ADAPTIVE ERROR CONTROL FOR FINITE ELEMENT

APPROXIMATIONS OF THE LIFT AND DRAG COEFFICIENTS IN

VISCOUS FLOW

MICHAEL GILES MATS G LARSON J MARTEN LEVENSTAM AND ENDRE SULI

Abstract We derive estimates for the error in a variational approximation of the lift

and drag co ecients of a b o dy immersed into a viscous ow governed by the Navier

Stokes equations The variational approximation is based on computing a certain weighted

average of a nite element approximation to the solution of the NavierStokes equations

Our main result is an a posteriori estimate that puts a b ound on the error in the lift

and drag co ecients in terms of the lo cal mesh size a lo cal residual quantity and a lo cal

weight describing the lo cal stability prop erties of an asso ciated linear dual problem The

weight may b e approximated by solving the dual problem numerically The error b ound is

thus computable and can b e used for quantitative error estimation we apply it to design

an adaptive nite element algorithm sp ecically for the approximation of the lift and drag

co ecients

Introduction

Often the purp ose of numerical computations in mathematical mo delling of physical

phenomena is to approximate a functional of the analytical solution to a dierential equa

tion represented as an integral average of the solution rather than to obtain accurate

p ointwise values of the solution itself in such instances solving the dierential equation

considered is only an intermediate stage in the pro cess of computing the primary quan

tities of concern For example in uid dynamics one may b e concerned with calculating

the lift and drag co ecients of a b o dy immersed into a viscous incompressible uid whose

ow is governed by the NavierStokes equations The lift and drag co ecients are dened

as integrals over the b oundary of the b o dy of the stress tensor comp onents normal and

tangential to the ow resp ectively Similarly in elasticity theory the quantities of prime

interest such as the stress intensity factor or the moments of a shell or a plate are derived

quantities

The ob jective of this pap er is to obtain a posteriori error b ounds for nite element ap

proximations of functionals that arise in uid dynamics sp ecically we shall b e concerned

with the construction and the a posteriori error analysis of nite element approximations

Mathematics Subject Classication N N N

Key words and phrases NavierStokes equations lift and drag co ecients adaptive nite element

metho d a posteriori error estimates mesh renement

Submitted to SIAM J Num Anal

M GILES M LARSON M LEVENSTAM AND E SULI

to the lift and drag co ecients in a viscous incompressible ow and the implementa

tion of these b ounds into an adaptive nite element algorithm with reliable and ecient

quantitative control of the error

It is frequently the case that the functional under consideration may b e expressed in

various forms which are mutually equivalent at the continuous level but result in very

dierent approximations under discretisation Thus it is imp ortant to select the appropriate

representation of the functional b efore formulating its discretisation While this basic

idea has b een widely exploited in structural mechanics see and and heat

conduction see in p ostpro cessing nite element approximations it do es not seem to

have p enetrated the eld of computational uid dynamics

We approximate the solution u u p of the NavierStokes equations where u rep

resents the velo city of the uid and p its pressure by means of a nite element metho d

using piecewise p olynomials of degree k for u and degree k for p Let N u denote the

b oundary integral representing the lift or drag co ecient dep ending on the choice of the

function dened on the b oundary of the computational domain the precise denition of

N u will b e given in Section Using the dierential equation we may express N u

in a variational form involving test functions that are equal to on the b oundary of the

h

domain this form will b e the basis for constructing an accurate approximation N u to

h

N u Provided the b oundary is suciently smo oth and there are no variational crimes

h

the order of convergence of N u to N u is k while for the naive direct approximation

h

N u the order of convergence is typically only k

h

h

Our main result is a weighted a posteriori estimate for the error N u N u of the

h

form

X

h

R u h u j c jN u N

h h

T

h

where the sum is taken over the elements in a triangulation T of the computational

h

domain h is the diameter of given that is a real numb er with k R u is

h

an element residual quantity is a lo cal weight and c c is a p ositive constant The

element residual quantity R u is a computable b ound on the actual residual obtained

h

by inserting the approximate solution into the dierential equation The derivation of

h

u in terms the estimate is based on a representation formula for the error N u N

h

of the computed solution u and the solution to an asso ciated linear dual equation with

