A multiscale/stabilized ﬁnite element method for the...

Comput. Methods Appl. Mech. Engrg. 193 (2004) 1997–2018

www.elsevier.com/locate/cma

A multiscale/stabilized finite element methodfor the advection–diffusion equation

A. Masud *, R.A. Khurram

Department of Civil and Materials Engineering, The University of Illinois at Chicago, 842 West Taylor Street,

2095 Engineering Research Facility, Chicago, IL 60607-7023, USA

Received 2 January 2003; received in revised form 9 September 2003; accepted 8 December 2003

Abstract

This paper presents a multiscale method that yields a stabilized finite element formulation for the advection–dif-

fusion equation. The multiscale method arises from a decomposition of the scalar field into coarse (resolved) scale and

fine (unresolved) scale. The resulting stabilized formulation possesses superior properties like that of the SUPG and the

GLS methods. A significant feature of the present method is that the definition of the stabilization term appears

naturally, and therefore the formulation is free of any user-designed or user-defined parameters. Another important

ingredient is that since the method is residual based, it satisfies consistency ab initio. Based on the proposed formu-

lation, a family of 2-D elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed.

Numerical results show the good performance of the method on uniform, skewed as well as composite meshes and

confirm convergence at optimal rates.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

The advection–diffusion equation governs several important phenomena in physics and engineering, and

also serves as a vehicle to study the more advanced Navier–Stokes equations. Therefore it has been thefocus of intense research for quite some time (see e.g., [7,16,18,22] and references therein).

For the advection dominated case this equation becomes hyperbolic and develops sharp features in the

solution. Classical numerical methods for the advection–diffusion equation result in non-convergent ele-

ments. Specifically, methods based on the standard Galerkin finite element approach lack stability that

manifests itself in terms of nonphysical oscillations. In order to correct the deficiencies in the standard

Galerkin approach, Hughes and colleagues introduced streamline-upwind-Petrov–Galerkin (SUPG) tech-

nique [7,16]. This technique turned out to be the fore-runner of a new class of stabilization schemes, called

the Galerkin/least-square (GLS) stabilization methods [18]. In the context of the advection–diffusionequation, the essential feature of this method is the stabilization of the advection operator without

* Corresponding author. Fax: +1-312-996-2426.

E-mail address: [email protected] (A. Masud).

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.12.047

mail to: [email protected]

1998 A. Masud, R.A. Khurram / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1997–2018

upsetting consistency or compromising accuracy. The key idea in the GLS stabilization approach is toaugment the Galerkin finite element formulation with a least-squares form of the residuals that are based

on the corresponding Euler–Lagrange equations. These least-squares integrals contain stabilization

parameters that are designed so that the method achieves exact solution in the case of one-dimensional

model problems. A generalization of the nodaly exact stabilization parameters in 1-D to the case of nodaly

exact parameters in multi-dimensions has been a challenging task and a literature review reveals various

attempts in designing the optimal stabilization parameters [12,13]. GLS stabilization was followed by the

‘‘unusual stabilized methods’’ introduced by Franca and coworkers [8–12]. Concurrently, another class of

stabilized methods that was based on the idea of augmenting the Galerkin method with virtual bubblefunctions was introduced by Brezzi and coworkers [1–6].

In [14,17], Hughes revisited the origins of the stabilization schemes from a variational multiscale ap-

proach. In the Hughes� variational multiscale (HVM) method different stabilization techniques appear as

special cases of the underlying sub-grid scale modelling concept. In this paper we follow this line of thought

and present a multiscale/stabilized finite element formulation for the advection–diffusion equation. The new

formulation, hereafter termed as the HVM formulation, has improved stability properties as compared to

the Galerkin form of the problem. The main idea in the present method is a multiscale decomposition of the

scalar field in the coarse and the fine scales. This decomposition is based on the assumption that the finescale structures of the solution may not be captured by a given mesh. However, the influence of the fine

scales on the coarse scales may not be negligible and therefore it must be accounted for. It is important to

note that within the context of the Galerkin methods, the only possible way to capture the fine scale

features is to successively refine the mesh, i.e., reduce the characteristic length scale of discretization.

However this line of thought is extremely limited in its scope because it can quickly exhaust any given

computational platform. In fact the question one needs to consider is as follows: Is it possible to capture all

the essential features in the solution, both coarse and fine, while still using cruder meshes? In other words, is it

possible to develop highly accurate numerical schemes on cruder discretizations? Taking this line of thought,which is based on the notion of existence of the fine scales in the problem, we proposed a stabilized for-

mulation for the Darcy flow [19], the linearized incompressible Navier–Stokes equations [20], and for the

convective–diffusive heat transfer equations [21]. The present paper also aims at taking this viewpoint and

presents the HVMmethod for the advection–diffusion equation. However, in this paper we do not deal with

either the discontinuity-capturing operators or the nonlinear methods that are at times employed to model

sharp layers in the solution.

