Dipartimento di Matematica

S. BERRONE, M. MARRO

SPACE-TIME ADAPTIVE SIMULATIONS FOR UNSTEADY

NAVIER-STOKES PROBLEMS

Rapporto interno N. 11, luglio 2008

Politecnico di Torino

Corso Duca degli Abruzzi, 24-10129 Torino-Italia

Space-Time adaptive simulations for unsteady

Navier-Stokes problems

S. Berrone a, M. Marro b

aDipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24,

10129 Torino, Italy

bDipartimento di Ingegneria Aerospaziale, Politecnico di Torino, Corso Duca degli

Abruzzi 24, 10129 Torino, Italy

Abstract

In this paper we apply to the unsteady Navier-Stokes problem some results concern-ing a posteriori error estimate and adaptive algorithms known for steady Navier-Stokes, unsteady heat and reaction-convection-diffusion and unsteady Stokes equa-tions. Our target is to evaluate the real viability of a fully combined space andtime adaptivity for engineering problems. The comparison between our numericalsimulations and literature results demonstrates the accuracy and efficiency of thisadaptive strategy.

Key words: Navier Stokes equations, space-time adaptive techniques, a posteriorierror estimators

1 Introduction

Adaptivity is starting to be an important aspect in many engineering fields andthe large amount of theoretical results and numerical experiments providedin many recent papers (mainly for steady problems) suggests its viability forpractical computations.

Starting from the early paper on adaptivity [2] many fundamental resultsconcerning the derivation of robust estimators and convergent algorithms werepresented. The situation is much more complete for steady problems, whereasfor unsteady problems the situation is still much less settled.

Email addresses: sberrone@calvino.polito.it (S. Berrone),massimo.marro@polito.it (M. Marro).

7 July 2008

The adaptive strategy, especially adaptive in space, is used in many fluiddynamics applications. Cao [14,15] has simulated compressible and incom-pressible Navier-Stokes flows, using mesh adaptation. In particular, boundarylayers, the backward facing step and the inclined air inlet have been simulated.Prudhomme and Oden [43] have tested their adaptive in space methodologyfor the channel flows past a cylinder, considering as a sequent target the esti-mate of the numerical errors with respect to the time-discretization, in orderto control the timestep-length.

Many other discretization techniques, like spectral methods or wavelets dis-cretizations, were developed to highly improve the spatial accuracy for fluiddynamics problems (see for example the recent survey books [13,21,34] or[20]), nevertheless very few results concerning both space and time accuracyare known.

In this paper we try to extend and apply to the unsteady Navier-Stokes prob-lem some results known for the steady Navier-Stokes equation [43,7], the un-steady heat equation [42,50,5,9], the unsteady reaction-convection-diffusionequation [51] and the unsteady Stokes problem [6]. Our target is to evaluatethe real viability of a combined space and time adaptivity for some commontest flows. We do not perform any rigorous analysis and we do not addresssome important problems as: rigorous proof of equivalence between the pro-posed error estimator and the true error (see for example [49] and referencestherein), robustness of the estimates with respect to the Reynolds number[49,7,8], rigorous proof of the splitting between the space error estimator andthe time error estimator [42,5,9] and convergence of the proposed adaptive al-gorithm [22,41]. The proposed error estimators are inherited from the analysisperformed for problems that have many mathematical and physical commonproperties with the unsteady Navier-Stokes equations (steady Navier-Stokes,unsteady heat and reaction-convection-diffusion and unsteady Stokes equa-tions) and from the conclusions driven from numerical investigation performedfor these problems. Another aspect that we do not tackle is the control of theinterpolation error due to the transmission of the solution computed at the endof previous timestep on the new mesh used for the next timestep. A possiblecontrol for this error is provided in [10] using a suitable projection operatorinstead of the simple interpolation operator.

We remark that, although some a posteriori error estimates for unsteady linearproblems are proposed in bibliography, to our knowledge also for these simplerproblems very few numerical experiments are performed to address the con-crete viability of a space-time adaptive approach. For this reason we considerthe present tests with the unsteady Navier-Stokes model of space-time errorestimators and a space-time adaptive algorithm interesting.

The article is organized as follows: in Section 2 we briefly present the con-

2

tinuous problem and the discrete stabilized formulation we apply; in Section3 we propose our space and time error estimators; in Section 4 the adaptivealgorithm is described and, finally, in Section 5 we present numerical resultsobtained with the proposed error estimators and adaptive algorithm and com-pare these results with well settled bibliographic results.

