45 mm

A NEW STABILIZED FINITE ELEMENT FORMULATION FOR SCALARCONVECTION-DIFFUSION PROBLEMS AND ITS UP-WIND FUNCTION USING

LINEAR AND QUADRATIC ELEMENTS

Eduardo Gomes Dutra do CarmoDepartment of Nuclear Engineering – COPPE – Federal University of Rio de JaneiroIlha do Fundão – 21945-970 – P.B. 68509 – Rio de Janeiro, RJ, Brazilcarmo@lmn.con.ufrj.brGustavo Benitez AlvarezAbimael Fernando Dourado LoulaNational Laboratory of Scientific Computation – LNCCGetulio Vargas 333 – Quitandinha – 25651-070 – Petrópolis, RJ, Brazilbenitez@lncc.braloc@lncc.br

Abstract. A new stabilized and accurate finite element formulation for convection dominatedproblems is presented. The new method uses the GLS formulation as starting point. The paperalso presents a study on the function of the Péclet number that appears in the upwind functionof the GLS, CAU and SAUPG methods, using linear and quadratic triangular elements, aswell as bilinear and biquadratic quadrilateral elements. The numerical experiments indicatethat in the case of elements with faces on the "outflow boundary", the upwind function isstrongly dependent on the geometry and on the degree of polynomial interpolation. A newfunction of the Péclet number is then proposed, to take these effects into consideration,creating a new method of stabilization for higher order elements.

Keywords: Stabilized FEM, Diffusion-Convection, Stabilization

1. INTRODUCTION

It is well known that Galerkin finite element method is unstable for convection dominatedconvection-diffusion problems for presenting spurious oscillations. During the last twodecades, innumerous attempts to obtain new methods have been carried out to avoid suchspurious oscillations. These new methods are known as stabilized finite element methods.References about some of them can be found in Carmo et all (2003). Among them, the SUPG- Streamline Upwind/Petrov-Galerkin method (Brooks et all, 1982) and later the GLS -Galerkin-Least-Squares method (Huhes et all, 1989) are two methods that stand out. Bothmethods add stabilization terms to the Galerkin formulation, providing good stability andaccuracy to the numerical solution, when the exact solution of the problem is smooth(Johnson, 1984). However, the spurious oscillations remain when the exact solution possessesboundary layers.

Many attempts have been used the SUPG or GLS methods as starting point for newstabilized formulations. In particular, the CAU - Consistent Approximate Upwind method(Galeão et all, 1988) maintain the term of SUPG and adds a nonlinear term providing an extracontrol concerning the function’s derivative in the direction of the approximate gradient, andeliminates the spurious oscillations. Nevertheless, when the exact solution of the problem issmooth, the CAU’s approximate solution presents undesirable crosswind diffusion (Carmo etall, 1991). Building up a stable method that captures discontinuities and also preserves theaccuracy of the SUPG or GLS methods for problems with smooth solution has been a greatchallenge. Recently, the authors develop the SAUPG method (Carmo et all, 2003), whichtakes advantage of the best qualities of the SUPG and CAU methods.

In this work, based on the idea if the SAUPG method, we introduces a new stabilizedmethod using the GLS method as a starting point. Also, we studied the upwind functiondependent on the geometry of the element (triangular and quadrilateral elements) and on thedegree of the polynomial in the shape functions.

2. THE STATIONARY SCALAR DIFFUSION-ADVECTION EQUATION

2.1 The boundary value problem

Let nR⊂Ω be an open bounded domain, whose boundary Γ is a piecewise smoothboundary. The unit outward normal vector n̂ to Γ is defined almost everywhere. We shallconsider the problem:

Ω=∇⋅+∇⋅∇− in)( fuD φφ , (1)Γ= ongφ , (2)

where φ denotes the unknown quantity of the problem, D is the diffusivity tensor,),,( 1 nuuu K= is the transport advective field, f is the volume source term and g is the

boundary value prescribed for φ on Γ .