An outline of the paper is as follows: Section 2 presents the governing equations and the standard

Galerkin form. Emphasis in the paper is on the variational multiscale approach for the development of anew stabilized form, which is presented in Section 3. Section 4 presents the numerical results, and con-

clusions are drawn in Section 5.

2. The advection–diffusion equation

Let X � Rnsd be an open bounded region with piecewise smooth boundary C. We assume nsd P 2. The

advection–diffusion equation is

a � rv� jDv ¼ f in X; ð1Þ

v ¼ w on C; ð2Þwhere v is the unknown scalar field, aðxÞ is the given flow velocity which is assumed solenoidal, i.e.,r � a ¼ 0 in X, j ¼ jðxÞ > 0 represents diffusivity, f ðxÞ is the prescribed source function, and w is the

boundary condition.

A. Masud, R.A. Khurram / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1997–2018 1999

2.1. The standard weak form

Let V � H 1ðXÞ \ C0ðXÞ denote the space of trial solutions and weighting functions for the unknown

scalar field. The weak form is obtained by multiplying the governing equation with an admissible weighting

function and integrating it over the domain, resulting in the following form.

ðw; a � rvÞ þ ðrw;jrvÞ ¼ ðw; f Þ; ð3Þwhere ð�; �Þ ¼

RX ð�ÞdX is the L2ðXÞ inner product and w is the weighting function for v.

The standard Galerkin form is obtained by substituting the discrete counterparts for the weighting

functions and the trial solutions, i.e., wh and vh, respectively. However, in the rest of the paper we have

dropped the superposed h to keep the notation simple.

Remark 1. The standard Galerkin form suffers from numerous technical drawbacks [7,16,18]. The most

significant being the lack of stability for the advection dominated problems. In order to overcome this issue,

stabilized methods such as the SUPG and the GLS methods have been developed.

Remark 2. The trial solutions and the weighting functions are usually piecewise polynomials and can only

capture variability at spatial scales that are larger than the characteristic length scale of the mesh. All spatial

features that are smaller than the element size and represent the sub-grid variability are neglected in the

standard Galerkin approach. The lack of stability of the Galerkin method for the advection dominated

problems can be understood in this context. As the sub-grid scales are not captured adequately, rather they

are completely ignored in the Galerkin method, their effect propagates to larger scales and deteriorate the

coarse-scale calculations.

3. The variational multiscale method

3.1. Multiscale decomposition

In this section we present the Hughes� variational multiscale approach [14] for the development of a

stabilized formulation for the advection–diffusion equation. A fundamental principle of multiscale ap-proach is to acknowledge the presence of the fine scales that cannot be captured by a given spatial dis-

cretization. This is particularly important for the advection dominated problems where the solution

develops sharp features that require an impractical grid resolution. Therefore, from the outset, we work

with the premise of the existence of multiple scales in the problem.

We consider the bounded domain X discretized into non-overlapping regions Xe (element domains) with

boundaries Ce, e ¼ 1; 2; . . . ; numel, such that

X ¼[numel

e¼1

Xe: ð4Þ

We denote the union of element interiors and element boundaries by X0 and C0, respectively.

X0 ¼[numel

e¼1

ðintÞXe ðelem: interiorsÞ; ð5Þ

C0 ¼[numel

e¼1

Ce ðelem: boundariesÞ: ð6Þ


We assume an overlapping sum decomposition of the scalar field into the coarse scales or the resolvable

scales and the fine scales or the sub-grid scales.

vðxÞ ¼ �vðxÞ|{z}coarse scale

þ v0ðxÞ|ffl{zffl}fine scale

: ð7Þ

Likewise, we assume an overlapping sum decomposition of the weighting function into the coarse and the

fine scale components indicated as w and w0, respectively.

wðxÞ ¼ wðxÞ|ffl{zffl}coarse scale

þ w0ðxÞ|fflffl{zfflffl}fine scale

: ð8Þ

We further make an assumption that the sub-grid scales although non-zero within the elements, vanish

identically over the element boundaries.

v0 ¼ w0 ¼ 0 on C0: ð9Þ

Remark 3. Another viewpoint to look at this decomposition is as follows. Following (7) we can write

v ¼ vh þ ve where vh is the solution that can be obtained with a good numerical scheme on a given mesh, and

ve is the part of the solution that is lost because its scale is smaller than the characteristic length scale of the

discretization. In other words ve represents the error in the solution. The method presented below can beconsidered as a systematic procedure whereby one can rebuild the error term in the weak form of the

problem. This procedure then automatically yields a stabilized form of the problem that shows higher

accuracy on cruder discretizations.