h

homogeneous righthand side and nonhomogeneous b oundary data The weight

describ es the lo cal size of derivatives of order of the solution to the dual problem The

data for the dual problem is known and therefore the weight can b e computed by solving

the dual problem numerically Therefore the righthand side of the estimate is computable

and can b e applied to design an adaptive nite element algorithm for approximating N u

h

u The fact stated and to ensure ecient quantitative control of the error N u N

h

h

u is k follows from the error representation ab ove that the order of convergence of N

h

formula together with an a priori estimate of the global error u u in the nite element

h

metho d Indeed for a linear elliptic mo del problem it can b e shown using results obtained

in that the a posteriori estimate is sharp

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

In a weighted a posteriori estimate is given for the error in the lift and the drag

however unlike our approach it is based on a direct approximation using N u For a

h

posteriori estimates of the error u u in nite element approximations of the solution to

h

the NavierStokes equations in various norms see and references therein

a general intro duction to a posteriori error analysis is given in Finally we mention

that an a priori error analysis of the approximation metho d employed here for a linear

convectiondiusion equation is presented in

The remainder of this pap er is organized as follows in order to illustrate the key ideas

in Section we outline the theory for a linear elliptic mo del problem In Section we

consider the extension of this analysis to the Stokes system which mo dels the ow of a

viscous incompressible uid at low velo cities we derive an a posteriori estimate of the error

in a nite element approximation to the lift and drag co ecients In Section we further

extend our results to the NavierStokes equations and present numerical computations for

the drag co ecient of a cylinder in a channel these illustrate the quality of the adaptive

error control based on the a posteriori error b ound Finally in Section we summarise

our results and draw some conclusions

A model problem

The mo del problem and the nite element metho d We b egin by intro ducing

d

some notation Let b e a b ounded domain in R d or with Lipschitzcontinuous

d 2

b oundary For an op en set K in R let L K signify the space of realvalued square

s

integrable functions on K with norm k k H K will denote the Sob olev space of real

K

s s

index s equipp ed with the norm k k and corresp onding seminorm j j with a

H (K ) H (K )

s

s

slight abuse of the notation we shall frequently write kD uk instead of juj s

K

H (K )

12 1 1

Given that is an element of H we let H H denote the space of all v in

1 1

H H which satisfy the Dirichlet b oundary condition v Finally we adopt

j

the following notational convention generic constants that are indep endent of the problem

and the mesh size are denoted by c while constants that dep end on the problem but not

on the mesh size are lab elled C with a subscript when necessary

As a rst mo del problem we consider the b oundaryvalue problem

r u f in u on

2 2

where f L L and u Aru with A a uniformly p ositive denite d d

matrix with continuous realvalued entries dened on This problem has a unique

1

In order to dene the corresp onding nite element approximation weak solution u H

0

h 1

we intro duce a family of nitedimensional spaces V contained in H which consist of

continuous piecewise p olynomials of degree k dened on a triangulation T of We denote

h

the diameter of a triangle T by h It will always b e assumed that the triangulation is

h

d

shap eregular ie there exists a p ositive constant c such that vol c h where vol is

12

the ddimensional volume of Further for each function H such that v j

h h h h

for some v V we let V V b e the space of all w V with w j In particular

h h

V is the space of all v V which vanish on the b oundary 0

M GILES M LARSON M LEVENSTAM AND E SULI

h

The nite element approximation of reads nd u V such that

h

0

h

au v f v for all v V

h

0

2

Here and b elow denotes the scalar pro duct in L and av w v rw

T 1 T

rv A rw for v w H where A is the transp ose of A

Approximation of the b oundary ux In this section we shall construct a nite

element approximation to the b oundary ux

Z

N u n u ds

1

where n denotes the unit outward normal vector to In order to do so for u H

0

12

denoting the weak solution to problem and H we consider the weighted

b oundary ux dened by

Z

N u n u ds

2 d 2

We note that since u L and r u L according to the trace theorem

12

see Theorem in n uj is correctly dened as an element of H and

N u is meaningful provided that the integral over is interpreted as a duality pairing

12 12

b etween H and H Moreover applying a generalisation of Greens Identity

1

see Theorem in we deduce that for any v H

N u u rv f v au v f v

Clearly the value of the expression au v f v on the righthand side is indep endent of

