study the efficiency (convergence characteristics) of the two differentnonlinear iterative processes. In the last paragraph, we focus on the applicability of the method forsolving realistic flow problems on domains with non-convex corners.

5.1. Convergence studies

All the numerical examples presented here have been performed using the Newton nonlinear iter-ative method. The domain is � WD Œ�2; 2� � Œ�2; 2�. We set �D D @� and the data f;uD of (1)are specified by the given exact solution. Every test problem has been solved up to the final timeT D 0:25, the error is computed at the final time step and is given by the expression [23]

k:kLDG WD�kF.L/ � F.Lh/k

2L2.�/

C kF�.A/ � F�.Ah/k2L2.�/

C�hkF.h�1�.uh � u/˝ n�/k2L2.EI

SED/

� 12

;

(28)

where F.A/ D .ı C jAj/p�22 A and F�.A/ D .ıp�1 C jAj/

p0�22 A with 1

pC 1

p0D 1. We use k D 2

and k D 1 for the local polynomial spaces. For every test problem, the values of the parameters isgiven in Table I.

Smooth problem. In this test the exact solution is

u.x; t / D²u.x; t / D B.t/ sin.x/;v.x; t / D B.t/ cos.y/;

(29)

where B.t/ D 1C exp.�100t/ and x D .x; y/. The initial unstructured mesh Th0 is generated bya triangular mesh generator with h0 D 1, and the next finer meshes Thi are obtained by subdividingthe triangles to four equal triangles, hiC1 D

hi2

. The numerical convergence rates r are computed

by the ratio r Dln.ehi =ehiC1 /

ln.2/ , where ehi is the error (28) computed on the mesh Thi . In Figure 1,



Table I. Numerical examples with the parameter values, graphs, and tables.

Numerical example Parameter values Figure/table

Smooth problemp 1.5 2 2.5� 0.2 2 2 ı D 0:001 Figure 1

Natural regularity,Numerical test 1Numerical test 2

p 1.5 2 2.5� 0.2 2 2 ı D 0:001 Figure 2(a,b,c)

Natural regularity,Numerical test 3

p 1.5 2 2.5� 0.2 2 2 ı D 0,ı D 1 Figure 2(d,e)

Natural regularity,Numerical test 4

p 1.5 2 2.5� 1 1 1 ı D 1 Figure 2(f), Table II

Numerical fluxes,test (i), test (ii),test (iii)

Flux (11b) Flux (13)- ı D 0:001 ı D 0:001

p 1.5 2.5 1.5 2.5� 0.2 2 0.5 2

Figure 3, Table III

Numerical fluxes,test(iv)

Flux (13), ı D 1p 1.5 2.5� 1 1

Table III

we plot the exact solutions (left column) side by side with the corresponding LDG solutions (rightcolumn) computed using k D 2; p D 2:5 and the numerical flux (11b). The rates r are presentedin Figure 1(e,f). We observe that the rates are the expected according to the regularity of u and aresimilar to the corresponding convergence rates that have been presented in [14, 15, 17] for p D 2

structure linear–nonlinear elliptic problems.

Natural regularity problem. In the following numerical experiments (tests 1, 2, and 3), we assesthe convergence rate of the proposed method in case where the solution of (1) has poor regularity.We consider a solution u of (1) with a point singularity at the origin given by

u.x; t / D²u.x; t / D B.t/jxja.p/y;v.x; t / D B.t/jxja.p/.�x/;

(30)

where the exponent a.p/ will be specified in the test cases. An easy computation shows that jrujbehaves like jxja.p/ and F.ru/ behaves like jxj

a.p/p2 . The numerical flux (11b) has been used.

Numerical test 1. In this example, we set a.p/ D 0:01 and the regularity of F.ru/ is close toW 1;2.�/. Because of the reduced regularity of the exact solution, we expect theconvergence rates r for the k D 2 solution to be reduced in comparison with theresults of Figure 1(e). Indeed, in Figure 2(a), we plot the error versus grid sizein the case k D 2 and we can see that all error rates r are less than two even inthe linear case p D 2. In the case k D 1, Figure 2(b), all error rates are optimalr D 1 with respect to the regularity of the exact solution.

Numerical test 2. In this example, we increased the regularity of the exact solution (30), by set-ting a.p/ D 2

pC 0:01, which means F.ru/ 2W 2;2.�/. The rates are presented

in Figure 2(c) and p ¤ 2 are suboptimal (less than 2) and only for the linear casep D 2, the rate r tends to 2, because we have ru 2W 2;2.�/.