We now introduce the appropriate spaces of functions for the coarse and the fine scale fields and specify

a direct sum decomposition on these spaces.

V ¼ V�V0; ð10Þ

where V in (10) is the space of trial solutions and weighting functions for the coarse scale velocity field and

is identified with the standard finite element space.

V ¼ f�vj�v 2 H 10 ðXÞ;�vðXeÞ ¼ PkðXeÞg; ð11Þ

where PkðXeÞ denotes complete polynomials of order k over Xe.

On the other hand, various characterizations of V0 are possible, subject to the restriction imposed by the

stability of the formulation that requiresV andV0 to be linearly independent. Consequently, in the discrete

caseV0 can contain various finite dimensional approximations, e.g., bubble functions or p-refinements, thatsatisfy (9).

V0 ¼ fv0jv0 ¼ 0 on C0g: ð12Þ

Furthermore, because of (9), w0 2 V0.

3.2. The multiscale variational problem

We now substitute the trial solutions (7) and the weighting functions (8) in the standard variational form

(3), and this becomes the point of departure from the conventional Galerkin formulations.

ðwþ w0; a � rð�vþ v0ÞÞ þ ðrðwþ w0Þ; jrð�vþ v0ÞÞ ¼ ðwþ w0; f Þ: ð13Þ


With suitable assumptions on the fine scale field, as stipulated in (9), and employing the linearity of the

weighting function slot, we can split the problem into the coarse and the fine scale parts, indicated as Wand W0, respectively.

The coarse scale sub-problem W can be written as

ðw; a � rð�vþ v0ÞÞ þ ðrw; jrð�vþ v0ÞÞ ¼ ðw; f Þ: ð14ÞThe fine scale sub-problem W0 can be written as

ðw0; a � rð�vþ v0ÞÞ þ ðrw0; jrð�vþ v0ÞÞ ¼ ðw0; f Þ: ð15ÞThe general idea at this point is to solve the fine scale problem to obtain the fine scale solution v0. Thissolution can then be substituted in the coarse scale problem (14), thereby eliminating the fine scales,yet retaining their effect.

3.3. The fine scale sub-problem (W0)

Let us now consider the fine scale part of the weak form W0, which, because of the assumption on the

fine scale field, is defined over X0. Exploiting linearity of the solution slot in (15) we have

ðw0; a � r�vÞX0 þ ðw0; a � rv0ÞX0 þ ðrw0; jr�vÞX0 þ ðrw0; jrv0ÞX0 ¼ ðw0; f ÞX0 ; ð16Þwhere ð�; �ÞX0 ¼

Pnumel

e¼1

RXe ð�ÞdX is the L2ðXeÞ inner product.

Applying integration-by-parts to the third term on the left hand side and employing (9)

ðw0; a � r�vÞX0 þ ðw0; a � rv0ÞX0 � ðw0; jD�vÞX0 þ ðrw0; jrv0ÞX0 ¼ ðw0; f ÞX0 : ð17ÞRearranging terms, the fine scale problem reduces to

ðw0; a � rv0ÞX0 þ ðrw0;jrv0ÞX0 ¼ ðw0; f � a � r�vþ jD�vÞX0 : ð18ÞTo crystallize ideas, and without loss of generality, we assume that the fine scales are represented via

bubbles over element domains, i.e.,

v0jXe ¼ be1v0e on Xe; ð19Þ

w0jXe ¼ be2w0e on Xe; ð20Þ

where be1 and be2 represent the bubble shape functions for the fine scale trial solutions and the fine scaleweighting functions, respectively. Furthermore, v0e and w0

e represent the coefficients for the fine scale trial

solutions and the weighting functions, respectively. Substituting (19) and (20) in the fine scale problem (18)

we get

ðbe2w0e; a � rbe1v

0eÞ þ ðrbe2w

0e; jrbe1v

0eÞ ¼ ðbe2w0

e; f � a � r�vþ jD�vÞ: ð21ÞTaking the constant coefficients w0

e and v0e out of the integral expressions and employing arbitrariness of w0e,

we get

v0e ¼�1

½ðbe2; a � rbe1Þ þ ðrbe2; jrbe1Þ�ðbe2; a � r�v� jD�v� f Þ: ð22Þ

We now reconstruct the fine scale field via recourse to (19)