1

the choice of v H From now on for the sake of simplicity we shall always assume that

h

v j for some v V in other words will b e supp osed to b e a continuous piecewice

p olynomial of degree k dened on This assumption will b e satised in our applications

h

Motivated by the identity we dene the approximation N u to N u as follows

h

h h

N u au v f v v V

h h

h h

u is indep endent of the choice of v V Furthermore We note that b ecause of N

h

we observe that in general

Z

h

N u n u ds N u

h h h

in contrast with identity satised by the analytical solution u However we shall

h

show b elow that N u is the appropriate approximation for N u rather than N u

h h

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

Error representation using duality In order to derive a representation formula

h

for the error N u N u in the b oundary ux we intro duce the following dual problem

h

1

in variational form nd H such that

1

av for all v H

0

Consider the global error e u u Setting v e in we obtain

h

ae ae ae

1 h

where we made use of the fact that the error e is zero on the b oundary here H V

is a linear op erator satisfying the approximation prop erty b elow Since the denitions

h 1 h

of N u and N u are indep endent of the choice of v H and v V resp ectively

h

we deduce that

h

u ae au f au f N u N

h h

Thus we arrive at the error representation formula

h

N u N u ae au f

h h

where to obtain the last equality we exploited the fact that u is the weak solution of

1

and that b elongs to H

0

The a posteriori and a priori error estimates In this section we intro duce

a residual quantity asso ciated with the approximate solution u and derive a weighted

h

a posteriori estimate for the error in the approximation to the b oundary ux We then

investigate the order of convergence of the approximation through an a priori error analysis

Two basic hyp otheses will b e required First we assume the following lo cal approxima

1 h

tion prop erty there is a linear op erator H V such that

s s

kv v k h kD v v k ch kD v k s k

for each T and c a p ositive constant indep endent of v and h We note here

h

that without altering the basic error analysis that will follow the righthand side in this

s s

kD v k where S is the union of all elements in inequality can b e replaced by ch

S ( )

S ( )

the partition whose closure has nonempty intersection with the closure of and

h max h

S ( )

T ; S ( )

h

an approximation prop erty of this kind would arise if one to ok v to b e the quasi

interp olant of v see for example For the sake of notational simplicity we shall refrain

from doing this and assume instead Second we make an assumption concerning the

regularity of the dual problem there is a t such that for every s s t there

is a constant C such that the solution to the dual problem satises the following

s

estimate

s s12

kD k C kD k

s

M GILES M LARSON M LEVENSTAM AND E SULI

s12 s11

whenever H For instance this b ound holds when C and the entries

[s]

of A b elong to C see For each triangle T and u denoting the solution to

h h

h

in V we intro duce the residual quantity R u by

h

0

12

R u kr u f k h kn u k

h h h

n

where w is the jump in w across the faces of elements in the partition and n is the unit

outward normal vector to

Starting from the error representation and estimating the righthand side using the

CauchySchwarz inequality elementbyelement together with the approximation prop erty

we arrive the following theorem its pro of will b e given in the next section

12

Theorem Let and hold and suppose that H then we

have that

X

h

h R u mint k jN u N u j c

h h

T

h

where c is a constant the local weight is dened by kD k and is the

weak solution of Recal l that the boundary ux N u is dened in the discrete

h

approximation N u in and the residual quantity R u in

h h

Note in particular that the p ower of h max h is determined by the approximation

prop erties of the nite element space the parameter t in the regularity assumption

and the smo othness of the dual solution To highlight the quality of the approximation

h

N u to N u we state the following a priori error estimate its pro of is p ostp oned

h

until the next section

12

Theorem Assume that and hold Supposing that H

and u H k we have that

h + 2

jN u N u j ch kD uk kD k mint k

h

where is the weak solution of c is a constant and h max h

T

h

h

u This estimate shows that for suciently smo oth data the order of convergence of N

h

2k

to N u is O h In general this high order of convergence is not achieved for the naive

R

approximation n u ds At least on quasiuniform meshes the a posteriori b ound

h

stated in Theorem can b e proved to b e sharp by applying the a priori residual estimate

obtained in this shows that

12

X

2 k 1

R u O h

h

T

h

2k

and thus the righthand side of the a posteriori estimate is O h in agreement with