Numerical test 3. In order to see the influence of the parameter ı (see (2)) to the accuracy ofthe LDG method, we performed the numerical test 2, setting ı D 0 and ı D1. The corresponding convergence rates are plotted in Figure 2(d,e). Althoughthe rates in Figure 2(d,e) are slightly increased in comparison with the rates in



Figure 1. Convergence studies, smooth problem. First and second lines: the contours of the exact solutions(29) side by side with the contours of the LDG solutions in case of P2.E/ and p D 2:5. Last line: (e) rates

r for P2.E/; (b) rates r for P1.E/.



Figure 2. Convergence studies, natural regularity problem, (a) test 1: rates r for P2.E/ and F.ru/ 2W 1;2.�/; (b) test 1: rates r for P1.E/ and F.ru/ 2 W 1;2.�/; (c) test 2: rates r using P2.E/ andF.ru/ 2 W 2;2.�/; (d) test 3: rates r for P2.E/, ı D 0 and F.ru/ 2 W 2;2.�/. (e) test 3: rates r forP2.E/, ı D 1 and F.ru/ 2W 2;2.�/. (f) test 4: rates r for P2.E/, ı D 1; � D 1 and F.ru/ 2W 2;2.�/.



Figure 2(c), the overall behavior is the same. Thus, the numerical convergencerate of the LDG method, that we found in this example, is compatible with theregularity properties of the solution and is not strongly affected by the change ofthe perturbation parameter ı.

Numerical test 4. In Section 3, we proved that the proposed LDG method is stable for any � > 0.In order to see numerically the influence of � , we performed again the last testsetting � D 1, see Table I. The rates are presented in Figure 2(f) and appear to bethe same with the rates presented in Figure 2(e), as it was expected. A comparisonof the CPU time for the two different choices of � (but the same ı D 1) is givenin Table II. In general, for the p > 2 tests cases, the CPU times are very closed,but for p D 1:5, the CPU time increases when the value of � is increased. Thisexplains our initial choice (see first lines of Table I) to perform the p D 1:5 testsusing a small value for the parameter � .

Numerical flux comparisons. Here, we investigate the effect of the penalty jump terms of thenumerical fluxes (11b), (13) to the convergence rate of the error

kF.L/ � F.Lh/kL2.�/: (31)

The parameter values are given in Table I.

Test (i). We solved problem (1) with exact solution (29) applying k D 2 polynomial space.Figure 3(a,b) presents the error (31) versus the grid size. As we can observe, the error ofboth fluxes (11b) and (13) converges with (the expected) rate r D 2.

Test (ii). We solved problem (1) with exact solution (30) in the case of a.p/ D 0:01 (F.ru/ 2W 1;2.�/) using k D 1 for the polynomial space. Figure 3(c,d) shows the convergencerates of the error (31). In both graphs, we can see that the rates r are similar and remainclose to the optimal rate r D 1, as the mesh is progressively subdivided. The errormagnitude of flux (11b) is greater than the error magnitude of (13).

Test (iii). For problem (1) with exact solution (29), we present the CPU times for every mesh inthe first four columns of Table III. As we can see, for p D 2:5 test cases the CPU time isalmost the same for both numerical fluxes. For the test case with p D 1:5, the CPU timeof the flux (13) is less than the CPU time of flux (11b), specially for the finer meshes.This shows that for the p D 1:5 test cases, the flux (13) produces a faster iterative methodthan the flux (11b).

Test (iv). The last two columns of Table III show the CPU times of the solution of problem (30)utilizing the flux (13) and using � D 1 and ı D 1. A comparison with the correspondingresults in Table II shows that the CPU time of p D 1:5 test is lees for flux (13), but theCPU times for the test p D 2:5 are very close. Thus, the CPU times of the numericalfluxes have similar behavior as in test (iii).

5.2. Picard performance

In this paragraph, we present results related to the performance of the Picard nonlinear iterativeprocess and make comparisons with the results of the Newton iterative method. Both Picard and

Table II. CPUs for the two different choices of � .

p D 1:5 p D 2 p D 2:5

[0.5ex] Thi � D 0:2 � D 1 � D 2 � D 1 � D 2 � D 1

- CPU time, ı D 1i D 0 1.76 0.92 0.81 0.71 1.93 1.96i D 1 8.79 15.35 12.47 12.30 16. 15.25i D 2 46.20 72.08 43.75 45.38 71.11 89.71i D 3 334.24 528.9 360.12 347.88 559.08 702.89i D 4 4531.8 5928.25 3693.2 3605.34 6192.34 6751.8



Figure 3. Numerical flux comparison. (a) test (i) exact solution (29): rates r for P2.E/; (b) test (i) exactsolution (29): rates r for P2.E/; (c) test (ii) exact solution (30): rates r for P1.E/, and F.ru/ 2W 1;2.�/;

(d) test (ii) exact solution (30): rates r for P1.E/, and F.ru/ 2W 1;2.�/.

