Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999

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Comput. Methods Appl. Mech. Engrg.

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Bubble stabilization of linear finite element methods for nonlinearevolutionary convection–diffusion equations

Javier de Frutos a,1, Julia Novo b,*,2

a Departamento de Matemática Aplicada, Universidad de Valladolid, Spainb Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

a r t i c l e i n f o

Article history:Received 19 October 2007Received in revised form 18 March 2008Accepted 22 March 2008Available online 11 April 2008

Keywords:Nonlinear evolutionary convection–diffusion equationsStabilizationBubble functions

0045-7825/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.cma.2008.03.028

* Corresponding author. Tel.: +34 914977635; fax:E-mail address: julia.novo@uam.es (J. Novo).

1 Research supported by Spanish MEC, Grant MTMCastilla y León, Grant VA079A06 (cofinanced by FEDER

2 Research supported by Spanish MEC, Grant MTMCastilla y León, Grants VA045A06 and VA079A06 (cofin

a b s t r a c t

A spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionaryconvection–diffusion equations is discussed. The method of lines with SUPG discretization in space leadsto numerical schemes that are not only difficult to implement, when considering nonlinear evolutionaryequations, but also do not produce satisfactory results. The method we propose can be seen as an alter-native to this kind of methods. Once the numerical approximation belonging to a linear finite elementspace enriched with bubble functions is obtained, the bubble part is discarded. The linear part is shownto give a stabilized approximation to the solution being approached. The bubble functions are deducedusing a linear steady convection–diffusion model in such a way that the linear part of the approximationto the linear steady model (after static condensation of bubbles) reproduces the SUPG method in the con-vection-dominated regime. However, for the nonlinear evolutionary equations we consider in the paperthe method we propose is not equivalent to the SUPG method. Some numerical experiments are providedin the paper to show the efficiency of the procedure.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

It is well known that standard finite element methods usinglow order piecewise polynomials perform poorly for advection–diffusion problems in the case in which the advective term ismuch stronger than the diffusive one. Spurious oscillations maycontaminate the approximations in the advection-dominated re-gime corresponding to high Reynolds numbers. Stabilized finiteelement methods have been introduced in order to avoid oscilla-tions and to get improved, physically sound, numerical approxi-mations, see for example [27,26] and the references therein. Oneof the most popular methods is the SUPG method introduced byHughes and Brooks [25,9]. Several authors have established a rela-tionship between Galerkin methods with bubble functions andstabilized finite element methods for linear stationary advec-tion–diffusion equations. Essentially, it has been shown in the lit-erature that the linear part of the finite element plus bubbleapproximation satisfies the equations of a stabilized method fora particular choice of the stability parameter emanating from the

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specific definition of the bubble function used. In [4], the Galerkinmethod employing piecewise linear elements with bubble func-tions is shown to be equivalent to the SUPG method in the socalled diffusive limit. The advection-dominated regime is alsostudied in [4]. In this case, the authors suggest to redefine the bub-ble functions in order to reproduce SUPG. The results of [4] are ex-tended to the linearized incompressible Navier–Stokes equationsin [15,28].

The main problem with the established connections betweenclassical stabilized methods and bubbles and multiscale analysisin the linear steady case is that they apparently do not bring muchmore insight into the methods. The selection of the optimal stabi-lized parameter is translated into the problem of the selection ofthe optimal bubble space. This problem is solved by the residual-free bubble approach where the stabilizing mechanism is con-tained in the enrichment of the space. In the residual-free bubblemethod the optimal parameter is determined through the solutionof a suitable boundary value problem in each element. In thissense, the residual-free bubbles are optimal. However, in order todevelop a practical algorithm the infinite dimensional space ofbubbles must be approximate with a finite number of degrees offreedom, see [7,18,5,6,30,11,31,29]. Many other references addressthe question of how to enrich the finite element space with localfunctions in order to improve the resolution in the context of stea-dy convection-dominated advection–diffusion problems, see forexample [19,33,31,17,11,16,29].

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999 3989

In the case of evolutionary convection–diffusion equations theliterature concerning stabilization techniques to treat the convec-tion-dominated regime is rather scarce, specially when consideringnonlinear equations. Linear parabolic initial boundary value prob-lems in one space dimension are studied in detail in [27] (see alsothe references therein). We refer to [24] where the semi-discreteSUPG formulation is extended to the fully discrete space–time for-mulation using a discontinuous Galerkin method in time to treatlinear symmetric multidimensional advective–diffusive systems.In [20,21], the authors study the stability and accuracy of severalstabilized nonstationary finite element methods applied to a linearone dimensional advection–reaction–diffusion equation while in[12,22,23] the linear multidimensional case is considered. Somenumerical schemes are considered in [2] for the unsteady advec-tion–diffusion–reaction linear problem in one space dimension.The authors investigate two possible different ways of discretizingfirst in space and then in time or first in time and then in space. Thestability of the SUPG method applied to linear evolutionary con-vection–diffusion equations is studied in [8] where implicit timeintegration is coupled with SUPG discretization in space. In [32],a fully discrete scheme to solve the incompressible Navier–Stokesequations is proposed. Semi-implicit time integrators of first andsecond order in time are combined with a linear finite elementmethod with scaled bubble functions in space. Both, the bubblesproposed in [32] and the bubbles proposed in this paper, whilebeing different, reproduce the SUPG method in the advection-dom-inated regime in a linear steady model. In the so-called scaled bub-bles of [32] the height of the bubble increases with the Pecletnumber as the support of the bubble dismisses while in the bub-bles proposed in this paper the height of the bubbles is fixed equal1. However, the implementation of the two methods in a nonlinearevolutionary problem is completely different. In the approach pro-posed in [32] many of the terms appearing in the Galerkin formu-lation of the method using the bubble enriched space aredisregarded so that the final formulation, and also the performanceof the method, is very similar to the SUPG method. Finally, we referto [13,14] (see also the references therein) where the authors pres-ent a finite element model to solve incompressible Navier–Stokesequations based on the stabilization with orthogonal subscales.

Although it is generally agreed that time–space elements arethe most natural setting to develop stabilized methods for evolu-tionary equations effective algorithms for treating time-dependentproblems can be defined separating temporal and spatial discreti-zations. Moreover, the increased cost in the number of unknownsfor coupled time–space formulations is a significant drawback. Inthe sequel, we will consider methods in which spatial and tempo-ral discretization are separated. As we will show in the numericalexperiments of Section 3 the SUPG method does not produce com-pletely satisfactory results in the linear evolutionary case bothwhen discretizing first in space or in time, see also [2,8]. In thenonlinear case, we have studied the application of the SUPG meth-od using the method of lines (discretizing first in space and then intime). The main difficulty in the numerical implementation of themethod is the treatment of the time derivative appearing in theadded term that provides an additional diffusion in the streamlinedirection, see Section 3. In the approach followed in [8] for the lin-ear evolutionary case the mass matrix is modified in order to incor-porate the time derivative coming from the residual. However, inthe nonlinear case this procedure gives rise to a time-dependentmass-matrix of the form M þ LðuhÞ where M is the standardmass-matrix of the finite element space and LðuhÞ is a matrix thatdepends on the approximation being computed. This modifiedmass matrix not only has to be recomputed at every time stepbut also could be a source of numerical instabilities.