v0ðxÞ ¼ be1�1

½ðbe2; a � rbe1Þ þ ðrbe2; jrbe1Þ�ðbe2; a � r�v

�� jD�v� f Þ

�: ð23Þ

In order to keep the presentation simple, and for the case where the residual of the coarse scales over

element interiors can be considered constant, we can simplify Eq. (23) as follows:


v0ðxÞ ¼ �s½a � r�v� jD�v� f �; ð24Þ

where s is defined as the stability function written as follows:

s ¼ be1

ZXebe2 dX½ðbe2; a � rbe1Þ þ ðrbe2; jrbe1Þ�

�1: ð25Þ

Remark 4. Note that in the diffusion dominated case s is Oðh2=jÞ, and in the advection dominated case sis Oðh=jajÞ. Consequently in both the limit cases s possesses the right order [10,18].

3.4. The coarse scale problem (W)

Employing the linearity of the solution slot in the coarse scale sub-problem (14) we get

ðw; a � r�vÞ þ ðw; a � rv0Þ þ ðrw; jrvÞ þ ðrw; jrv0Þ ¼ ðw; f Þ: ð26ÞApplying integration by parts to the second and the fourth terms, applying conditions (2) and (9), and then

combining all the v0 terms we get

ðw; a � r�vÞ þ ðrw; jr�vÞ � ða � rwþ jDw; v0Þ ¼ ðw; f Þ: ð27Þ

Substituting v0 from (24) in (27)

ðw; a � r�vÞ þ ðrw; jr�vÞ þ ða � rwþ jDw; sða � r�v� jD�vÞÞ ¼ ðw; f Þ þ ðsða � rwþ jDwÞ; f Þ: ð28Þ

3.5. The HVM form

The stabilized form (28) is completely expressed in terms of the coarse/resolvable scales in the problem.

Therefore, in order to keep the notation simple we drop the superposed bars and write the resulting HVM

form as

ðw; a � rvÞ þ ðrw; jrvÞ þ ða � rwþ jDw; sða � rv� jDvÞÞ ¼ ðw; f Þ þ ðsða � rwþ jDwÞ; f Þ: ð29Þ

Remark 5. It is remarkable that up to the definition of s, Eq. (28) derived above is identical to the modified

GLS form first proposed in [10].

Remark 6. When compared with the standard Galerkin method, the multiscale approach involves addi-

tional integrals that are evaluated element wise. These additional terms represent the effects of the sub-gridscales that are being modeled in (29) in terms of the residuals of the coarse scales of the problem.

Remark 7. The sub-grid scales are proportional to the residual of the coarse scales, i.e., it is a residual based

method and therefore satisfies consistency ab initio.

3.6. The stability function

The stability function is given in Eq. (25). We can write (25) as

s ¼ be1

ZXebe2 dXs; ð30Þ

Fig. 1. 1-D bubbles for the weighting function and trial solution.


where

s ¼ ½ðbe2; a � rbe1Þ þ ðrbe2; jrbe1Þ��1; ð31Þ

where be1 is the bubble function for the trial solution and be2 represents the bubble for the weighting function.In our numerical implementation we have used the absolute value of the advection velocity, i.e., ja1j andja2j. Furthermore, in order to keep s positive, we take the absolute values of both the terms in the brackets.This restriction can however be relaxed by a better choice of the bubble for the fine scale weighting function

w0.

The bubble functions for 1-D trial solutions v0 and weighting function w0 are presented in Fig. 1, and

are given as follows.

b1ðxÞ ¼ ð1� x2Þ; �16 x6 1 ð32Þand

b2ðxÞ ¼

1

1þ Xb

� �ðxþ 1Þ; �16 x6Xb;

�1

1� Xb

� �ðx� XbÞ þ 1; Xb 6 x6 1;

8>>><>>>:

ð33Þ

where Xb is the location of the internal virtual node for the piecewise linear bubble for w0.

Remark 8. The bubble for the fine scale weighting function is piecewise continuous over Xe, but is not in C1

[15]. Therefore this function is differentiable in a distributional sense only.

We have employed this idea to develop piecewise bilinear functions for the fine scale weighting functionfor quadrilaterals and triangles. These functions are presented in Appendix A.

4. Numerical examples

We have developed a family of 2-D elements comprising 3 and 6 node triangles and 4 and 9 node

quadrilaterals (see Fig. 2). The bubble functions employed to compute s in (31) are given in Appendix A.