2 Incompressible Navier-Stokes problem

2.1 The continuous problem

Let us denote by Ω ∈ R2 the computational domain with boundary ∂Ω and

(0, Ξ) the time interval of interest. We consider a viscous incompressible New-tonian fluid whose flow is described by time-dependent Navier-Stokes equa-tions, given here in the non-dimensional form:

∂ u

∂t− 1

Re4 u+

(

u · ∇)

u +∇p= f, in Ω× (0, Ξ), (1)

∇ · u = 0, in Ω× (0, Ξ), (2)

where u, p, f represent the velocity, the pressure and the body force, respec-tively; Re is the Reynolds number.

In order that the problem is well-posed, initial and boundary conditions areneeded:

u(·, 0)=u[0], in Ω, (3)

u = gD, on ΓD × (0, Ξ), (4)

1

Re

∂ u

∂n− pn = gN , on ΓN × (0, Ξ), (5)

where ∂Ω = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅; gD, gN and u[0] are given functions; nis the outward unit vector normal to ∂Ω.

2.2 Semidiscrete variational formulation

Let us consider a partition of (0, Ξ) into intervals (t[n−1], t[n]) of length ∆t[n] =t[n] − t[n−1], with 0 = t[0] < t[1] < · · · < t[N ] = Ξ. We perform the timediscretization through a semi-implicit Euler method, i.e. we consider all the

3

terms of the time-discretized equation evaluated at the end of each timestep,but the convective velocity of the non-linear term

(

u · ∇)

u evaluated at thebeginning of the timestep. The main computational advantage of this approachwith respect to the fully implicit method is the linearization of the problemat each timestep. The semi-discrete version of problem (1)-(5) becomes asfollows:

For all n = 1, ..., N, find [u[n], p[n]] ∈ V×Q such that

(

u[n]−u[n−1]

∆t[n], v

)

+1

Re

(

∇u[n],∇ v)

+(

(u[n−1] ·∇) u[n], v)

−(

p[n],∇ · v)

=(

f [n], v)

+(

g[n]N , v

)

ΓN

, ∀ v ∈ V, (6)(

q,∇·u[n])

= 0, ∀q ∈ Q, (7)

where (., .) denotes the usual inner product in L2(Ω) or in [L2(Ω)]2 and (., .)ΓN

denotes the inner product in [L2(ΓN)]2 and

H10,D(Ω) =

v ∈ H1(Ω) : v|ΓD= 0

, L20(Ω) =

q ∈ L2(Ω) :∫

Ωq dΩ = 0

,

V = [H10,D(Ω)]2, Q =

L20(Ω) if |ΓN | = 0,

L2(Ω) if |ΓN | > 0.

We use the symbol ‖·‖0 to denote the L2-norm and |·|1 for the H10,D semi-norm.

We do not discuss important aspects like definition of the suitable functionalsetting and existence and uniqueness of the solution of the continuous problemneither of the semi-discrete problem (6)-(7). For these aspects we refer toDautray and Lions [19], Ciarlet and Lions [18], Temam [47,48].

2.3 Space discretization

Let Ω[n]h be a standard finite element triangulation of the computational do-

main Ω with h[n] the maximum of the diameters h[n]T of each triangle T ∈ Ω

[n]h .

We assume the triangulation satisfying the standard regularity and conformityconditions [17].

We define the following conforming finite element spaces:

V[n]h =

vh ∈ V ∩[

C0(Ω)]2

: v|T ∈ [Pk(T )]2, ∀T ∈ Ω[n]h

⊂ V,

Q[n]h =

qh ∈ Q ∩ C0(Ω) : qh|T∈ Pl(T ), ∀T ∈ Ω

[n]h

⊂ Q,

4

where Pi(T ) is the space of polynomials of degree i ≥ 1.

The fully discrete Galerkin formulation of the Navier-Stokes equations guar-antees the existence and uniqueness of the solution [u

[n]h , p

[n]h ] ∈ V

[n]h ×Q

[n]h for

each timestep if the velocity and the pressure satisfy the discrete coercivityinequality and the inf-sup or Babuska-Brezzi condition [12], respectively. Inorder to guarantee always the discrete inf-sup condition for any choice of fi-nite elements for the velocity and the pressure and to damp the oscillations inthe velocity field generated by the advective operator (u

[n−1]h · ∇) u

[n]h , we sta-

bilize the problem with the Streamline Upwind/Petrov-Galerkin formulation(SUPG) [25,24,32].