In this paper, we study a stabilization technique via bubblefunctions to be applied to nonlinear evolutionary convection–dif-

fusion equations. The method we propose can be seen as an alter-native to classical stabilized methods to treat nonlinearevolutionary equations. For the construction of the bubble func-tions we follow a completely standard approach imposing thatthe linear part of the approximation reproduces the SUPG method,in the convection-dominated regime, for steady linear equations.However, when applied to nonlinear evolutionary convection–dif-fusion equations the method proposed in this paper is not equiva-lent to a SUPG method. The method of lines we use to solve theevolutionary equations has the advantage of decoupling the proce-dures of spatial and temporal discretization allowing the use ofstate-of-the-art numerical integrators for the system of ordinarydifferential equations resulting from the spatial discretization. Inthe numerical experiments of this paper we have used the linearlyimplicit Runge–Kutta method of order 4 introduced in [10] to inte-grate in time. For the spatial discretization the linear finite elementspace is enriched with bubble functions that are eliminated afterthe numerical integration has been performed. That is, if a stabi-lized approximation is required at a given time t� a Galerkinapproximation based on an enhanced space is computed until timet�. Then, the linear part of this approximation gives the desired sta-bilized solution. Of course the linear part of the computed Galerkinapproximation gives a stabilized approximation at any intermedi-ate time 0 < t < t�. As we explained before, the bubbles, that areeasily computable, are deduced from a steady convection–diffu-sion model in such a way that the linear part of the approximationreproduces the SUPG method in the convection-dominated regime.The numerical experiments show that the proposed proceduregives a stabilized approximation that avoids the Galerkin oscilla-tions appearing in the plain linear finite element method. Althoughfor simplicity most of the numerical experiments in the paper arecarried out using Burgers’ equation the method we propose canequally be applied to more complicated models in two or threedimensions and even extended to Navier–Stokes equations. Anumerical example in 2-D is shown at the end of the paper con-firming the good behavior of the method. The practical implemen-tation of the method proposed in this paper and theimplementation of the so-called mini-element [1] only differs inan up-date of the mass matrix every time step due to the changein the support of the bubbles as the time integration proceeds.The performance of the method applied to Navier–Stokes equa-tions will be subject of future research. The outline of the paperis as follows. In Section 2, we state some preliminaries and nota-tions. In Section 3, we study the application of the SUPG methodto nonsteady convection–diffusion equations. The bubble methodis introduced in Section 4. Finally, in Section 5 some numericalexperiments in 1 and 2 spatial dimensions show the performanceof the new method.

2. Preliminaries and notation

We will consider the numerical approximation of nonlinearconvection–diffusion equations that can be written in the form

utðt; xÞ � mDuðt; xÞ þ Rðuðt; xÞÞ ¼ f ðt; xÞ; x 2 X; 0 6 t 6 T;

uð0; xÞ ¼ u0ðxÞ; x 2 X;

uðt; xÞ ¼ 0; x 2 oX;

ð1Þ

where X � Rn, n ¼ 2;3, is a domain with smooth boundary, m is apositive constant, and R is a nonlinear convection term,RðuÞ ¼ ðu � rÞu. In the one dimensional case we will consider Bur-gers’ equation with nonlinearity RðuÞ ¼ uux in the domainX ¼ ða; bÞ � R. For 1 6 q 61 and l P 0, we consider the standardSobolev spaces, Wl;qðXÞn, of functions with derivatives up to orderl in LqðXÞn, and HlðXÞn ¼Wl;2ðXÞn. We will denote by ð�; �Þ the innerproduct in L2ðXÞn and by k � k0 the associated norm. The norm in

3990 J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999

the Sobolev space H1ðXÞn will be denoted by k � k1. It is well knownthat for a function u 2 H1

0ðXÞn the norm kuk1 is equivalent to

ðru;ruÞ1=2. In the sequel we will use the notation k � k1 to denoteany of the two equivalent norms. Finally, H1

0ðXÞn will be the closure

of the set of infinitely differentiable functions with compact supportrespect to the H1ðXÞn-norm. The standard weak form of problem (1)consists in finding u : ð0; T� ! H1

0ðXÞn with uð0Þ ¼ u0, such that

ðut ;uÞ þ aðu;uÞ þ ðRðuÞ;uÞ ¼ ðf ;uÞ 8u 2 H10ðXÞ

n; ð2Þ

where aðu; vÞ denotes the Dirichlet bilinear form induced byA ¼ �mD

aðu; vÞ ¼ ðA1=2u;A1=2vÞ ¼ mðru;rvÞ u; v 2 H10ðXÞ

n:

Let us assume that the solution u of (1) belongs to H10ðXÞ

n \ HsðXÞn,s P 2 and assume for simplicity that X is a convex polygon and Th

is a regular, quasiuniform partition of X. We consider the linear fi-nite element method. Let us denote by

VhðXÞ ¼ fvh 2 ðC0ðXÞÞn j vhjK 2 ðP1ðKÞÞn 8K 2Th; vh ¼ 0 in oXg;

where ðP1ðKÞÞn is the space of linear polynomials over K. We shalldenote by Ah the positive self-adjoint operator defined by

aðvh;whÞ ¼ ðAhvh;whÞ 8vh;wh 2 Vh;

and by Ph the standard L2ðXÞn orthogonal projection. We denote byuh : ð0; T� ! VhðXÞ the semi-discrete Galerkin approximation satis-fying uhð0; �Þ ¼ u0

h 2 VhðXÞ and

ððuhÞt ;uhÞ þ aðuh;uhÞ þ ðRðuhÞ;uhÞ ¼ ðf ;uhÞ; 8uh 2 VhðXÞ: ð3Þ

It is well known that under the conditions assumed above the semi-discrete Galerkin approximation satisfies the optimal a priori errorbound

sup06t6T

kuðtÞ � uhðtÞkl 6 Ch2�l; l ¼ 0;1: ð4Þ

However, in the convection-dominated regime the Galerkin approx-imation uh performs poorly and spurious oscillations appear formoderate values of the mesh diameter h. Although a well resolvedapproximation can be reached for h small enough the tiny size of hrequired for high Reynolds numbers strongly recommends the useof stabilization techniques. In Section 4 we introduce a numericalscheme that can be used to get stabilized approximations to thesolution of (1) in the advection-dominated regime.

3. The SUPG method for nonsteady convection–diffusionequations

3.1. Linear case

Let us consider the following linear one dimensional transientconvection–diffusion equation

ut � muxx � bux ¼ f ; uð0; xÞ ¼ u0ðxÞ; x 2 ð0;1Þ;uðt; 0Þ ¼ uðt;1Þ ¼ 0; ð5Þ

where m and b are positive constants. For simplicity we will take inthe sequel f ¼ 0. Following [2], we consider two possible ap-proaches using the SUPG method: discretize first in space and thenin time or first in time and then in space. We use linear finite ele-ments in space. To show the differences between the two proce-dures we use in both cases the implicit Euler method to integratein time. Similar conclusions can be reached using other time inte-grators. Let us denote by uST

h the SUPG approximation in Vhð0;1Þ ob-tained discretizing first in space, i.e., using the method of lines. Tosimplify the exposition we consider a uniform partition of ð0;1Þ ofsize h. We will use the notation uST;n

h ðxÞ ¼ uSTh ðtn; xÞ, tn ¼ nk,

n ¼ 0;1; . . ., k being the time step. It is easy to show that uSTh satisfies

the following equations:

uST;nþ1h � uST;n

h

k;uh

!þ mððuST;nþ1

h Þx; ðuhÞxÞ � bððuST;nþ1h Þx;uhÞ

¼XK2sh

d �uST;nþ1h � uST;n

h

kþ bðuST;nþ1

h Þx;�bðuhÞx

!K

8uh 2 Vh: ð6Þ

We are going to study the advection-dominated regime and, in con-sequence, we take the stabilization parameter d ¼ h=2b. The aboveequations can also be written as follows:

uST;nþ1h � uST;n

h

k;uh

!þ mþ hb

2

� �ððuST;nþ1

h Þx; ðuhÞxÞ

� bððuST;nþ1h Þx;uhÞ

¼XK2sh

duST;nþ1

h � uST;nh

k; bðuhÞx

!K

8uh 2 Vh; ð7Þ

which shows that the SUPG method, apart from adding the term onthe right hand side involving an approximation to the time deriva-tive, is also adding, as in the steady case, some numerical diffusionof size h. In terms of finite differences the added diffusion can alsobe interpreted as follows: we are using central differences to dis-cretize the second derivative �muxx and upwind for the first orderterm �bux instead of central differences for the two terms as inthe classical Galerkin method based on linear elements. Let us de-note by uST;n

h;j ¼ uST;nh ðjhÞ, then, Eq. (7) gives the recurrence

1k

46

uST;nþ1h;j þ 1

6uST;nþ1

h;j�1 þ16

uST;nþ1h;jþ1 �

46

uST;nh;j �

16

uST;nh;j�1 �

16

uST;nh;jþ1

� �

¼ m

h2 ðuST;nþ1h;jþ1 � 2uST;nþ1

h;j þ uST;nþ1h;j�1 Þ þ

bhðuST;nþ1

h;jþ1 � uST;nþ1h;j Þ

þ 14kððuST;nþ1

h;j�1 � uST;nþ1h;jþ1 Þ � ðu

ST;nh;j�1 � uST;n

h;jþ1ÞÞ:

If we denote by r ¼ k=h2 and l ¼ k=h we can write the above equa-tions as follows:

812þ 2rmþ bl

� �uST;nþ1

h;j þ � 112� mr

� �uST;nþ1

h;j�1

þ 512� mr� bl

� �uST;nþ1

h;jþ1

¼ 812

uST;nh;j �

112

uST;nh;j�1 þ

512

uST;nh;jþ1:

Then, for N ¼ 1=h and denoting by �ul ¼ ½uST;lh;1 ; . . . ;uST;l

h;N�1�t, l ¼ 0;1; . . .

we have to solve at each time step a linear system of the form

A�unþ1 ¼ B�un; ð8Þ

where A and B are tridiagonal matrices. One can not prove thatkA�1Bk1 6 1. Indeed, in the numerical experiments we have carriedout this norm is greater than one. As we will see in the experiments,a bound of the form

juST;nh;j j 6 K max

06i6NjuST;0

h;i j 8j; n;

with K ¼ 1 is not satisfied and the numerical approximation pro-duces some spurious oscillations.

Let us now denote by uTSh the SUPG approximation in Vhð0;1Þ

obtained discretizing first in time and then in space. As before,we consider a uniform partition of ð0;1Þ of size h. Following [2]the method we obtain can be written as follows:

uTS;nþ1h � uST;n

h

k;uh

!þ mððuTS;nþ1

h Þx; ðuhÞxÞ � bððuTS;nþ1h Þx;uhÞ

¼XK2sh

~d �uTS;nþ1h � uTS;n

h

kþ bðuTS;nþ1

h Þx;�bðuhÞx

!K

8uh 2 Vh;

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

x

u (x

)

ν=1e−6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

x

ν=1e−6

u (x

)

Fig. 1. SUPG methods of first order in time on the left and second order in time on the right against exact solution, T ¼ 0:2.

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999 3991

where ~d ¼ ð2=kþ b=hÞ�1. As we can observe, the above equationsare the same as before, see (6), up to a change in the stabilizationparameter.

In fact if one uses the time step restriction k ¼ 2h=b then bothparameters are the same, i.e., ~d ¼ d. If one uses another time inte-grator such as the Crank–Nicolson or the BDF (backward differen-tiation formulae) of order 2, as in the numerical experiments weshow in this section, the only difference between discretizing firstin time and then in space or first in space and then in time is againa change in the stabilization parameter.

In Fig. 1, on the left, we have represented the SUPG approxima-tions based on linear finite elements in space and the implicit Eulermethod in time discretizing first in space and then in time (inblue3) and first in time and then in space (in green) against the exactsolution (in black). We solve Eq. (5) with m ¼ 1e� 6, b ¼ 1, f ¼ 0 andu0ðxÞ ¼ expð�100ðx� 0:5Þ2Þ. The final time is t ¼ 0:2, we takeh ¼ 1=100 and k ¼ h=5 for both methods. We can observe that bothmethods give very similar approximations, being slightly better theuTS

h approximation. However, although the oscillations present inboth approximations are very slight they are two much diffusive.On the right of Fig. 1 we have represented the SUPG approximationsbased on linear finite elements in space and the BDF of order 2 intime (in blue) and the Crank–Nicolson method (in green) discretizingfirst in space and then in time. The results discretizing first in timeand then in space are very similar. The values of all the parametersare the same as before. We can observe that although the use of asecond order method in time gives less diffusive approximations(both approximations are very similar) the oscillations increase.

In Fig. 2, we have repeated the same experiment but with finaltime t ¼ 0:5 in order to observe the behavior of the different meth-ods when the theoretical solution develops a steep gradient at theboundary x ¼ 0. On the left of Fig. 2, we observe that now there is adifference when we discretize first in space and then in time (blueline) or first in time and then in space (green line) using the impli-cit Euler method. Both approximations are too much diffusive butthe approximation obtained discretizing first in time producessome spurious oscillations near the boundary that can be avoidedusing the method of lines. On the other hand, for the second ordermethods, on the right of Fig. 2, we get the same behavior as inFig. 1. As we said in the introduction of the paper, we have notachieved completely satisfactory results with the SUPG methodin the numerical experiments of this section.

3 For interpretation of color in Figs. 1 and 2, the reader is referred to the webversion of this article.

3.2. Nonlinear case

In this section, we consider two stabilized methods to approxi-mate the solution u of (1) for the advection-dominated regimeusing the method of lines: i.e., discretizing first in space and thenin time. We first consider a simplified SUPG method. As a first ap-proach, in this method we do not include the contribution of thetime derivative in the residual. We find us1

h : ð0; T� ! VhðXÞ satisfy-ing us1

h ð0; �Þ ¼ u0h 2 VhðXÞ and for all uh 2 VhðXÞ,

ððus1h Þt ;uhÞ þ aðus1

h ;uhÞ þ ðRðus1h Þ;uhÞ

¼ ðf ;uhÞ þXK2sh

dKðf þ mDus1h � ðu

s1h � rÞu

s1h ; ðu

s1h � rÞuhÞK ; ð9Þ

where

dK ¼hk

2kus1h k0;h

; ifkus1

h k0;hhk

6mP 1:

We denote by kus1h k0;h a discrete L2-norm of the stabilized approxi-

mation us1h . Let us observe that for the scalar advection–diffusion

equation:

b � rU� mDU ¼ f ;

(b being a divergence free velocity field) the value of dK is hk2kbkK

in theadvection-dominated regime. In this case we take kus1

h k0;h instead ofkbkK . Since, in the definition of us1

h , we have not included the timederivative in the residual, the solution u of (1) does not satisfy theweak formulation (9), i.e., this method is not strongly consistent(see [26, Chapter 8]). This fact can be a drawback if one wants to ex-tend the method to other than linear elements since the methodcannot be better than first order accurate. This method can be seenas an artificial diffusion method in which the added term is nonlin-ear: in the one dimensional case the added term isP