Fig. 2. A family of 2-D linear and quadratic elements.


In these functions Xb and Yb represent the location of the internal virtual node. In the numerical simulations

presented here, we have used ðXb; YbÞ ¼ ð�0:99;�0:99Þ for quadrilaterals, and ðXb; YbÞ ¼ ð0:34; 0:34Þ for

triangles.

4.1. Rate of convergence study

The first numerical simulation is a study of convergence rates on a model problem. The domain under

consideration is a biunit square with origin at ðx; yÞ ¼ ð0; 0Þ. The exact solution is assumed of the following

form.

u ¼ sin2pxL

sin2pyL

: ð34Þ

The forcing function as evaluated from Eq. (1) is given below.

f ðxÞ ¼ 2pL

a1 cos2pxL

sin2pyL

�þ a2 sin

2pxL

cos2pyL

þ 4pjL

sin2pxL

sin2pyL

�: ð35Þ

In specifying the boundary-value problem, the forcing function is integrated over X while w ¼ ujC is

prescribed nodally at the boundary.

The rate of convergence study is divided into two parts. In the first part we show the rates for low Peclet

number flows, and in the second part we present convergence rates for the high Peclet number flows.

Throughout these convergence rate studies, we have taken j ¼ 1. Fig. 3a and b present the elevation plots

of the forcing functions for low and high Peclet numbers, respectively.

4.1.1. Convergence study at low Peclet numbers

This section presents the rate of convergence study for the (i) uniform, (ii) graded, and (iii) composite

mesh cases, at low Peclet number. The convective velocity in this study is a1 ¼ a2 ¼ 10. The corresponding

forcing function is presented in Fig. 3a.

Case 1. Uniform meshes

Fig. 4a and b show representative meshes for the linear quadrilateral and triangular elements, while Fig.

5a and b show the representative meshes for quadratic elements. For linear quadrilateral elements, themeshes employed consisted of 100, 400, 1600 and 6400 elements. The linear triangular element meshes

consisted of exactly twice as many elements. For quadratic quadrilateral elements, the meshes employed

consisted of 100, 400, 1600 and 6400 elements. Again, the quadratic triangular element meshes consisted of

Fig. 3. Elevation plot of the forcing function for (a) low Pe (a1 ¼ a2 ¼ 10) and (b) high Pe (a1 ¼ a2 ¼ 106).

(a) (b)

Fig. 4. Meshes of (a) 800 3-node triangles and (b) 400 4-node quadrilaterals.

(a) (b)

Fig. 5. Meshes of (a) 200 6-node triangles and (b) 100 9-node quadrilaterals.


Fig. 6. Convergence rates for linear elements (uniform meshes, low Pe).


twice as many elements. The element mesh parameter, h, is taken to be the edge length of the elements for

the quadrilaterals, and the short-edge length for triangles.

Convergence rates are presented in Figs. 6 and 7. Fig. 6 shows the L2-norm and H 1-seminorm of the

scalar field for the linear quadrilateral and triangle. We see that optimal rates are achieved. Fig. 7 presents

the corresponding rates for the quadratic elements. Once again we see optimal convergence rates. Fig. 8a

and b present the elevation plots of the exact and the computed fields for a representative mesh of 4-nodeelements. We have also plotted the contours of the scalar field for 3-node triangles and the 9-node quad-

rilaterals in Fig. 9a and b, respectively. These plots present a qualitative comparison of the computed

solution with the corresponding exact solution, and show a stable response of the elements. Similar plots

were obtained for the 6-node triangles and are therefore not shown here.

Case 2. Distorted structured meshes

We have performed convergence study over structured, distorted, graded meshes composed of 3- and 6-

node triangles and 4- and 9-node quadrilaterals. The representative graded meshes for linear and quadratic

Fig. 7. Convergence rates for quadratic elements (uniform meshes, low Pe).

Fig. 8. (a) Elevation plot of the scalar field (exact solution). (b) Elevation plot of the scalar field for 4-node elements (20· 20 element

mesh).

Fig. 9. Contour plot of the scalar field for (a) 3-node triangles and (b) 9-node quadrilateral (low Pe).


elements are shown in Figs. 10a,b and 11a,b, respectively. The degree of distortion has been kept the samefor all the element types. In this study the convective velocity a1 ¼ a2 ¼ 10, and the corresponding forcing

function is presented in Fig. 3a. In all these cases convergence is attained at optimal rates and is presented in

Figs. 12 and 13 for the linear and the quadratic elements, respectively. Fig. 14a and b presents the contours

of the scalar field for the 4-node and the 9-node quadrilaterals, respectively. These plots present a quali-

tative comparison of the computed solution and show a stable response of the elements. Similar plots were

obtained for the triangular elements and are therefore not shown here.