For all n = 1, ..., N, find [u[n]h , p

[n]h ] ∈ V

[n]h ×Q

[n]h such that

u[n]h −u

[n−1]h

∆t[n], vh

+1

Re

(

∇u[n]h ,∇ vh

)

+(

(u[n−1]h ·∇) u

[n]h , vh

)

−(

p[n]h ,∇· vh

)

+∑

T∈Ω[n]h

τT

u[n]h −u

[n−1]h

∆t[n]− 1

Re4 u

[n]h +(u

[n−1]h · ∇) u

[n]h +∇p

[n]h , (u

[n−1]h · ∇) vh

T

=(

ΠT f[n]

, vh

)

+(

ΠE g[n]N , vh

)

ΓN

+∑

T∈Ω[n]h

τT

(

ΠT f[n]

, (u[n−1]h · ∇) vh

)

T

, ∀ vh ∈ V[n]h

(8)

(

qh,∇ ·u[n]h

)

+∑

T∈Ω[n]h

τT

u[n]h −u

[n−1]h

∆t[n]− 1

Re4 u

[n]h +(u

[n−1]h · ∇) u

[n]h +∇p

[n]h ,∇qh

T

=∑

T∈Ω[n]h

τT

(

ΠT f[n]

,∇qh

)

T

, ∀qh ∈ Q[n]h . (9)

We consider approximations of the data f[n]

, g[n]N by some projections ΠT f

[n],

ΠE g[n]N . The stability parameter τT is a positive constant for each triangle T

and depends on the local conditions of the flow in each element [24,31].

τT =

mk

(

h[n]T

)2

8Re, 0 ≤ ReT < 1,

h[n]T

2∥

∥

∥u[n]h

∥

∥

∥

∞,T

, ReT ≥ 1,(10)

where

ReT = mk

∥

∥

∥ΠT u[n]h

∥

∥

∥

∞,Th

[n]T

4 1Re

(11)

5

and mk = min

13, 2

C∗

, C∗ being the constant of the inverse inequality [31]:(

h[n]T

)2 ‖4vh ‖20,T ≤ C∗ ‖∇vh ‖20,T , ∀ vh ∈ V[n]h , ∀T ∈ Ω

[n]h .

3 A posteriori error estimation

The discretization error of the numerical solution is due to both space andtime discretization. In order to control these two sources of error separately,we need to suitably choose the space discretization via a suitable mesh foreach timestep and the timestep-length ∆t[n]. To reach this target we use tworesidual-based error estimators.

The fully discrete stabilized scheme (8)-(9) defines the solution(

u[n]h , p

[n]h

)

that

is the approximation of the continuous solution (u, p) at the time t[n]. Let usintroduce the piecewise affine in time functions uh,∆t and ph,∆t respectively

defined as the affine functions assuming the values u[n]h and p

[n]h at the times

t[n].

Now let us define the error we are aiming to control for each timestep withour a posteriori error indicator:

(E[n]tot)

2 =∫ t[n]

t[n−1]| u−uh,∆t |21,Ω dt +

∫ t[n]

t[n−1]‖ p− ph,∆t ‖20,Ω dt. (12)

Theoretical and numerical investigations [42,50,5,9,6] have showed that whenthe mesh and the timestep-length are not too coarse, we can define an errorestimator that mainly give information on the quality of the mesh and adifferent error estimator related to the quality of the timestep-length. Thefirst error estimator is the natural extension to the unsteady problem of thecorresponding error estimator of the steady problem and it is based on theresiduals of the strong form of the equations, whereas the time error estimatoris related to the changes in the timestep of the gradient of u. Assuming thatthis splitting is viable for Navier-Stokes equations, we define the followingestimators.

3.1 Space error estimator

According to the analytical results of [42,50,5,9,6], the space error estimatorwe are using is derived by the steady error estimators that has been analyzedin many publications [7,43,6]. Here we do not address the important problemof robustness with respect to the Reynolds number for the estimators givenin [7] and [43] assuming that, although this aspect requires much more heavy

6

computations to be tackled [51], these estimators give reliable information onthe quality of the space discretization [7].

For any T ∈ Ω[n]h we denote by E(T ) the set of its edges; we denote by E [n]

h =⋃

T∈Ω[n]h

E(T ) the set of all edges of the triangulation. Moreover, we split E [n]h into

the form E [n]h = E [n]

h,Ω ∪E [n]h,D ∪E [n]

h,N with E [n]h,Ω =

E ∈ E [n]h : E 6⊂∂Ω

, E [n]h,D =

E ∈ E [n]h : E⊂ΓD

, E [n]h,N =

E ∈ E [n]h : E⊂ΓN

.