Table III. CPUs for the two different numerical fluxes.

Problem (29) Problem (30)

p D 1:5 p D 2:5 p D 1:5 p D 2:5

Thi Flux (11b) Flux (13) Flux (11b) Flux (13) Flux (13) ı D 1; � D 1

- CPU timei D 0 2.18 1.9 2.2 2.186 2.01 2.06i D 1 14.78 14.63 16.48 16.82 9.0 11.11i D 2 150.026 120.446 162.6 141.24 68.07 99.67i D 3 1504 1098.4 604.2 576.588 548.58 790.7i D 4 16040.4 11091.1 4995.44 4854.34 4885.01 6540.3

Newton methods are utilized for the LDG method with numerical flux (13). We do not presentconvergence rate graphs, because the rates have been found to be similar with the rates presented inthe previous paragraph.



Table IV. CPUs for the Picard nonlinear iterative processes.

Picard ExplJc–Picard Jc–Picard

Thi p D 1:5 p D 2:5 p D 1:5 p D 2:5 p D 1:5 p D 2:5

- CPU timei D 0 6.48 6.17 6.93 6.24 4.78 4.26i D 1 58 55.36 58.05 55.2 38.36 36.4i D 2 534.3 527.2 518.85 545.8 420.5 412.5i D 3 5207.3 5087.6 5058.3 4756.07 4065.6 3966i D 4 51361.4 49780 48628.9 44936 38462 37880

Table V. First lines: total number of iterations for the nonlinear iterative processes.Last line: Min–Max iterations of the GMRES solver.

Picard Jc–Picard Newton

Thi p D 1:5 p D 2:5 p D 1:5 p D 2:5 p D 1:5 p D 2:5

- Total number of iterationsi D 0 135 117 125 90 100 80i D 1 330 262 295 245 263 201i D 2 1406 1120 1183 1012 654 537i D 3 10055 7675 6545 5165 2202 1806i D 4 50412 46020 35770 26697 6640 5500

- GMRES iterationsMin–Max 5-17 3-15 5-17 3-15 14-35 11-24

We solved problem (1) with exact solution (29), using k D 1 for the local space and � D 2 for allp test cases, see the first line of Table I. Table IV shows the CPU time of the Picard iterative methodsfor every Thi . The CPU times of ExplJc–Picard and Picard methods are quite close. It seems that theimplementation of the interior method (27) for updating the nonlinear entries of the Picard matrixdoes not remarkably improve the performance of the whole iterative method. Conversely, the CPUtime values decrease quite when the Jc–Picard method is applied. The decrease is higher for thefine meshes. In general, the CPU time values of the same Picard method for the two different ptest cases are close. We cannot see the same relation of the Newton CPU time values, which arepresented in Tables III and II. Further, we observe that the increasing rate of the Picard CPU timefor both p cases is almost 9.80 and the corresponding rate of the Jc–Picard is around 9.55 (even forthe fine meshes), this shows a stable behavior for the method. On the other hand, the results of theNewton method in Table III show that the CPU time of p D 1:5 test case is increasing with a rate ofabout 11.12, but the increasing rate for the p D 2:5 case is about 9.4, hence, the performance of theNewton iterative method varies with the choice of p. A comparison of the convergence behavior ofthe methods with respect to the total number of iterations is presented in the first lines of Table V.The Newton methods appear to have better computational efficiency due to the lower number of totaliterations needed for the full solution of the problem. The iterations of the Jc–Picard are reducedenough in comparison with the iterations of Picard method, (specially for the fine meshes), showingthe effectiveness of this method over the last method. The last line of Table V shows the minimum–maximum number of the GMRES iterations during the solution of the problem on the last threemeshes. The numbers confirm, the comments in Section 5 (paragraph Newton iterative process)that for p D 1:5 case, the resulting Jacobian matrix of the Newton method has higher conditionnumber than the Jacobian matrix of p D 2:5 case. On the other hand, the results indicate that thecondition numbers of the produced Picard matrices are less sensitive to the choice of the p, becausethe GMRES solver has similar behavior for the two p test cases.

For the numerical examples that we present here, we point out that for the Newton method, weused a matrix-free implementation code (the Jacobian matrix is not explicitly stored), [36, 38],achieving better overall performance than the applied Picard methods, where the matrices are stored.