K2shdKððus1

h Þ2ðus1

h Þx; ðuhÞxÞK .We next consider a SUPG method in which we take into account

all the contributions in the residual, including the time derivative.The approximation reads as follows: find us2

h : ð0; T� ! VhðXÞ satis-fying us2

h ð0; �Þ ¼ u0h 2 VhðXÞ and for all uh 2 VhðXÞ

ððus2h Þt ;uhÞ þ aðus2

h ;uhÞ þ ðRðus2h Þ;uhÞ � ðf ;uhÞ

¼X

K2shdKðf � ðus2

h Þt þ mDus2h � ðu

s2h � rÞu

s2h ; ðu

s2h � rÞuhÞK ; ð10Þ

where the value of the parameter dK is the same as before. Unlikethe method (9), the solution of (1) satisfies the weak formulation(10) and then the method is strongly consistent and can equallywell be applied to other than linear elements without loosing accu-racy. The main difficulty in the numerical implementation of thismethod is the treatment of the term

PK2sh

dKð�ðus2h Þt; ðu

s2h � rÞuhÞK .

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

0.2

0.4

0.6

0.8

1

x

u (x

)

ν=1e−6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

u (x

)

ν=1e−6

Fig. 2. SUPG methods of first order in time on the left and second order in time on the right against exact solution, T ¼ 0:5.

3992 J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999

We have followed the following approach that considerably simpli-fies the implementation: we have replaced the time derivative ðus2

h Þtin this term by the Galerkin time derivative:

�Ahus2h � PhRðus2

h Þ þ Phf :

To finish this section, we show some numerical experiments in or-der to study the performance of these methods. We consider theone dimensional Burgers’ equation

ut � muxx þ uux ¼ 0; 0 < x < 1;

subject to homogeneous Dirichlet boundary conditions. We take asinitial condition uðx; 0Þ ¼ sinð2pxÞ. A linearly implicit Runge–Kuttamethod of order 4 implemented in the variable step mode is usedto integrate in time. This Runge–Kutta method has shown in [10]to be well designed for the numerical integration of the semi-dis-crete equations arising after the spatial discretization of advec-tion–reaction–diffusion equations. Since we are studying spatialdiscretization errors a small enough tolerance is chosen for the timeintegrator in all the computed approximations in order to have neg-ligible temporal errors. To get the ‘‘exact” solution we compute theGalerkin linear finite element approximation over a sufficientlysmall spatial mesh size and with a sufficiently small value of the tol-erance in the time integrator. For the numerical experiments ofFig. 3 we have taken m ¼ 5e� 4 and we have represented theapproximations obtained at time t ¼ 1. The black line correspondsto the exact solution while the blue, red and green lines are usedfor the SUPG approximations over a uniform partition of size

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

u (x

)

ν=0.0005

Fig. 3. SUPG methods against the exa

h ¼ 1=20, 1/40 and 1/80, respectively. The approximations on theleft corresponds to the first approach us1

h and those on the right tothe second one us2

h . The Galerkin approximations for the same val-ues of h can be seen in Fig. 7 where they are compared with thenew bubble method we propose. We can observe in Fig. 3 thatthe first approximation annihilates the spurious oscillations of theGalerkin method although the behavior of the exact solution isnot very well reproduced. On the other hand, the second stabilizedapproximation on the right, reproduces the behavior of the exactsolution better than the previous one but presents some spuriousoscillations that should not appear in a well suited stabilized meth-od. The presence of the time derivative in the residual of the meth-od seems to generate spurious oscillations that were alreadyappreciated in the linear case shown before, see Figs. 1 and 2. Innext section, we propose an alternative method that solves thedrawbacks of these two methods to approximate equation (1) inthe convection-dominated regime. We want to remark that othertreatments of the time derivative in the residual could also be con-sidered opening the possibility to design methods working muchbetter. We refer the reader to [8] where the performance of theSUPG method in space and a h-method in time is studied for a tran-sient linear convection–diffusion equation. In this work the massmatrix is modified in order to incorporate the time derivative com-ing from the residual. However, the numerical approximationsshown in the numerical experiments of [8] can not completelyavoid spurious oscillations, in agreement with the experiments ofFigs. 1 and 2. The examples we have included here only illustrate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

u (x

)

ν=0.0005

ct solution for Burgers’ equation.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

K

Ki

00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.5

1

Fig. 5. Bubble function in the two dimensional case.

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999 3993

the difficulties inherent to the SUPG method when applied to non-linear evolutionary equations. In next section we propose an alter-native procedure to get stabilized approximations.

4. Bubble stabilization method

Let us denote by

Xh ¼ VhðXÞ � BhðXÞ

where BhðXÞ is a bubble space of functions in H10ðXÞ

n such that anyof the components of the functions belong to the space:

XK2Th

aK bK ; aK 2 R; bKðxÞ ¼ ðnþ 1Þnþ1Ynþ1

j¼1

kjðxÞ 2 H10ðKiÞ; Ki � K

( ):

Here kjðxÞ denote the barycentric coordinates of each x 2 Ki. In thedefinition of the bubble space Ki is a subset of K that will be definedlater. We will define the bubbles in such a way that the size of Ki

decreases as the Peclet number increases. On the other side, inthe diffusive limit, Ki ¼ K and BhðXÞ is the usual bubble space usedfor example in the so called mini element (see for example [1]). Twotypical functions of the above defined space are shown in Figs. 4 and5 in the one dimensional and two dimensional cases, respectively.For uh 2 Xh, we will use the notation

uh ¼ ulh þ ub

h; ulh 2 VhðXÞ; ub

h 2 BhðXÞ:

Let us observe that the above decomposition is orthogonal with re-spect to the H1

0ðXÞn-inner product. The method we propose reads as

follows. If a stabilized approximation is required at a given time t�

we first compute usbh : ð0; t�� ! Xh satisfying usb

h ð0; �Þ ¼ u0h 2 Xh and

ððusbh Þt ; vhÞ þ aðusb

h ; vhÞ þ ðRðusbh Þ; vhÞ ¼ ðf ; vhÞ 8vh 2 Xh: ð11Þ

Then, usb ;lh ðt

�Þ, the linear part of usbh ðt

�Þ ¼ usb ;lh ðt

�Þ þ usb ;bh ðt

�Þ, is a stabi-lized approximation to the solution u of (1) at time t�. Indeed, thelinear part of the computed approximation at any intermediatetime 0 < t < t�, usb ;lðtÞ, gives also a stabilized approximation to thesolution u of (1) at time t. Next, we show the way in which we de-duce the size of the support of the bubble function, Ki, in order toget a stabilized approximation in the advection-dominated regime.For simplicity we first consider the one dimensional case, thededuction of the size of Ki in the two or three dimensional casesis analogous. As we can see in Figs. 4 and 5 the element Ki is cen-tered in the original element K and both elements share the samebarycenter. One only need to know the size (longitude, area or vol-ume) of Ki in order to define the bubble function that takes the va-

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

xi x

i+1K

i

Fig. 4. Bubble function in the one dimensional case.

lue 1 at the barycenter and vanishes at the boundary of Ki. Let usconsider the linear one dimensional convection–diffusion equation