Theorem Furthermore we observe that since the data for the dual problem is

known we may calculate the weight by approximating the solution of the dual problem

numerically The righthand side of the a posteriori error estimate is thus computable and

can b e used for direct quantitative error estimation

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

Pro ofs of Theorems and In this section we present the pro ofs of Theorems

and

Proof of Theorem Starting from the error representation and integrating triangle

bytriangle using Greens identity we have

h

N u N u u r f

h h

Z

X

r u f dx

h

T

h

Z

X

n u ds

h

n

T

h

I I I

where we made use of the fact that the weak solution of the dual problem is in

0

C for some in see Theorem in so that across n

for each element in the partition We now turn to estimating I and II For I we apply

the CauchySchwarz inequality and use the approximation prop erty to obtain

X

s s

h kr u f k kD k jI j c

h

T

h

Next we consider II Applying the multiplicative trace inequality see followed by

the approximation prop erty yields

s12 s

k k ch kD k

Thus

X

s12 s

h kn u k kD k jII j c

h

n

T

h

Substituting the b ounds on I and II into and recalling the denition of the

residual quantity we deduce that

X

s s h

h R u kD k jN u N u j c

h h

T

h

which is the desired estimate

Proof of Theorem It follows from the error representation formula that

h h

jN u N u j ckD ek kD k

A standard energy norm error estimate gives

1

kD ek ch kD uk k

Further using the approximation prop erty we obtain

h + 2

jN u N u j ch kD uk kD k

h

for mink t

M GILES M LARSON M LEVENSTAM AND E SULI

The Stokes problem

We now turn to the approximation of the lift and drag co ecients for a ow governed

by the stationary incompressible NavierStokes equations First we develop the theory

for the Stokes problem which is a linear elliptic system that approximates the Navier

Stokes equations at low velo cities In Section we extend our results to the NavierStokes

equations

The Stokes problem and its nite element approximation The Stokes prob

d

lem describing the stationary motion of a uid in a b ounded domain R d

d

has the form nd u R and p R such that

r u p f in

r u in

u on

where the comp onents of the stress tensor u p are dened by

u p u p i j d

ij ij ij

and u is the strain tensor with entries

u u u i j d

ij j i i j

Here and b elow x In these equations u denotes the velo city vector of the uid

j j

is the constant viscosity co ecient p is the pressure and f denotes the b o dy

1 d 2

c

forces Intro ducing the space W W M where W H and M L R and the

0 0 0

0

bilinear forms

d

X

v w bv q r v q av w

ij ij

ij =1

c

the variational form of the Stokes problem reads nd u u p W such that

0

c

Au v au v bv p bu q f v for all v W

0

Here we made use of the fact that v w v w which follows from the

ij j i ij ij

c c

symmetry of the strain tensor Clearly A is a bilinear form on W W

0 0

The nite element approximation of the Stokes problem has the form nd u

h

h h h

c

u p W W M such that

h h

0 0

h

c

Au v f v for all v W

h

0

d h h h h

and M V In order to ensure that this problem has a unique V where W

0 0

h h

solution it suces to supp ose that the spaces W and M satisfy the infsup condition

0

we note however that the infsup condition do es not enter explicitly into the a

posteriori error analysis that will b e describ ed b elow

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

Approximation of the lift and drag co ecients The traction vector on the

P

d

n where n is the unit outward normal to and b oundary has comp onents

ij j

j =1

12 d

thus given that H the force in the direction on is given by

Z

d

X

N u u n ds

ij j i

ij =1

If is a unit vector parallel with the direction of the ow N u is called the drag on

and if is a unit vector p erp endicular to the direction of the ow N u is referred to

as the lift on If only a part of the b oundary is of concern then can b e taken

0

to have its supp ort in Indeed in most problems of practical interest is a closed

0 0

surface representing the b oundary of an ob ject immersed into the uid and n will

0

then denote the b oundary of the container In addition in the rest of the pap er we shall

assume that

Z

X

d

n ds

i i

i=1

in order to ensure that represents the trace on of a divergencefree function that is in

1 d

H

In complete analogy with the derivation in Section we note that up on multiplying