Case 3. Composite meshes

This study is interesting from a practical viewpoint where different element types are used in the same

discretization. Fig. 15a presents a composite mesh composed of linear elements, i.e., combination of 3-node

triangles and 4-node quadrilaterals in the same computational domain. Similarly, Fig. 15b presents the

(a) (b)

Fig. 11. Meshes of (a) 200 6-node triangles and (b) 100 9-node quadrilaterals (distorted meshes).

(a) (b)

Fig. 10. Meshes of (a) 800 3-node triangles and (b) 400 4-node quadrilaterals (distorted meshes).

Fig. 12. Convergence rates for linear elements (distorted meshes, low Pe).


Fig. 13. Convergence rates for quadratic elements (distorted meshes, low Pe).

Fig. 14. Contour plot of the scalar field for (a) 4-node quadrilateral and (b) 9-node quadrilateral (distorted meshes, low Pe).


corresponding composite mesh for the quadratic elements. Fig. 16 presents the contour plot of the scalar

field for the linear composite mesh. A similar plot was obtained for the composite mesh composed ofquadratic elements and is therefore not shown here. The rate of convergence is a function of the order

of interpolation polynomials and once again we get optimal rates as presented in Figs. 17 and 18.

4.1.2. Convergence study at high Peclet numbers

In this section we repeat the rate of convergence study for the (i) uniform, (ii) graded, and (iii) composite

mesh cases, at high Peclet number. The convective velocity in this study is a1 ¼ a2 ¼ 106. The corresponding

forcing function is presented in Fig. 3b.

Case 1. Uniform meshes

Meshes for this study are shown in Figs. 4 and 5. Convergence rates are presented in Figs. 19 and 20. Fig.

19 shows the L2-norm and H 1-seminorm of the scalar field for the linear quadrilateral and triangle. We see

Fig. 16. Contour plot of the scalar field for composite mesh of linear elements (low Pe).

Fig. 17. Convergence rates for composite mesh of linear elements (low Pe).

(a) (b)

Fig. 15. Composite mesh of (a) linear elements and (b) quadratic elements.


Fig. 18. Convergence rates for composite mesh of quadratic elements (low Pe).

Fig. 19. Convergence rates for linear elements (uniform meshes, high Pe).


that optimal rates are achieved. Fig. 20 presents the corresponding rates for the quadratic elements thatshow some degradation in the rate because of the high Peclet number.

Case 2. Distorted structured meshes

The representative graded meshes for linear and quadratic elements are shown in Figs. 10a,b and 11a,b,

respectively. Figs. 21 and 22 present the convergence rates. Once again we obtain optimal convergence for

the low order elements. Fig. 23a and b presents the contours of the scalar field for the 4-node and the 9-

node quadrilaterals, respectively, and show a stable response of the elements.

Case 3. Composite meshes

The composite linear- and quadratic meshes are shown in Fig. 15a and b. The rate of convergence is a

function of the order of interpolation polynomials and once again we obtain optimal convergence rates for

Fig. 20. Convergence rates for quadratic elements (uniform meshes, high Pe).

Fig. 21. Convergence rates for linear elements (distorted meshes, high Pe).


the linear composite mesh composed of 3-node triangles and 4-node quadrilaterals (see Fig. 24). Theconvergence rates for the quadratic mesh are shown in Fig. 25.

4.2. Advection in a rotating flow field

This numerical simulation is a test of very high Peclet number flows and is used to assess solutions which

are essentially purely advective in nature. The problem is defined on a unit square of coordinates �0:56 x,y6 þ 0:5, where the flow velocity components are given by

a1 ¼ �y; a2 ¼ x: ð36ÞAlong the external boundary v ¼ 0 and along the internal boundary (OA),

v ¼ 12ðcosð4py þ pÞ þ 1Þ; �0:56 y6 0: ð37Þ

Fig. 22. Convergence rates for quadratic elements (distorted meshes, high Pe).

Fig. 23. Contour plot of the scalar field for (a) 4-node quadrilateral and (b) 9-node quadrilateral (distorted meshes, high Pe).


A schematic diagram of the problem statement is shown in Fig. 26. The diffusivity is j ¼ 10�6. A uniform

mesh with 30 · 30 elements is employed. Elevation plots are shown in Fig. 27a and b for the HVM and the

Galerkin methods, respectively. This problem has a smooth exact solution and therefore both methods

perform well. There are however small amplitude oscillations in the Galerkin method that can be seenin Fig. 27b.