�muxx þ aux ¼ f ; ð12Þ

in the interval (0,1) subject to homogeneous Dirichlet boundaryconditions. We assume that a and f are piecewise constant. Theapproximation obtained using the SUPG method is the following.Find us

h 2 Vhð0;1Þ such that

mððushÞx; ðuhÞxÞ þ ðaðus

hÞx;uhÞ þXK2sh

dKðaðushÞx; aðuhÞxÞK

¼ ðf ;uhÞ þXK2sh

dKðf ; aðuhÞxÞK ; ð13Þ

where

dK ¼hK

2kakK; if PeK ¼

kakK hK

6mP 1;

dK ¼h2

K

12m; if PeK ¼

kakK hK

6m< 1;

hK being the maximum diameter of the element K 2 sh, kakK a normof the function a in K and PeK the Peclet number. On the other side,an approximation belonging to the enhanced space Xhð0;1Þ is ob-tained in the following way. Find usb

h ¼ usb ;lh þ usb ;b

h 2 Xhð0;1Þ ¼Vhð0;1Þ � Bhð0;1Þ satisfying

mððusb ;lh Þx þ ðu

sb ;bh Þx; ðu

lhÞx þ ðub

hÞxÞ þ ðaððusb ;lh Þx þ ðu

sb ;bh ÞxÞ;u

lh þ ub

hÞ¼ ðf ;ul

h þ ubhÞ 8uh ¼ ul

h þ ubh 2 Xhð0;1Þ: ð14Þ

Taking in (14) ulh ¼ 0 and ub

h ¼ bK and writing usb ;bh ¼

PK2sh

cK bK weget

mcKððbKÞx; ðbKÞxÞK þ ðaððusb ;lh Þx þ cKðbKÞxÞ; bKÞK ¼ ðf ; bKÞK ;

so that

mcKððbKÞx; ðbKÞxÞK þ ðaðusb ;lh Þx; bKÞK ¼ ðf ; bKÞK :

Then, the value of the coefficient cK is

cK ¼ðf ; bKÞK � ðaðu

sb ;lh Þx; bKÞK

mððbKÞx; ðbKÞxÞK: ð15Þ

Taking in (14) ulh 2 Vhð0;1Þ and ub

h ¼ 0, we get the equation thatsatisfies the linear part of usb

h

mððusb ;lh Þx; ðu

lhÞxÞ þ ðaðu

sb ;lh Þx;u

lhÞ þ

XK2sh

cKðaðbKÞx;ulhÞK ¼ ðf ;ul

hÞ: ð16Þ

3994 J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999

Integrating by parts and using (15) we deduce that the stabilizationterm in (16) can be written in the form:XK2sh

cKðaðbKÞx;ulhÞK ¼ �

XK2sh

cKðabK ; ðulhÞxÞK

¼XK2sh

ðaðusb ;lh Þx; bKÞK � ðf ; bKÞKmððbKÞx; ðbKÞxÞK

ðabK ; ðulhÞxÞK :

Since we are assuming that a and f are piecewise constant (and tak-ing into account that ðul

hÞx is piecewise constant too) we obtain

XK2sh

cKðaðbKÞx;ulhÞK ¼

ðaðusb ;lh Þx � f Þð1; bKÞK

mððbKÞx; ðbKÞxÞKaðul

hÞxð1; bKÞK

¼ ðaðusb ;lh Þx � f ; aðul

hÞxÞK h�1K

ð1; bKÞ2KmððbKÞx; ðbKÞxÞK

:

Then, in view of (16), the linear part usb ;lh of the finite element

approximation usbh is the solution of a SUPG method with a value

of the stabilization parameter dK defined as

dK ¼ h�1K

ð1; bKÞ2KmððbKÞx; ðbKÞxÞK

:

Let us now observe that for K ¼ ðxi; xiþ1Þ, hK ¼ xiþ1 � xi, takingbK ¼ 4ðx� xiÞðxiþ1 � xÞ=h2

K , i.e. taking the support of the bubbleequal to the whole interval K, we get

dK ¼ h�1K

16h2K=36

m16=3hK¼ h2

K

12m;

so that usb ;lh is the solution of the SUPG method in the diffusive limit,

see (13) and (4). This is exactly the conclusion reached in [4]. In theone dimensional case this approximation, usb

h , is just the quadraticfinite element approximation to (12) and then the linear part ofthe quadratic approximation behaves as a stabilized solution forsmall values of the Peclet number PeK . Our aim is now to showhow the bubble functions bK can be redefined in order to reproducethe value of the stabilization parameter dK in the convective limit.Let us take a bubble like that shown in Fig. 4 and let l be the lengthof the interval Ki � K . Then we obtain

dK ¼ h�1K

ð1; bKÞ2KmððbKÞx; ðbKÞxÞK

¼ h�1K

4l2

9m163l

¼ h�1K

l3

12m;

so that taking

l ¼ 6mh2K

kakK

!1=3

we get dK ¼ hK2kakK

, i.e., the value of the stabilization parameter in theconvective limit (see (4)). Let us observe that a modified bubble

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

t

rela

tive

leng

th o

f K

i

ν=0.0005

N=160

N=320

N=640

Fig. 6. Evolution in time of the relative size of the support of the b

function is used whenever l < hK . Moreover, let us note that thecondition l ¼ hK is equivalent to

PeK ¼kakK hK

6m¼ 1;

i.e. the value of the Peclet number that determines the change of theregime. Then, the change of the regime from diffusion-dominated toconvection-dominated coincides with the change of the support ofthe bubbles (that passes from the whole interval hK to a subintervalof hK ). Let us note that a stabilized method valid at all values of PeK

can be easily programmed by choosing bubble functions as those inFig. 4 with lengthðKiÞ ¼ ð6mh2

K=kakKÞ1=3 when PeK P 1 and Ki ¼ K

when PeK < 1. For the bidimensional case the situation is analogous.We choose a cubic bubble function as shown in Fig. 5. In the diffu-sive limit the support of the bubble is the whole triangle K while inthe convective limit the support is the triangle Ki contained in K. Weonly need to determine the size of Ki. Let us assume for simplicitythat a uniform partition is being considered so that K is a rectangletriangle with area h2

K=2 and let us denote by l the longitude of any ofthe two catheti of Ki. Reasoning as before, taking as a modelequation

�mDuþ a � ru ¼ f ;

and assuming a and f are piecewise constant we get

dK ¼ 2h�2K

ð1; bKÞ2KmðrbK ;rbKÞK

¼ 2h�2K

81l4

402

m8110

¼ l4

80mh2K

;

so that taking

l ¼ h3K 40mkakK

!1=4

we get dK ¼ hK2kakK

, i.e., the value of the stabilization parameter in theconvective limit as before. Now, we get l ¼ hK for

kakK hK

40m¼ 1;

which does not coincide exactly with the change of regime as be-fore, i.e. with PeK ¼ 1, but it is not far away from it. Also in this casea method valid for both regimes can be programmed. Our recom-mendation would be to take l ¼ ðh

3K 40mkakKÞ1=4 whenever kakK hK

40m P 1 andl ¼ hK for kakK hK

40m < 1. Now we apply the bubble stabilization methodto the model problem (1). Since the convective term is u � ru, i.e.a ¼ u, from the above definition we observe that the size of the sup-port of the bubble depends on the solution u. In the method we pro-pose we use a norm of usb ;l

h , the linear part of the approximation,instead of kakK to determine the bubble function. During the timeintegration of the semi-discrete Eq. (11) the bubble space BhðXÞ is

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34ν=0.000005

rela

tive

leng

th o

f K

i

t

N=160

N=320

N=640

ubble functions. On the left m ¼ 5e� 4, on the right m ¼ 5e� 6.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

u (x

)

ν=0.0005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

u (x

)

ν=0.0005

Fig. 7. Exact solution in black for m ¼ 5e� 4. On the left: Galerkin approximations, on the right: bubble stabilized approximations for N ¼ 20, 40 and 80, t� ¼ 1.