This fact and the quadratic (expected) convergence rate of Newton approach help this iterativemethod appear more efficient than the two Picard methods, even though as we observed in the lastline of Table V, the Newton method applies more GMRES iterations.

5.3. Realistic problems in non-smooth domains

In this paragraph, we test the proposed LDG method on two more realistic flow problems wherethe boundary of the computational domain has non-convex corners. We know by the theory of linearelliptic problems, [39], that the solution has less regularity (u 2 W lD1C"<2;2) around the cornerpoints and the numerical method may lose its optimal accuracy. A numerical treatment in order torecover the accuracy of the method is the use of locally graded mesh refinement technique. Thistechnique has been extensively studied for FEMs for linear elliptic problems, see, for example,[40–42]. The refinement is determined by studying the singular behavior of the solution aroundthe corners; and the FEM, applied on the resulting graded refined mesh, has optimal convergencerate properties.

Next, for the solution of the first problem, we just use a quite fine uniform mesh and k D 2for the local polynomial space. For the numerical solution of the second problem, we construct agraded mesh.

Problem 1. The computational domain � D �1 � �2 where �1 D Œ�2; 2� � Œ�2; 2�, �2 DŒ�1; 1�� Œ�1; 1�, with its triangulation is presented in Figure 4(a). The corners of the interior square�2 D Œ�1; 1� � Œ�1; 1� constitute the severe singularities of the computational domain and makethis problem challenging. Our particular interest here is to study if there is any effect of the singularboundary points to the symmetric structure of the solution. This means that the flow field producedby the LDG method in the upper channel must be the same as this of the lower channel.

On the boundary of the interior square as on boundary parts ¹�2 6 x 6 2; y D ˙2º,¹x D �2;�2 6 y 6 �1º; ¹x D �2; 1 6 y 6 2º (denoted in Figure 4(a) by �0),Dirichlet boundary conditions .u.x; y; t/; v.x; y; t// D .0; 0/ are imposed. On the part�ud D ¹x D �2;�1 6 y 6 1º, we set .u.x; y; t/; v.x; y; t// D .1; 0/, and on the Neumann part�N D ¹x D 2;�2 6 y 6 2º, we set aN D 0, see Equation (1d). The problem has been solved up tothe final time T D 50 using the numerical flux (11b). The values of the parameters are as in the firstline of Table I. In Figure 4(b,c,d), we plot the contour lines of the uh LDG solution for p D 2:5,p D 2, and p D 1:5, respectively. The contour lines show the symmetric flow field as we expected.Next, we examine the profiles of uh on the upper and lower channels. In Figure 4(e,f,g), we plotthe profiles of uh computed on the points of the upper line, Lup D ¹x D 0; 1 6 yupi 6 2, whereyupi D y

upi�1 C h; i D 1; : : :º and on the points of the lower line Llw D ¹x D 0;�2 6 ylwi 6 �1,

where ylwi D ylwi�1 C h; i D 1; : : :º. Note that in this graph, for plotting reasons, we have setylwi WD �y

lwi . The two profiles coincide for the three p test cases as expected for a symmetric flow

field. In Figure 4(h), we plot the uh profiles computed on points of the Lup line. All profiles areparabolic, depending on p, with the maximum value on the center point M D .0; 1:5/ of the axialdirection. In comparison with the linear case p D 2, the profile becomes flatter with increasing thediffusivity (p D 1:5) and conversely becomes sharper reducing the diffusivity (p D 2:5), similarwith the results that have been found for p-Navier–Stokes systems in [22].

Problem 2. We consider problem (1) in the computational domain � of Figure 5(a) with f D 0,where the boundary conditions are as follows: on �D;1, we set periodic Dirichlet conditions .u; v/ D.cos.2t/ C 2; 0/; on �D;0 Dirichlet conditions, .u; v/ D .0; 0/; and on �N Newman conditions,aN D 0 (see (1d)).

The domain has three singular corners Cs; s D 1; 2; 3. The initial mesh Th is graded in thefollowing way. For every Cs , we consider a region RC;s D ¹x 2 � W jx � Csj 6 0:5º. In everyRC;s , we construct N ring-type layers Li D ¹x 2 � W ri�1 < jx � Csj < ri ; i D 1; : : : ; N º

with ri D 0:5. iN/1� , where the parameter 2 .0; 1� is controlling the grading and we use 1

Nto be of order h. The layers Li are further partitioned to (approximately) equal side triangles ofhE2Li � ri � ri�1, see Figure 5(a) for an illustration of layer Li and the resulting corner mesh for



Figure 4. Corner singularity problem. (a) The computational domain with the uniform triangulation; (b) uhfield for p D 2:5; (c) uh field for p D 2; (d) uh field for p D 1:5. (e) Point value comparison of uh onLup; Llw lines for p D 2:5; (f) point value comparison of uh on Lup; Llw lines for p D 2; (g) point valuecomparison of uh on Lup; Llw lines for p D 1:5 (h) comparison of uh profiles computed on Lup for the

three values of p.