4.3. Thermal boundary layer problem

This problem can be viewed as the simulation of a thermal boundary layer on a fully developed flow

between two parallel plates, where the bottom plate is fixed and the top plate is moving with a unit velocity.

The computational domain has dimensions Lx ¼ 1:0 and Ly ¼ 0:5. Schematic diagram of the problem

is given in Fig. 28. Boundary conditions are given as follows.

Fig. 24. Convergence rates for composite mesh of linear elements (high Pe).

Fig. 25. Convergence rates for composite mesh of quadratic elements (high Pe).


u ¼ 1x ¼ 0; 0 < y6 0:5;y ¼ 0:5; 06 x6 1;

�ð38Þ

u ¼ 0; y ¼ 0; 06 x6 1:0; ð39Þ

u ¼ 2y; x ¼ 1; 0 < y < 0:5: ð40ÞThe flow components are

a1 ¼ 2y; a2 ¼ 0 in X ð41Þand the diffusivity is j ¼ 7� 10�4. Taking the top plate velocity as the characteristic flow velocity, we have

a Peclet number Pe ¼ uLy=j ¼ 714 for this flow.

Fig. 26. Schematic diagram of the rotating flow field problem.

Fig. 27. Rotating flow problem. Elevation plot of the scalar field for the (a) HVM method and (b) Galerkin method.

Fig. 28. Schematic diagram of the thermal boundary layer problem.


Fig. 29. Thermal boundary layer problem (a) HVM and (b) GLS.


We employ a mesh consisting of 21 equally spaced nodes in the x-direction, 11 nodes uniformly dis-

tributed in the interval 06 y6 0:1 and the same number of nodes equally spaced on 0:16 y6 0:5. Thisamounts to subdividing the domain into two different regions. Fig. 29a presents the elevation profile for the

HVM method. For comparison we have plotted the elevation profile for the GLS method [10] in Fig. 29b.

5. Conclusions

We have presented a new stabilized formulation for the advection–diffusion equation. The key point of

the proposed formulation is a multiscale decomposition of the scalar field in the coarse and the fine scales.

A significant feature of the present method is that it is free of user-defined stability parameters. The

multiscale approach has several advantages over other stabilized formulations; (i) the stabilization term

arises naturally, (ii) it is not restricted to a particular sub-grid model, and (iii) incorporates the effect of the

fine (sub-grid) scales onto the coarse (grid) scales. When compared with standard Galerkin method, themultiscale method involves additional integrals that are evaluated element wise and incorporate the effect of

the sub-grid scales on the coarse scales. The sub-grid scales are proportional to the residual of the coarse

scales, i.e., the method is residual based and therefore, it is automatically consistent. We have developed

a family of triangular and quadrilateral elements based on the proposed formulation. Numerical results

show the good performance of the method and yield optimal convergence rates in the norms considered.

Acknowledgements

Support for this work was provided by the ONR grant N00014-02-1-0143. This support is gratefully

acknowledged. The authors wish to thank the anonymous reviewers for helpful comments.

Appendix A. Bubble functions for the 2-D cases

A.1. Quadrilaterals

The bubble for the fine scale trial solution is the usual quadratic bubble defined as

b1ðx; yÞ ¼ ð1� x2Þð1� y2Þ: ðA:1Þ

Fig. 30. Schematic diagram showing the domains for the piecewise functions for (a) quadrilaterals and (b) triangles.


For the fine scale weighting function, the element domain is divided into four regions (see Fig. 30a) and thepiecewise functions on these regions are given as follows.

b2ðx; yÞ ¼

xXbð1þ YbÞ

� �ð1þ yÞ for x; y in region 1;

yYbð1� XbÞ

� �ð1� xÞ for x; y in region 2;

xXbð1� YbÞ

� �ð1� yÞ for x; y in region 3;

yYbð1þ XbÞ

� �ð1þ xÞ for x; y in region 4;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ðA:2Þ

where Xb and Yb represent the location of the internal virtual node in Xe.

A.2. Triangles

The bubble for the fine scale trial solution is a quadratic bubble defined as

b1ðx; yÞ ¼ 27xyð1� x� yÞ: ðA:3ÞFor the fine scale weighting function, the element domain is divided into three regions (see Fig. 30b) andthe piecewise functions on these regions are given as follows.

b2ðx; yÞ ¼

xyXbYb

� �for x; y in region 1;

1� x� y1� Xb � Yb

� �for x; y in region 2;

xyXbYb

� �for x; y in region 3:

8>>>>>>><>>>>>>>:

ðA:4Þ

References

[1] C. Baiocchi, F. Bezzi, L.P. Franca, Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S), Comput. Methods Appl.