4 For interpretation of color in Fig. 7, the reader is referred to the web version othis article.

Table 1Errors for the Galerkin method (left) and the bubble stabilization method (right) forBurgers’ equation; m ¼ 5e� 4, t ¼ 1

N L2ð0;1Þ H1ð0;1Þ L2ð0;1Þ H1ð0;1Þ

10 1.994 1.721 0.446 0.98920 1.048 1.463 0.312 0.96840 0.295 1.220 0.197 0.93180 0.133 1.087 0.116 0.852160 0.048 0.782 0.057 0.675320 0.011 0.379 0.015 0.366640 0.002 0.171 0.003 0.169

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999 3995

computed at each time step. The linear approximation usb ;lh that has

just been computed at a given step is used to determine the size ofthe bubble functions to be used in the next step. In Fig. 6, we canobserve the evolution in time of the relative size of the support ofthe bubble functions for Burgers’ equation (with forcing termf ¼ 0) and initial condition uðx;0Þ ¼ sinð2pxÞ for two different val-ues of the diffusive parameter m. On the left we have plotted the re-sults for m ¼ 5e� 4 and on the right for m ¼ 5e� 6 for a fixeduniform partition of the spatial interval (0,1) into N ¼ 160,N ¼ 320 and N ¼ 640 subintervals, respectively. We integrate intime from t ¼ 0 to t� ¼ 1. In this experiment all the bubble functionsused at a given partition have the same support. As can we can seein Fig. 6, in both cases, as expected, the relative size of the supportof the bubble increases as N increases. On the other size, the relativesize of the support dismisses as the diffusion parameter tends tozero. For m ¼ 5e� 4 and N ¼ 320 we can observe that a change ofthe regime from convection-dominated to diffusion-dominated oc-curs during the time integration. Approximately at time 0:4 the sup-port of the bubble changes to be the whole interval and continuesbeing the same till the final time. This situation is completely differ-ent from the linear steady case in which for a given m the value ofh ¼ 1=Ndictates the regime. In next section we show some numer-ical experiments to study the behavior of the bubble stabilizationmethod we propose.

5. Numerical experiments

To conclude, some numerical experiments using Burgers’ equa-tion are shown in the first subsection. In the second subsection weconsider a two dimensional problem to show that our methodmaintains its good properties in two spatial dimensions.

5.1. Numerical experiments in 1-D

As in the numerical experiments of Section 3 we consider theone dimensional Burgers’ equation subject to homogeneousDirichlet boundary conditions. We take the same initial conditionas before and use the same linearly implicit Runge–Kutta methodof order 4 implemented in the variable step mode to integrate intime. For the bubble stabilization method we consider a uniformpartition of [0,1] into subintervals of length h and enrich the lin-ear finite element space with bubble functions as those in Fig. 4taking

lengthðKiÞ ¼6mh2

2kusb ;lh k0;h

!1=3

; if2kusb ;l

h k0;hh6m

P 1

and Ki ¼ K ¼ ðxi; xiþ1Þ whenever2kusb ;l

hk0;hh

6m < 1. The discrete norm ofusb ;l

h , the linear part of usbh 2 Xhð0;1Þ, is defined as

kusb ;lh k0;h ¼

XN�1

i¼1

hðusb ;lh ðxiÞÞ2

!1=2

; xi ¼ihN:

Let us observe that we have slightly changed the definition of thesupport of the bubble and the change of the regime since we take2kusb ;l

h k0;l instead of kusb ;lh k0;l. The only reason is that we have checked

experimentally that this turns out to be a more convenient selec-tion. On the other side, we take kusb ;l

h k0;h, a discrete norm in the fullinterval (0,1) for all the elements, instead of, for example, kusb ;l

h k1;Kthe L1-norm in the element, since in our numerical experience thechoice of a different support for the bubbles at every element hasnot led to a better performance of the method while considerablyincreases the computational cost. In Fig. 7, we have plotted the ex-act solution for m ¼ 5e� 4 at time t� ¼ 1 using black line and theGalerkin (on the left) and bubble stabilized approximations (onthe right) for different values of N. The blue4 color is for N ¼ 20,red for N ¼ 40 and green for N ¼ 80. On the left we can see the spu-rious oscillations that produces the plain Galerkin method. Althoughthe oscillations dismiss as N increases the numerical approximationsof the picture do not give satisfactory results. The situation on theright is completely different. As we can observe, the bubble stabiliza-tion procedure completely annihilates oscillations. The solution isvery well approximated out of a region around the steep gradientat x ¼ 0:5 while the gradient at this point is increasingly betterreproduced as N increases. Comparing the plain Galerkin and thenew bubble approximations for N ¼ 80 we conclude that while theGalerkin one is still too oscillatory the bubble stabilized approxima-tion reproduces very well the behavior of the exact solution. On theother side, with only N ¼ 20 or N ¼ 40 degrees of freedom the new

f

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

x

u (x

)

ν=0.000005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

u (x

)

ν=0.000005

Fig. 8. Exact solution in black for m ¼ 5e� 6. On the left: Galerkin approximations, on the right: bubble stabilized approximations for N ¼ 160, 320 and 640.

Table 2Errors for the Galerkin method (left) and the bubble stabilization method (right) forBurgers’ equation; m ¼ 5e� 6, t ¼ 1

N L2ð0;1Þ H1ð0;1Þ L2ð0;1Þ H1ð0;1Þ

80 2.362 1.518 0.230 0.999160 1.582 1.543 0.147 0.998320 0.414 1.401 0.095 0.995640 0.189 1.378 0.062 0.9901280 0.094 1.357 0.040 0.9782560 0.046 1.301 0.026 0.9545120 0.022 1.188 0.016 0.90510240 0.009 0.972 0.009 0.803

3996 J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999

method gives pretty good approximations to the solution. In Table 1,we have represented the relative errors for the Galerkin and the bub-ble stabilized approximations in the L2ð0;1Þ and H1ð0;1Þ norms form ¼ 5e� 4 at time t� ¼ 1. The first two columns (after column 1)gathers the Galerkin errors while the last two correspond to the er-rors of the bubble stabilized method. We can observe that asymptot-ically (last three errors) both methods achieve quadratic converge inL2ð0;1Þ and linear convergence in H1ð0;1Þ. The errors of second andforth columns are asymptotically divided by 4 from one row to thenext one as the errors of the third and fifth columns are dividedby 2. It is easy to prove the quadratic (respectively, linear) conver-gence of the new method in theL2ð0;1Þ (respectively, H1ð0;1Þ) normin the diffusive limit using the techniques of [3]. However, we arenow interested in the convective limit. In this regime, as can be ob-served in the table, the bubble stabilization procedure achieves bet-ter errors in both norms, specially in L2. Let us remark that, forexample, for N ¼ 10 the L2 norm of the error of the plain Galerkinapproximation is four times larger than the error of the bubble sta-bilized approximation. As in Figs. 7 and 8, we have plotted the exactsolution at time t� ¼ 1 using black line and the Galerkin (on the left)and bubble stabilized approximations (on the right) for different val-ues of N. The results of this figure are for m ¼ 5e� 6. The blue5 coloris for N ¼ 160, red for N ¼ 320 and green for N ¼ 640. Similar com-ments to those stated for Fig. 7 can be applied in this case with a sub-stantial difference: while the situation for the Galerkin methodbecomes worse due to the spurious oscillations that lead to uselessapproximations the results for the bubble stabilized method consid-erably improve. In Table 2, as in Table 1, we have represented the er-rors for the two methods in the L2 and H1 norms for the value ofm ¼ 5e� 6. Let us observe that, in agreement with our earlier com-