2.2 Variational Formulation

Consider the set of all the admissible functions S and the space of the admissiblevariations V defined as: { }Γ=Ω∈= on:)(1 gHS ψψ , { }Γ=Ω∈= on0:)(1 ηη HV , where

)(1 ΩH is the standard Sobolev space. The variational formulation of the boundary value

problem defined as Eq. (1) and Eq. (2), involves finding S∈φ that satisfies the variationalequation:

VdfduD ∈∀Ω=Ω∇⋅+∇⋅∇ ∫∫ΩΩ

ηηηφηφ ])()[( . (3)

2.3 Finite element formulations

Consider },,{ 1 nehM ΩΩ= K a finite element partition of Ω , such that: U

ne

ee

1=Ω=Ω ,

UΓΩ=Ω , U eee ΓΩ=Ω and ≠ 0/=ΩΩ/ se/ eeee I . The respective finite element set

and space of S and V are defined as: }on , )(:)({ 1, Γ=Ω∈Ω∈= hhekh

ehkh gPHS φφφ

and }on 0 , )(:)({ 1, Γ=Ω∈Ω∈= hekh

ehkh PHV ηηη , where )( e

kP Ω being the space ofpolynomials of degree k , hg denotes the interpolation of g and heφ denotes the restriction of

hφ to eΩ .Galerkin, Streamline Upwind/Petrov-Galerkin (SUPG), Galerkin Least-Squares (GLS)

and Consistent Approximate Upwind (CAU) finite element formulations of problem Eq. (3)consist in: finding khh S ,∈φ that satisfies khh V ,∈∀η ,

Galerkin method)(),( hG

hhG FA ηηφ = , (4)

∑ ∫ ∑ ∫= Ω = Ω

Ω=Ω∇⋅+∇⋅∇=ne

e

ne

ee

heGe

he

he

he

heG

e e

dfFduDA1 1

,])()[( ηηφηφ , (5)

SUPG method)()(),(),( hSUPG

hG

hhSUPG

hhG FFAA ηηηφηφ +=+ , (6)

∑ ∫= Ω

Ω∇⋅∇⋅+∇⋅−∇=ne

ee

hee

e

eehee

heeSUPG

e

duu

huDA

1

)(2

)]()([ ητ

φφ , (7)

∑ ∫= Ω

Ω∇⋅=ne

ee

hee

e

eeeSUPG

e

duu

hfF

1)(

2η

τ, (8)

e

ee b

uh 2= , je

n

j j

iie ux

b ,1

, ∑= ∂

∂=

ξ,

−=e

e PCC

2

11,0maxτ , Duh

P eee 2= , (9)

GLS method)()(),(),( hLS

hG

hhLS

hhG FFAA ηηηφηφ +=+ , (10)

∑ ∫∑ ∫= Ω= Ω

Ω=Ω∇⋅+∇⋅−∇=ne

ee

heeLS

ne

ee

he

hee

heeLS

ee

dpfFdpuDA11

,)]()([ φφ , (11)

[ ]heeheee

eehe uDu

hp ηη

τ∇⋅+∇⋅∇−= )(

2 he M∈Ω∀ , (12)

CAU method)()()),(),(),( hSUPG

hG

hhCAU

hhSUPG

hhG FFAAA ηηηφηφηφ +=++ , (13)

∑ ∫= Ω

Ω∇⋅∇=ne

ee

he

he

heeCAU

e

dCA1

)( ηφφ , (14)