Let us define the following notation for the momentum residual R[n]T , ∀T ∈ Ω

[n]h ,

the stress-jump J[n]E on the internal edge, ∀E ∈ E [n]

h,Ω and the residual in the

Neumann boundary conditions R[n]E,N , ∀E ∈ E [n]

h,N :

R[n]T =

u[n]h −u

[n−1]h

∆t[n]− 1

Re4 u

[n]h +

(

u[n−1]h · ∇

)

u[n]h +∇p

[n]h −ΠT f

[n]

∣

∣

∣

∣

∣

∣

T

,

J[n]E =

[[

nE ·(

1

Re∇u

[n]h −p

[n]h I

)]]

E

,

R[n]E,N = n ·

(

1

Re∇u

[n]h −p

[n]h I

)

−ΠE g[n]N .

We denote by [[ϕ]]E the jump of ϕ across the generic internal edge E alongthe orientation of nE. Following the results in [7], we define the local-in-space

error estimator η[n]R,T as follows:

(

η[n]R,T

)2=∫ t[n]

t[n−1]

(

h[n]T

)2∥

∥

∥

∥

R[n]T

∥

∥

∥

∥

2

0,T

+1

2

∑

E∈E(T )∩E[n]h,Ω

h[n]E

∥

∥

∥

∥

J[n]E

∥

∥

∥

∥

2

0,E

+∑

E∈E(T )∩E[n]h,N

h[n]E

∥

∥

∥

∥

R[n]E,N

∥

∥

∥

∥

2

0,E

+∥

∥

∥∇ ·u[n]h

∥

∥

∥

2

0,T

dt. (13)

Then, we define the global-in-space error estimator:

η[n]R =

√

√

√

√

∑

T∈Ω[n]h

(

η[n]R,T

)2(14)

that gives an evaluation of the quality of the space discretization for the n-thtimestep.

7

3.2 Time error estimator

Based on the analysis performed for the unsteady heat and the Stokes equa-tions [42,50,9,6] we use the following quantity as local in time error estimatorfor the unsteady Navier-Stokes equations:

(

η[n]∇

)2=

∑

T∈Ω[n]h

∫ t[n]

t[n−1]

∥

∥

∥

∥

∥

1√Re∇(

u[n]h − u

[n−1]h

)

∥

∥

∥

∥

∥

2

0,T

dt

. (15)

The time error estimator allows to evaluate the quality of the timestep-length∆t[n]: if η

[n]∇ is large, ∆t[n] has to be shortened; otherwise, if it is small, we can

enlarge the timestep-length.

4 Adaptive algorithm

We use the information given by the error indicators to develop an adaptivestrategy, that allows to modify the mesh and the timestep-length separately,in order to optimize the computational resources. In the following we use

the quantities(

σ[n]Ω

)2and

(

σ[n]I

)2to normalize the spatial and temporal error

indicators, respectively:

(

σ[n]Ω

)2=∫ t[n]

t[n−1]

(

∣

∣

∣u[n]h

∣

∣

∣

2

1+∥

∥

∥p[n]h

∥

∥

∥

2

0

)

dt,(

σ[n]I

)2=∫ t[n]

t[n−1]

∣

∣

∣u[n]h

∣

∣

∣

2

1dt.

4.1 Mesh adaptation

In order to modify the mesh we use the strategy of equidistribution of theerror indicator [2,33,39].

Equidistribution strategy aims at uniformly distributing the space discretiza-tion error over all the elements of our discretization. To reach this target weact to equidistribute our error estimator over all the elements requiring thaton the mesh all the elements satisfy the following inequalities:

8

(1− α)2 TOL2Ω

(

σ[n]Ω

)2

NT

≤(

η[n]R,T

)2, (16)

(

η[n]R,T

)2 ≤(1 + α)2 TOL2

Ω

(

σ[n]Ω

)2

NT

, (17)

where TOLΩ is a given tolerance, NT is the number of elements in the triangu-lation, α is a given parameter in the range (0, 1] used to control the refinementand the coarsening.

We use the target relations (16), (17) in order to adapt the mesh withineach timestep in the following way. At first, on a given mesh and with a giventimestep-length, we compute the solution and, for all the triangles, the element

residuals η[n]R,T . If

(

η[n]R,T

)2is larger than the upper bound in (17), that means

this element has a spatial estimated error larger than the admitted estimatederror by the equidistribution strategy, so we decide to refine this triangle;

whereas if(

η[n]R,T

)2is smaller than the lower bound in (16), the element can

be coarsened.