N D 3; D 0:4. Such refinements (with the same grading parameters) were applied in [43] for thenumerical solution of steady linear elliptic problems (p D 2).

The problem has been solved up to the final time T D 10 using the numerical flux (11b)and k D 1 for the local polynomial space. For making comparisons, the problem has been solvednumerically on a sequence of Thi ; i D 0; : : : ; 3 non-graded meshes, too. Because there is no knownexact solution, we check the numerical results using a reference solution, Ur , which has beenobtained on a fine graded mesh and k D 3 for local polynomial space. The uh fields computedon the graded ThiD1 mesh at the final time step are presented in Figure 5(b,c,d). All the fields aresymmetric with slow diffusion phenomena for p D 1:5 and more intense for p D 2:5. The diffu-sivity function for this test case is of the form QA.ru.x; t// D .ı C jru.x; t/j/p�2, one can expectthat the uh.:; t/ will strongly vary with respect to t . For every time step tn, we compute the valuesof uh.P; tn/, where the mesh point P D .0:368; 0/ is located in the vicinity of the middle corner.In Figure 5(e,f,g), the values of uh and Ur for every p test case are plotted versus the time. Good



Figure 5. Time dependent corner singularity problem: (a) The computational domain with the graded meshregions near the corner singularities; (b) uh flow field for p D 1:5; (c) uh flow field for p D 2; (d) uhflow field for p D 2:5; (e) comparisons between uh.P; tn/ and Ur .P; tn/ point values for p D 1:5; (f)comparisons between uh.P; tn/ and Ur .P; tn/ point values for p D 2; (g) comparisons between uh.P; tn/and Ur .P; tn/ point values for p D 2:5; (h) experimental convergence rates using graded meshes; and (i)

experimental convergence rates using uniform meshes.

agreement between the two solutions can be seen, both values uh and Ur have the same periodicevolution. The amplitude of the obtained p D 2:5 solution is higher compared with the amplitudeof p D 1:5 solution.

In Figure 5(h), we display the convergence rates of the error for the graded ( D 0:4) meshesand in Figure 5(i) the rates for uniform meshes ( D 1). The experimental results show that forthe graded mesh, the numerical solutions can approximate quite well the singular behavior of thesolution, because the rates r of every p case approach the optimal convergence rate r D 1, (seethe numerical examples in paragraph 6.1). On the other hand, the convergence rates measured onnon-graded meshes are modulated by the poor regularity behavior of the solution. Here, we have tomention that the convergence rates of p D 2 test presented in Figure 5(i) are slightly higher thanthe rates found in [43].



6. CONCLUSIONS

We have presented an LDG scheme for discretizing .p; ı/-structure systems. The proposed schemeutilizes two different nonlinear jump terms in the viscous numerical flux, which exhibits the .p; ı/-structure of the diffusion term. We have proven an a priori bound for a simplified scalar problem. Forthe time integration, diagonal implicit Runge–Kutta methods have been utilized. We have consideredNewton and Picard iterative processes for solving the resulting nonlinear algebraic systems. Thedeveloping of further local interior Jacobi-type iterations accelerated the overall performance of thePicard method. The main advantage of the last method is that it uses a better solution to updatethe nonlinear entries of the main iterative matrix. The performance of the iterative processes wascompared on several test cases. The Picard methods are less affected by the change of the parametervalues than the Newton method. We have discussed in detail through the numerical examples theconvergence rates of the proposed LDG method and the rates were found to be optimal with respectto the regularity of the exact solution. More realistic problems have been considered in domainswith non-smooth boundary in order to investigate the performance of the method. The problemswere solved on graded meshes and the experimental convergence rates for all p cases, found to besimilar with the rates of the FEMs applied on similar linear problems.

ACKNOWLEDGEMENTS

The first two authors were supported by the German Research Foundation under the project SFB TR 71Geometric Partial Differential Equations, C2 Fluid structure interaction. The third author was partiallysupported by the German Research Foundation under the project SFB TR 71 Geometric Partial Differen-tial Equations, C2 Fluid structure interaction and by the Austrian Academic Foundation under the projectFWF-NFN S11703-N23.

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