1 1

integrating over and using the rst equation in by v W H H

1

d

Greens identity we obtain

N u u rv f v Au v f v

where in the last equality we made use of the fact that bu q Further we note that

c

the righthand side is indep endent of the choice of v W Motivated by this formula and

our analysis in Section we dene the following approximation to N u

h

N u u rv f v Au v f v

h h h

h h h h

c

M Again the righthand side is indep endent where v v q W V V

1

d

h

c

of the choice of v W and thus the denition is correct

Error representation using duality For the representation of the error N u

h

c

N u we intro duce the following dual problem in variational form nd W

h

such that

c

Av for all v W

0

12 d

This problem has a unique weak solution provided that H Setting v u u

h

in we obtain

Au u Au u Au u

h h h

M GILES M LARSON M LEVENSTAM AND E SULI

h

c

where we added and subtracted W with and denoting two

1 2 1 2

b ounded linear op erators to b e dened b elow Next we observe that

h

Au u N u N u

h h

Thus we obtain the following error representation formula

h

N u N u Au u Au f

h h h 1

where in the last equality we used the fact that u is the weak solution of and that

1 d

H

0

The a posteriori estimate In order to derive an a posteriori error estimate we

shall make two assumptions First we adopt the following approximation prop erty there

h

c c

exists a linear op erator W W with v v q such that for each T

1 2 h

we have

s s

kv v k h kD v v k ch kD v k s k

1 1

s s

kq q k ch kD q k s k

2

Next we make the following assumption concerning the regularity of the dual problem

there exists t such that for every s s t there is a constant C such that

s

the solution of the dual problem satises the following b ound

s s1 s12

kD k kD k C kD k

s

s12 s11

whenever H For instance this estimate is valid when C see

Starting from the error representation formula and estimating the righthand

side using the same technique as in Theorem we arrive at the following result

Theorem Let u and u be the solutions of and respectively and suppose

h

12 d

that H and that and hold with t then

X

h

h R u mint k jN u N u j c

h h

T

h

where for each triangle T the residual quantity R u is dened by

h h

1 12

R u kr u f k h kr u k h kn u k

h h h h

and the local weight is given by

1

kD k kD k

where is the solution to

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

Proof By considering the error representation formula and integrating triangleby

triangle using Greens identity we have

Z

X

h

r u f dx N u N u

h 1 h

T

h

Z

X

r u dx

h 2

T

h

Z

X

n u ds

h 1

n

T

h

I II I I I

where we made use of the fact that the weak velo citysolution of the dual problem

0

is in C for some in so that across n for each element

1

in the partition Now let us estimate I II and III For I we apply the CauchySchwarz

inequality and use the approximation prop erty to obtain

X

s s

h kr u f k kD k jI j c

h

T

h

Similarly

X

s1 s1

h kr u k kD k jII j c

h

T

h

To deal with III we argue in the same manner as for Term II in the pro of of Theorem

to arrive at the following b ound

X

s12 s

h kn u k kD k jIII j c

h

n

T

h

Substituting the b ounds on I II and III into and recalling the denition of the

residual quantity we deduce the desired a posteriori error b ound

The a priori estimate Pro ceeding in the same manner as in the case of the scalar

elliptic equation discussed in the previous section it is also p ossible to derive an a priori

estimate for the error in the lift and drag co ecients Indeed it follows from the error

representation formula that

h

N u N u au u b p p bu u

h h 1 1 h h 2

where is the dual velo citypressure solution pair Thus by the CauchySchwarz

inequality

h

jN u N u j kD u u k kD k

h h 1

kD k kp p k kD u u k k k

1 h h 2

M GILES M LARSON M LEVENSTAM AND E SULI

h h

Assuming that the pair of nite element spaces W M satises the infsup condition

0

k +1 d

see and that the solution to the Stokes problem and its dual are in H

k

H it follows from this inequality and the approximation prop erties that

h 2k

the error b etween N u and N u is of size O h

h

The incompressible Navier Stokes Equation

Analysis The stationary incompressible Navier Stokes equations have the form nd

d

u R d and p R such that

r u p u ru f in

r u in

u on

P

d

where u rv u v and is the density of the uid The corresp onding weak

i j j i

j =1

c

form reads nd u W such that

0

c

Au u v f v for all v W

0

where

Aw u v au v bv p cw u v bu q

Here the bilinear forms a and b are dened in and the trilinear form c

is given by

cw u v w ru v

We now discretize this problem analogously to the Stokes problem The convection term