<∇∇

∇−

≥∇

=

ee

ce

ce

hee

he

he

he

hee

he

ee

ce

ceee

ee

ce

ce

hee

he

hee

hh

ui

uhhh

hh

ui

C

ττ

φ

φ

φ

φ

φ

φ

τττ

ττ

φ

φ

φ)Re(

f,)Re()Re(

2

)Re(f ,0

)( , (15)

ehee

hee

he fDu −∇⋅∇−∇⋅= )()Re( φφφ

he M∈Ω∀ , (16)

ce

heec

e b

vuh

−= 2 , )( ,,

1,

hjeje

n

j j

icie vux

b −∂∂

= ∑=

ξ,

−= ce

ce PC

C

2

11,0maxτ , (17)

e

hee

cec

e D

vuhP

2

−= ,

≠∇∇∇

−

=∇

=0 if ,

)Re(

0 if ,

2he

he

he

he

e

hee

he u

u

vφφ

φ

φ

φ

, (18)

where ),,1()( niei K=Ωξ is the local coordinate system that maps eΩ in the usual standard

elements eΩ̂ ; jeu , is the "j" component of eu ; and 1C and 2C are constants that depend of theelement type, eΩ and k ( 121 == CC for bilinear quadrilaterals elements).

When the advective term is dominant, the Galerkin method is totally unstable, unless themesh is highly refined such that heu

Ke Mh e

e ∈Ω∀≤ 2ˆ , where eĥ denote the diameter of eΩ .

The instability of the Galerkin method is due to the weak control on the gradient of thesolution. The term ),( hhLSA ηφ introduces control in the streamline direction. However, whenthe problem possesses boundary layers the GLS method presents spurious oscillations. Theapproximate upwind direction hee

he vuU −= is in the direction of

heφ∇ . The CAU method

possess a additional stability, in regions near internal and/or external boundary layers,compared to GLS or SUPG methods for adding another residual to control the gradient of thesolution in any direction.

3. THE GALERKIN LEAST SQUARES AND APPROXIMATE UPWIND METHOD

The main idea that supports the development of SAUPG method (Carmo & Alvarez,2003) is the performance of both SUPG and CAU methods. The SUPG method lacks stabilitywhen applied to problems with boundary layers, while the CAU method loses accuracy insmooth problems. In Carmo et all (2004) the stabilization term of SUPG was replaced by theGLS term to develop the GLSAU formulation. This formulation consists in finding khh S ,∈φthat satisfies khh V ,∈∀η :

)()(),(),(),( hLSh

Ghh

AUhh

LShh

G FFAAA ηηηφηφηφ +=++ , (19)

∑ ∫= Ω

Ω∇⋅∇=ne

ee

he

he

hee

hhAU

e

dBA1

)(),( ηφφηφ , (20)

)()]([)]([][)( 1211

21hee

hee

hee

hee CCBBCBBB φφφφ

γγγ −− == , (21)where 1B , 2B and γ are functions that depend on the regularity of the approximate solutionand were determined for the SAUPG method. In Carmo et all (2003) the non-dimensional

variable ||||

|)Re(|||

||hee

he

e

hee

e uuvu

φφ

α∇

=−

= was define to evaluate the regularity of the approximate

solution, and the functions 1B and 2B were defined as:

)1)(1(11

1][)( +−= γγβα eB , eee hu

Bτ||

22 = ,

≥<

=1 se 01 se )(

)( 21e

eee

qααα

αγ (22)

<

≥+

<=

≥

=1 if

if ][ if

1 if 1

3321

332

1e

ee

e

e

ααββααββ

β

α

β , (23)

)(3

1][ eqehαβ = , )}(adim ,1min{ ee hh =

he M∈Ω∀ , (24)

where )(adim ⋅ is a function that turns its argument dimensionless. Following the proceduredescribed in Carmo et all (2003) the function γ is given by:

<

≤=

1]Re[ if ]Re[

1]Re[ if ][)()(

)()(

433

33

ee

ee

qqe

qqe

αα

αα

α

αγ , (26)

where |)||(|adimRe heeu φ∇= . Also, the functions 1q , 2q and 3q dependent on the parameter

eα are obtained with the procedure described in Carmo et all (2003),

221 )(1)()( eee qq ααα −== ,

23 )(2

13)( eeeq ααα ++= . (27)

When the solution is very smooth, γ tends to 2 and the ),( hhLSA ηφ prevails on),( hhAUA ηφ in Eq. (19). In the case where the solution is not smooth, the contribution of the),( hhAUA ηφ term is then significant. For the new method, the parameter eτ defined by Eq.