5 For interpretation of color in Fig. 8, the reader is referred to the web version ofthis article.

ments about the numerical approximations represented in Fig. 8,the difference between the errors of the plain Galerkin methodand the bubble stabilized method increases. For example, forN ¼ 80 and N ¼ 160, the errors of the Galerkin method in the L2

norm are ten times larger than those of the stabilized method. More-over, for N ¼ 80 when passing from m ¼ 5e� 4 to m ¼ 5e� 6 theGalerkin error in L2 is increasing by a factor of 17:75 while the sta-bilized error only in 1:94 so that the method we propose seems tobe very robust respect to the changes in the diffusion coefficient.For all the values of N appearing in Table 2 the bubble methodachieves better errors than the Galerkin approximation in both theL2 and H1 norms. However, the reduction of the errors in the H1

norm is extremely slow due the steep gradient that presents the ex-act solution at x ¼ 0:5 as m decreases. On the left of Fig. 9, we haveplotted a zoom of the right picture of Fig. 8 to see the improvementin the approximation of the gradient at x ¼ 0:5 with the bubblemethod as N increases. Finally, the aim of the picture on the rightof Fig. 9 is to give a example of the efficiency of the method we pro-pose. In this figure, for m ¼ 5e� 6 we have represented the exactsolution in black and the bubble stabilized approximation withN ¼ 160 in blue as before. We have also plotted in the picture theplain Galerkin approximation with N ¼ 860 in red. The errors ofthe two approximations, bubble stabilized and Galerkin, in theL2ð0;1Þ norm are essentially the same. However, the Galerkin errorin H1ð0;1Þ is 1.381 while the energy norm of the error of the bubbleapproximation is 0.997. On the other hand, the computing time re-quired to obtain the Galerkin approximation is 3.5 times larger thanthe time needed to get the bubble approximation. We want to re-mark that we are studying spatial discretization errors but we donot want to unnecessary increase the computing time required toget the approximations. For this reason, both approximations arecomputed with a tolerance for the time integrator that is the biggestone between those given the same error so that for the chosen toler-ance the temporal error is negligible and then the remained error isonly due to the spatial discretization. Observing the oscillations thatare still present for the value N ¼ 860 in the Galerkin approximationwe conclude that the new method is by far more efficient than theplain Galerkin.

5.2. Numerical experiments in 2-D

We consider the equation

ut ¼ mDu� uux � uuy;

in the domain X ¼ ½�1;1� ½�1;1� subject to homogeneous Dirich-let boundary conditions and initial condition

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 10. ‘‘Exact” solution for m ¼ 0:001.

Table 3Relative errors for m ¼ 0:001

h Galerkin Bubble stabilized

L2 errors H1 errors L2 errors H1 errors

1/8 1.1433 1.5346 0.5594 0.96341/10 0.9789 1.4819 0.4410 0.95091/12 0.7933 1.3944 0.3873 0.9384

0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.520.2

0.25

0.3

0.35

0.4

0.45

x

u (x

)

ν=0.000005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

u (x

)

ν=0.000005

x

Fig. 9. Exact solution in black. On the left: detail of right of Fig. 8. On the right: Galerkin approximation for N ¼ 860 in red and bubble stabilized approximation for N ¼ 160 inblue. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999 3997

u0ðx; yÞ ¼ sinð2pxÞ sinð2pyÞ:

In our calculations we have taken the so-called regular pattern tri-angulations of X, which are induced by the set of nodes ði=N; j=NÞ,0 6 i; j 6 N, where N ¼ 1=h is an integer. We compare the Galerkinapproximation uh based on linear finite elements with the linearpart usb ;l

h of the bubble approximation usbh proposed in this paper.

As stated in Section 4 the bubbles are defined by

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

Fig. 11. Approximations for m ¼ 0:001 and h ¼ 1=10. On the left, Galer

l ¼ h340m

kusb ;lh k0

!1=4

; ifhkusb ;l

h k0

40mP 1;

l ¼ h; ifhkusb ;l

h k0

40m< 1:

Here, l denotes the longitude of any of the two catheti of the triangleKi contained in any triangle of the partition K, see Fig. 5. Let us ob-serve that, as before, we have replaced the local norm of the velocityvector field by the global L2-norm of kusb ;l

h k0 so that in this experi-ment the size of the bubbles is the same in all the triangles of thepartition. The final time chosen is T ¼ 1. All the experiments werecarried out using MATLAB. As time integrator we have used themid-point rule with fixed time step using fixed point iteration tosolve the nonlinear systems. To measure errors an ‘‘exact” solutionwas computed with the Galerkin method on a very fine mesh andwith sufficiently small time steps. Next, we show some numericalexperiments for m ¼ 0:001. The ‘‘exact” solution is shown in Fig. 10.

In Table 3, we show the relative errors in L2 and H1 of the Galer-kin method and the bubble stabilized method for h ¼ 1=8, 1=10

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

kin approximation, on the right, bubble stabilized approximation.

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.8−0.6−0.4−0.2

00.20.40.60.8

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.8−0.6−0.4−0.2

00.20.40.60.8

Fig. 12. Approximations for m ¼ 0:001 and h ¼ 1=12. On the left, Galerkin approximation, on the right, bubble stabilized approximation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

u h(x,x

)

ν=0.001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

u hs b,l (x,x

)

ν=0.001

Fig. 13. Sections with the plane y ¼ x for m ¼ 0:001, h ¼ 1=8, 1/10 and 1/12. Exact solution in black. On the left, Galerkin approximations, on the right, bubble stabilizedapproximations.

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1

Fig. 14. Approximations for m ¼ 0:0005 and h ¼ 1=14. On the left, Galerkin approximation, on the right, bubble stabilized approximation.

3998 J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999

and 1=12. For these values of the diffusion parameter m and meshdiameter h we are in the convection-dominated regime. We canobserve in the table that the bubble stabilized method producesbetter errors than the Galerkin method both in the L2 and H1

norms.In Figs. 11 and 12, we show the Galerkin approximations, on the

left, and the bubble stabilized approximations, on the right, forh ¼ 1=10 and 1/12, respectively. We can observe in both figuresthe spurious oscillations that completely contaminate the Galerkin

approximation based on linear elements. On the other hand, thebubble stabilized method is able not only to reduce the errors, aswe have already observed in Table 3, but also to drastically reducespurious oscillations reproducing very well the behavior of the ‘‘ex-act” solution (compare with Fig. 10).

A detail of the reduction of the oscillations achieved by the pro-posed bubble stabilized method can be observed in Fig. 13, wherewe represent the sections of the Galerkin and bubble stabilizedapproximations along the plane y ¼ x. The exact solution is plotted

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3988–3999 3999

in black. On the left we plot the Galerkin approximations forh ¼ 1=8, 1=10 and 1=12 in green,6 red and blue, respectively. Thesame colors are used for the bubble stabilized approximations onthe right. It is clear from the figure that the new method producesmuch more stable approximations for all the values of h in the figureeven for the largest one. Indeed, we can deduce from the experi-ments that the method we propose works pretty well for large Pecletnumbers for which the Galerkin method gives completely wrongapproximations.