Mech. Engrg. 105 (1993) 125–141.


[2] F. Brezzi, M.O. Bristeau, L.P. Franca, M. Mallet, G. Roge, A relationship between stabilized finite element methods and the

Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg. 96 (1) (1992) 117–129.

[3] F. Brezzi, L.P. Franca, T.J.R. Hughes, A. Russo, b ¼Rg, Comput. Methods Appl. Mech. Engrg. 145 (3–4) (1997) 329–339.

[4] F. Brezzi, L.P. Franca, A. Russo, Further considerations on residual-free bubbles for advective–diffusive equations, Comput.

Methods Appl. Mech. Engrg. 166 (1–2) (1998) 25–33.

[5] F. Brezzi, P. Houston, D. Marini, E. Suli, Modeling subgrid viscosity for advection–diffusion problems, Comput. Methods Appl.

Mech. Engrg. 190 (2000) 1601–1610.

[6] F. Brezzi, D. Marini, A. Russo, Application of the pseudo residual-free bubbles to the stabilization of convection–diffusion

problems, Comput. Methods Appl. Mech. Engrg. 166 (1998) 51–63.

[7] A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular

emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1–3) (1982) 199–259.

[8] L.P. Franca, C. Farhat, Bubble functions prompt unusual stabilized finite element methods, Comput. Methods Appl. Mech.

Engrg. 123 (1–4) (1995) 299–308.

[9] L.P. Franca, C. Farhat, M. Lesoinne, A. Russo, Unusual stabilized finite element methods and residual free bubbles, Int. J.

Numer. Methods Fluids 27 (2) (1998) 159–168.

[10] L.P. Franca, S.L. Frey, T.J.R. Hughes, Stabilized finite element methods: I. Application to the advective–diffusive model,

Comput. Methods Appl. Mech. Engrg. 95 (2) (1992) 253–276.

[11] L.P. Franca, A. Nesliturk, M. Stynes, On the stability of residual-free bubbles for convection–diffusion problems and their

approximation by a two-level finite element method, Comput. Methods Appl. Mech. Engrg. 166 (1-2) (1998) 35–49.

[12] L.P. Franca, F. Valentin, On an improved unusual stabilized finite element method for the advective–reactive–diffusive equation,

Comput. Methods Appl. Mech. Engrg. 190 (2000) 1785–1800.

[13] I. Harari, L.P. Franca, S.P. Oliveira, Streamline design of stability parameters for advection–diffusion problems, J. Comput. Phys.

171 (1) (2001) 115–131.

[14] T.J.R. Hughes, Multiscale phenomena: Green�s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles

and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401.

[15] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewoods

Cliffs, NJ, 1987, Dover edition, 2000.

[16] T.J.R. Hughes, A.N. Brooks, Streamline-upwind/Petrov–Galerkin methods for advection dominated flows, in: Proceedings of the

Third International Conference on Finite Element Methods in Fluid Flow, Banff, June 1980, pp. 283–292.

[17] T.J.R. Hughes, G.R. Feijoo, M. Luca, Q. Jean-Baptiste, The variational multiscale method––a paradigm for computational

mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1–2) (1998) 3–24.

[18] T.J.R. Hughes, L.P. Franca, G.M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The

Galerkin/least-squares method for advective–diffusive equations, Comput. Methods Appl. Mech. Engrg. 73 (2) (1989) 173–189.

[19] A. Masud, T.J.R. Hughes, A stabilized mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Engrg. 191/

39–40 (2002) 4341–4370.

[20] A. Masud, On a stabilized finite element formulation for incompressible Navier–Stokes equations, in: Proceedings of the Fourth

US–Japan Conference on Computational Fluid Dynamics, Tokyo, Japan, May 28–30, 2002.

[21] M. Ayub, A. Masud, A new stabilized formulation for convective–diffusive heat transfer, Numer. Heat Transfer Part B 44 (2003)

1–23.

[22] T.E. Tezduyar, Y.J. Park, Discontinuity-capturing finite element formulations for nonlinear convection–diffusion–reaction

equations, Comput. Methods Appl. Mech. Engrg. 59 (1986) 307–325.

A multiscale/stabilized ﬁnite element method for the...

Documents

Transcript of A multiscale/stabilized ﬁnite element method for the...