Every operation on the grid is performed by the open source numerical libraryALBERTA [45]. The refining strategy that ALBERTA implements is recursiverefinement. It consists of bisecting all the elements marked for refinementand also all the elements necessary to keep mesh conformity. The end of therecursion is guaranteed by the choice of the longest edge as refinement edge[40,37].

During the coarsening process used by ALBERTA, an element is coarsenedonly if all the elements involved in this operation (therefore also the elementsinvolved to keep the mesh conformity) are marked for coarsening.

This strategy of grid modification guarantees the refinement of all the elementswith a large local error indicator, but it allows to coarsen only the elementswith neighboring elements having a local error indicator small enough.

After each refining/coarsening, we solve the problem on the new mesh and werepeat the adaptive algorithm until on the new mesh the global-in-space errorestimator η

[n]R is bounded in the following way:

(1− α)TOLΩ ≤η

[n]R

σ[n]Ω

≤ (1 + α) TOLΩ. (18)

Obviously, decreasing the space tolerance the number of elements and, there-fore, the spatial accuracy increases. If (18) is satisfied, the mesh for the current

9

timestep is accepted and we proceed to the next timestep or with the adap-tation of the timestep-length.

4.2 Timestep size adaptation

The information given by the time error indicator are used to modify thetimestep-length. Our goal is to reduce the timestep-length ∆t[n] when the phe-nomenon has a fast evolution, and to enlarge ∆t[n] when the solution changesslowly. In order to get an optimal timestep-length, we require, likewise forthe space error estimator, that η

[n]∇ is bounded from above and from below in

terms of a given tolerance TOLI , as follows:

(1− α) TOLI ≤η

[n]∇

σ[n]I

≤ (1 + α) TOLI . (19)

If the double inequality (19) is satisfied the current timestep-length is suitable,otherwise ∆t[n] has to be modified: it is shortened if the time error estimatorexceeds the the upper bound and it is enlarged if η

[n]∇ is smaller than the lower

bound.

In the next subsection the complete space-time adaptive algorithm is resumed.

4.3 Adaptive in space and in time algorithm

The algorithm, we have implemented, is an extension of the adaptive algorithmfor the heat equation, reported in [9]. For each timestep the following processis performed:

1: do

2: compute the solution on a given mesh and a given timestep-length3: for all triangles T ∈ Ω

[n]h

4: compute the local-in-space error5: end for

6: compute the global-in-space error estimator and the time error estimator

7: if(

η[n]R

)2>(

σ[n]Ω

)2(1 + α)2 TOL2

Ω OR(

η[n]R

)2<(

σ[n]Ω

)2(1− α)2 TOL2

Ω

then

8: for all triangles T ∈ Ω[n]h

9: if(

η[n]R,T

)2>(

σ[n]Ω

)2(1− α)2

TOL2Ω

NTthen

10: mark T for refinement11: else if

(

η[n]R,T

)2<(

σ[n]Ω

)2(1 + α)2 TOL2

Ω

NTthen

12: mark T for coarsening

10

13: end if

14: end for

15: modify the mesh16: end if

17: if(

η[n]∇

)2>(

σ[n]I

)2(1 + α)2 TOL2

I then

18: shorten the timestep size∆t[n] ← ∆t[n]

ρ1

19: else if(

η[n]∇

)2<(

σ[n]I

)2(1− α)2 TOL2

I then

20: enlarge the timestep size∆t[n] ← ∆t[n]

ρ2

21: end if

22: while(

η[n]R

)2>(

σ[n]Ω

)2(1 + α)2 TOL2

Ω OR(

η[n]R

)2<(

σ[n]Ω

)2(1− α)2 TOL2

Ω

OR(

η[n]∇

)2>(

σ[n]I

)2(1 + α)2 TOL2

I

where

ρ1 = min

(

η[n]∇

)2

σ[n]I TOL2

I

, 2

, ρ2 = max

(

η[n]∇

)2

σ[n]I TOL2

I

, 0.5

.

The algorithm has been written in such way because it appears clearer. Indeed,in order to optimize the algorithm, the instructions implemented in the secondfor loop, i.e. the marking of the elements for refinement or coarsening, areperformed within the first for loop (instruction #3-5) and the second for loop(instruction #8-14) is obviously removed.