(9), as well as, the ceτ give by Eq. (17) are redefined as: }1,0max{ 1cePo

ce −= ττ and

}1,0max{ 1ePoe

−= ττ , where eP and c

eP are respectively given by Eq. (9) and Eq. (18). The

parameter oτ was determined through the numerical experiments for bilinear quadrilateralelement in Carmo et all (2003) as 1=oτ .

4. NUMERICAL RESULTS

In this section we present the numerical results obtained in several standard numericaltests with two types of problems presenting internal and/or external boundary layers and withsmooth solutions. In all cases, the medium is assumed homogeneous and isotropic with

1010−=D . The considered domain is a square of unitary sides (0,1)×(0,1). Five iterationswere accomplished for the CAU and GLSAU methods.

Example 1: advection skew to the mesh

In this problem we have 0),( =yxf and the following boundary conditions:0)0,( =xφ , 1)1,( =xφ

0),1( =yφ , ]1,0[∈∀y

0),0( =yφ , ]6.0,0[∈∀y

6.0),0( −= yyφ , ]65.0,6.0[∈∀y

05.0)65.0(18),0( +−= yyφ , ]70.0,65.0[∈∀y

95.0)70.0(),0( +−= yyφ , ]75.0,70.0[∈∀y

1),0( =yφ , ]1,75.0[∈∀y ,

and advectives field skew to the mesh are )2,1( −=u and )1,1( −=u .

Example 2: external boundary layers and semicircular inernal layer

In this example we have 0),( =yxf , where the Dirichlet boundary conditions are the same asin the latter example and the advective field is ),( xyu −= .

Example 3: advection of a sine hill in a rotating flow field

The problems statement is shown in Fig. 1. The flow's rotation is determined by theadvective field ),( xyu −= . Along the external boundary 0=φ , and on the internal‘boundary’ OA a sine hill function ]0,5.0[)2(),0( −∈∀= yysiny πφ is prescribed.

Figure 1. Advection in a rotating flow field: problem statement.

In Carmo et all (2004) was verified, numerically, that the functions )( eiq α (i=1,2,3) Eq.(27), determined through the numerical experiments for bilinear quadrilateral elements inCarmo et all (2003), are also valid for linear or quadratic triangular elements, as well as, forbiquadratic quadrilateral elements. Also, it was verified in all the examples that the threemethods introduce diffusivity in excess for the external boundary layers, when the oτ defined

for bilinear quadrilateral elements is used. At present, not a perfect theoretical support hasbeen obtained and not much has been studied about the parameter eτ dependence on theelement Péclet number for the higher order quadratic rectangular or triangular elements. Somefew attempts can be found in Heinrich (1980), Mizukami (1985) and Codina et all (1992). InCarmo et all (2004) was propose Eq. (28), based on a numerical analyses of the stability andon the fact that the internal and external boundary layers are of different physical nature(Johnson et all, 1984). The purpose was to avoid that the upwind function is not overestimatedor underestimated.

{ }

=

>⋅Γ∈=Γ

∅≠Γ∩Γ

∅=Γ∩Γ=

+

+

+

elementsular for triang 21

elements ralquadrilatefor 21

,

0ˆ: if

if 1

k

1)-(k

eoe

eo

eo

unx

τττ . (28)

All figures coming afterward show the numerical results for each example. In each figure,the cases (a) correspond to each one of the three methods, using oτ given by Eq. (28). Cases(b) refer to the solutions of the methods obtained with 1=oτ , that is to say, the upwindfunctions corresponding to the value determined for the bilinear quadrilateral elements.

4.1 Approximate solutions using quadratic quadrilateral elements (9-noded)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (a) CAU (a) GLSAU (a)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (b) CAU (b) GLSAU (b)

Figure 2- Example 1, )2,1( −=u , (a) 21=eoτ , (b) 1=

eoτ .