This fact can better be observed in Fig. 14, where we compareagain the Galerkin approximation, on the left, with the bubble sta-bilized approximation, on the right, for m ¼ 0:0005 and h ¼ 1=14.While the spurious oscillations of the Galerkin method increaseswith the increment in the Peclet number the bubble stabilizedmethod presents a robust behavior getting a very stableapproximation.

References

[1] D.N. Arnold, F. Brezzi, M. Fortin, A stable finite element for the Stokesequations, Calcolo 23 (1984) 337–344.

[2] M.I. Asensio, B. Ayuso, G. Sangalli, Coupling stabilized finite element methodswith finite difference time integration for advection–diffusion–reactionproblems, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3475–3491.

[3] B. Ayuso, J. de Frutos, J. Novo, Improving the accuracy of the mini-elementmethod for the Navier–Stokes equations, IMA J. Numer. Anal. 27 (2007) 198–218.

[4] F. Brezzi, M.O. Bristeau, L.P. Franca, M.M. Gilbert Rogé, A relationship betweenstabilized finite element methods and the Galerkin method with bubblefunctions, Comput. Methods Appl. Mech. Engrg. 96 (1992) 117–129.

[5] F. Brezzi, T.J.R. Hughes, D. Marini, A. Russo, E. Süli, A priori error analysis of afinite element method with residual-free bubbles for advection-dominatedequations, SIAM J. Numer. Anal. 36 (1999) 1933–1948.

[6] F. Brezzi, D. Marini, E. Süli, Residual-free bubbles for advection–diffusionproblems: the general error analysis, Numer. Math. 85 (2000) 31–47.

[7] F. Brezzi, A. Russo, Choosing bubbles for advection–diffusion problems, Math.Models Methods Appl. Sci. 4 (1994) 571–587.

[8] P.B. Bochev, M.D. Gunzburger, J.N. Shadid, Stability of the SUPG finite elementmethod for transient advection–diffusion problems, Comput. Methods Appl.Mech. Engrg. 193 (2004) 2301–2323.

[9] A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulationsfor convection dominated flows with particular emphasis on theincomprensibble Navier–Stokes equations, Comput. Methods Appl. Mech.Engrg. 32 (1982) 199–259.

[10] M.P. Calvo, J. de Frutos, J. Novo, Linearly implicit Runge–Kutta methods foradvection–reaction–diffusion equations, Appl. Numer. Math. 37 (2001) 535–549.

[11] A. Cangiani, E. Süli, Enhanced residual-free bubble method for con-vection–diffusion problems, Int. J. Numer. Methods Fluids 47 (2005) 1307–1313.

[12] R. Codina, J. Blasco, Analysis of a stabilized finite element approximation of thetransient convection–diffusion–reaction equation using orthogonal subscales,Comput. Visual Sci. 4 (2002) 167–174.

6 For interpretation of color in Fig. 13, the reader is referred to the web version ofthis article.

[13] R. Codina, A. Folch, A stabilized element predictor–corrector écheme for theincomprensible Navier–Stokes equations using a nodal-based implementation,Int. J. Numer. Methods Fluids 44 (2004) 483–503.

[14] R. Codina, J. Principe, O. Guasch, S. Badia, Time dependent subscales in thestabilized finite element approximation of incompressible flow problems,Comput. Methods Appl. Mech. Engrg. 196 (2007) 413–2430.

[15] L.P. Franca, S.L. Frey, Stabilized finite element methods: II. The incompressibleNavier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 99 (1992) 209–233.

[16] L.P. Franca, G. Hauke, A. Masud, Revisiting stabilized finite element methodsfor the advective–diffusive equation, Comput. Methods Appl. Mech. Engrg. 195(2006) 1560–1572.

[17] L.P. Franca, A.L. Madureira, F. Valentin, Towards multiscale functions:enriching finite element spaces with local but not bubble-like functions,Comput. Methods Appl. Mech. Engrg. 194 (2005) 3006–3021.

[18] L.P. Franca, A. Russo, Deriving upwinding, mass lumping and selective reducedintegration by residual-free bubbles, Appl. Math. Lett. 9 (1996) 83–88.

[19] L.P. Franca, A. Russo, Recovering SUPG using Petrov–Galerkin formulationswith adjoint residual-free bubbles, Comput. Methods Appl. Mech. Engrg. 182(2000) 333–339.

[20] G. Hauke, M.H. Doweidar, Fourier Analysis of semidiscrete and space–timestabilized methods for the advective–reactive–diffusive equation: I. SUPG,Comput. Methods Appl. Mech. Engrg. 194 (2005) 45–81.

[21] G. Hauke, M.H. Doweidar, Fourier Analysis of semidiscrete and space–timestabilized methods for the advective–reactive–diffusive equation: II. SGS,Comput. Methods Appl. Mech. Engrg. 194 (2005) 691–725.

[22] A. Huerta, J. Donea, Time-accurate solution of stabilized convection–diffusion–reaction equations: I, time and space discretization, Commun. Numer.Methods Engrg. 18 (2002) 565–573.

[23] A. Huerta, B. Roig, J. Donea, Time-accurate solution of stabilized convection–diffusion–reaction equations: II, accuracy análisis and examples, Commun.Numer. Methods Engrg. 18 (2002) 575–584.

[24] T.J.R. Hughes, L.P. Franca, M. Mallet, A new finite element formulation forcomputational fluid dynamics: VI. Convergence analysis of the generalizedSUPG formulation for linear time-dependent multidimensional advective–diffusive systems, Comput. Methods Appl. Mech. Engrg. 63 (1987) 97–112.

[25] T.J.R. Hughes, A.N. Brooks, A multidimensional upwind scheme with nocrosswind diffusion, in: T.J.R. Hughes (Ed.), Finite Element Methods forConvection Dominated Flows, ASME, 1979, pp. 19–35.

[26] A. Quarteroni, A. Valli, Numerical Approximation of Partial DifferentialEquations, Springer Series in Computational Mathematics, Berlin, 1997.

[27] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly PerturbedDifferential Equations–Convection–Diffusion and Flow Problems, SpringerSeries in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 1996.

[28] A. Russo, Bubble stabilization of finite element methods for the linearizedincompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg.132 (1996) 335–343.

[29] A. Russo, Streamline-upwind Petrov/Galerkin method (SUPG) vs residual freebubbles (RFB), Comput. Methods Appl. Mech. Engrg. 195 (2006) 1608–1620.

[30] G. Sangalli, Global and local error analysis for the residual-free bubblesmethod applied to advection-dominated problems, SIAM J. Numer. Anal. 38(2000) 1496–1522.

[31] G. Sangalli, A discontinuous residual-free bubble method for advection–diffusion problems, J. Engrg. Math. 49 (2004) 149–162.

[32] J.C. Simo, F. Armero, C.A. Taylor, Stable and time-dissipative finite elementmethods for the incompressible Navier–Stokes equations in advectiondominated flows, Int. J. Numer. Methods Engrg. 38 (1995) 1475–1506.

[33] X.Y. Yue, A note on practical bubbles for advection–diffusion problems, Calcolo39 (2002) 189–200.