may b e dealt with through the use of the streamline diusion metho d for example

indeed in our numerical exp eriments we employ the streamline diusion nite element

metho d However for simplicity of presentation we neglect the stabilizing terms in our

analysis and consider the standard Galerkin nite element metho d instead

The approximation of the lift and drag co ecients follows in exactly the same way as

for the Stokes problem To b e precise the approximation of N u Au u v f v

c

v W is dened by

h

N u Au u v f v

h h h

h

c

with v W where is as in Section

h

For the sake of representing the error N u N u we intro duce the following lin

h

c

earized dual problem in variational form nd W such that

c

Lu u v for all v v q W

h 0

where

Lu u v av b q cu u v bv

h h

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

and

cu u v u r ru v

h h

P

d

u This denition of the linearized dual problem is motivated with ru

j i j h i

j =1

by the following identity

Lu u u u Au u Au u

h h h h

Cho osing v u u in we thus obtain

h

Lu u u u

h h

Au u Au u

h h

Au u Au u

h h

Au u Au u

h h

c

where as ab ove we added and subtracted W Next observing that

h

Au u Au u N u N u

h h h

we nally arrive at the error representation formula

h

u Au u Au u N u N

h h h

Au u f

h h 1

c

where in the last transition we made use of the fact that W and that u is the

0

weak solution of

Estimating the righthand side in a similar fashion as in Theorem we obtain the

following a posteriori estimate

Theorem Let u and u be the solutions of and and assume that

h

12 d

and hold for the solution of Supposing that H

we have that

X

h

h R u mint k jN u N u j c

h h

T

h

where for each triangle T the residual quantity R u is dened by

h h

R u kr u u ru f k

h h h h

1 12

h kr u k h kn u k

h h

and the weight is dened by

s s1

kD k kD k

M GILES M LARSON M LEVENSTAM AND E SULI

A numerical example In this section we present a numerical example illustrating

the practical use of our estimates We consider the computation of the drag co ecient

for a cylinder immersed into a twodimensional viscous incompressible uid in a channel

whose ow is governed by the incompressible NavierStokes equations with prescrib ed

inow velo city noslip conditions on the walls of the channel and the cylinder and free

ow conditions at the outow This problem is one of the b enchmark problems presented

in and the value of the drag co ecient is determined exp erimentally

We approximate the exact ow in the channel by means of the streamline diusion nite

element metho d using stabilised piecewise linear approximation for b oth the velo city u and

the pressure p see for instance The discrete equations are solved using a multigrid

metho d and the adaptive algorithm is designed so that approximately of the triangles

are rened in each step dep ending on the size of h Ru In order to compute the

j

quantities h Ru for each triangle T we solve the dual problem numerically and

j h

approximate the weight using dierence quotients In this case ignoring variational

crimes we exp ect since the b oundary data for the dual problem is smo oth In

Figure we present the computed error b ound given in Theorem and the error in

the drag co ecient as functions of the numb er of degrees of freedom The constant c in

Theorem is chosen equal to The wiggles in the curves arise from the renement

h

u of the triangles and cancellation phenomena in the computation of N

h

In Figure we show the nal grid Note that the renement is lo cated close to the

cylinder which is what we exp ect Furthermore in Figures and we present the level

curves of the velo city parallel to the channel of the ow and the dual ow resp ectively

Note that the dual solution is large close to the cylinder indicating that it is imp ortant

that the residual is small in this area For simplicity we have neglected the inuence of

approximating the curved b oundary and b oundary conditions in the computations

1 10

0 10

−1 10

−2 10 3 4

10 10

Figure The error in the drag co ecient and the a p osteriori b ound in

Theorem as functions of the numb er of degrees of freedom

ERROR CONTROL FOR THE LIFT AND DRAG COEFFICIENTS

Figure The nal mesh

Figure The rst comp onent of velo city of the ow

Figure The rst comp onent of the velo city of the dual ow

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