-0.05

1.01

-0.05

1.01

...

-0.05

1.01

GLS (a) CAU (a) GLSAU (a)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (b) CAU (b) GLSAU (b)Figure 3- Example 1, )1,1( −=u , (a) 21=

eoτ , (b) 1=

eoτ .

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (a) CAU (a) GLSAU (a)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (b) CAU (b) GLSAU (b)Figure 4 - Example 2, (a) 21=

eoτ , (b) 1=

eoτ .

-0.05

0.99

-0.05

0.99

-0.05

0.99

GLS (a) CAU (a) GLSAU (a)

-0.05

0.99

-0.05

0.99

-0.05

0.99

GLS (b) CAU (b) GLSAU (b)

Figure 5 - Example 3, (a) 21=eoτ , (b) 1=

eoτ .

For the examples 1 and 2, we can observe in Figs. (2), (3) and (4) that the approximatesolutions of the GLSAU and CAU methods are very close. In the cases (b) undershoots areobtained for the external boundary layers. For the example 3, the GLS and GLSAU methodsyield to very good results (see Figure 5).

4.2 Approximate solutions using linear triangular elements

Here the observations done in the previous subsection continue being valid. Similarresults are obtained for quadratic triangular elements, so they are not shown here. Again, forthe examples 1 and 2, we can observe in Figs. (6), (7) and (8) the similarity between theapproximate solutions of the GLSAU and CAU methods. In the cases (b) undershoots areobtained for the external boundary layers. It should be highlighted, that the internal sharplayer of the example 1 (see Figure 7) is very well approached by the GLS, CAU and GLSAUsolutions. This is something expected, since in this case the advective field u is aligned to themesh. For the example 3, the GLS and GLSAU methods yield to very good results as showedin Figure 5.

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (a) CAU (a) GLSAU (a)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (b) CAU (b) GLSAU (b)Figure 6- Example 1, )2,1( −=u , (a) 21=

eoτ , (b) 1=

eoτ .

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (a) CAU (a) GLSAU (a)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (b) CAU (b) GLSAU (b)Figure 7 - Example 1, )1,1( −=u , (a) 21=

eoτ , (b) 1=

eoτ .

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (a) CAU (a) GLSAU (a)

-0.05

1.01

-0.05

1.01

-0.05

1.01

GLS (b) CAU (b) GLSAU (b)Figure 8 - Example 2, (a) 21=

eoτ , (b) 1=

eoτ .

0

0.95

0

0.95

0

0.95

GLS (a) CAU (a) GLSAU (a)

0

0.95

0

0.95

0

0.95

GLS (b) CAU (b) GLSAU (b)Figure 9 - Example 3, (a) 21=

eoτ , (b) 1=

eoτ .

5. CONCLUSIONS

In this work, we have presented a new stabilized finite element formulation to solvescalar convection diffusion problems. The formulation preserves the best qualities of the GLSand CAU methods, in terms of stability and accuracy. The numerical experiments showed thatthe functions )(1 eq α , )(2 eq α and )(3 eq α do not depend on the geometry of the element noron the degree of the polynomial in the shape functions.

We also studied the upwind function dependence on the geometry of the element and thedegree of the polynomial in the shape functions. The numerical results led us to define oτ asin Eq. (28), that is to say, value 1 is necessary for the internal sharp layer, while eoτ is theoptimum value to avoid overshoots and undershoots for the external boundary layer. Thismodification on stabilizing parameter near the outflow boundaries increases the quality of thenumerical solution for the GLS, CAU and GLSAU methods. Although we lack a theoreticalerror estimate, the simplicity of the new scheme, as well as the parameter τ must beconsidered for practical purposes.

Acknowledgements

The authors wish to thank the Brazilian research-funding agencies CNPq and FAPERJ fortheir support to this work. The second author would like to thank Maria Eugenia (Kena) forher helpful support.

REFERENCES

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