The choice of ρ1 and ρ2 is guided by the request to have a suitable change in∆t[n] with respect to our target, but to do not have these changes too abrupt.We want the largest enlargement of the the timestep-length to be the doubleof the previous one and the maximum shortening to be an half of the previous∆t[n].

Our aim is to go to the next timestep only if the solution has an adequateaccuracy, therefore only if the mesh and the timestep-length are such that η

[n]R

and η[n]∇ satisfy the inequalities (18)-(19).

If(

η[n]∇

)2 ≤ σ[n]I (1− α)2 TOL2

I and η[n]R satisfies (18), we proceed to the next

timestep anyway. We do that because we are sure this solution has a suitableaccuracy and re-computing a new solution with an enlarged ∆t[n] would giveno advantage; therefore, we enlarge ∆t[n] for the next timestep.

11

5 Numerical experiments: a backward facing step and a square

cylinder in a channel

We apply our adaptive finite element code to two test cases. Our goal is toshow that an adaptive strategy allows to have an optimal degrees of free-dom distribution in space and time and a substantial computational resourcessaving, giving accurate solutions.

We use P2−P1 elements (quadratic elements for the velocity and linear ele-ments for the pressure) satisfying the inf-sup condition. We solve the linearsystems given by the fully discrete formulation (8)-(9) with GMRES (Gener-alized Minimal RESidual) solver with the following parameters: relative toler-ance (relative decrease in the residual norm) equal to 1e-8, maximum numberof iterations to use equal to 10000 and 200 Krylov directions (restart). GMRESis applied with an ILU preconditioner with a level of fill-in equal to one. Thesolution of the linear system by preconditioned GMRES iterative algorithmis performed by the open source numerical library PETSc [3]. For each linearsystem to be solved to perform one timestep, we use the solution at the endof the previous timestep as initial guess for the GMRES method.

The parameter α, used in the adaptive algorithm, is chosen equal to 0.5. Thatmeans both refinement and coarsening are allowed.

5.1 Backward facing step

We consider the solution of the Navier-Stokes equation for an incompress-ible viscous fluid flow in a backward facing step. This test case is a classicalproblem and it has been deeply analyzed by many authors [1,27,28,36,38,46].Gresho et al., especially, through careful numerical analysis, conclude that atReynolds number 800, the flow behind a backward facing step with 1:2 ex-pansion ratio (ER) is steady. According to Gresho, if the numerical solutionis unsteady, this is due to a too coarse mesh that causes a large numericalerror contaminating the flowfield and causing unsteady spurious oscillationsin the numerical solutions [28,16]. We compute this problem treating it as anunsteady flow and verifying that it gets a final steady configuration.

The Reynolds number Re ≡ uH/ν, based on the full channel height H andaverage inlet velocity u, is 800. The geometry for the computation is shownin Figure 1. The domain extends from step face and inlet at x/H = 0 tothe outflow boundary at x/H = 15. The step and the inlet are taken to haveheights of s = h = 1/2. No-slip boundary conditions are applied on the channelwalls and step face, a parabolic velocity profile is applied at the inlet:

12

s

h

H

$%-

-

-

u

Fig. 1. Backward facing step geometry

ux = 24y(

1

2− y

)

,

uy = 0.

At the outflow the following Neumann condition is applied:

1

Re

∂ u[n]

∂n− p[n]n = 0. (20)

This condition ensures the velocity and pressure fields are not affected bysignificant disturbances on the outflow boundary.

In order to avoid the GMRES convergence problems, that can be caused bysudden variations in the velocity field, to get a suitable initial condition wesimulate the phenomenon at Re = 100 using as initial condition a stagnantflow inside all the domain; afterwards, we impose the computed steady solutionas initial condition for the simulation at Re = 800.

0 5 10 15−0.5

0

0.5

Fig. 2. Initial mesh of the backward facing step at Re = 100

0 5 10 15−0.5

0

0.5

Fig. 3. Final mesh of the backward facing step at Re = 100

The first simulation at Re = 100 starts with an uniform coarse mesh with3840 elements. (Figure 2). The timestep size is taken constant at 1e-3 and theTOLΩ is 0.01. The flow converges to the steady state very quickly and thegrid is basically coarsened (Figure 3).

13

0 5 10 15−0.5

0

0.5

(a)

0 5 10 15−0.5

0

0.5

(b)

0 5 10 15−0.5

0

0.5

(c)

0 5 10 15−0.5

0

0.5

(d)

0 5 10 15−0.5

0

0.5

(e)

0 5 10 15−0.5

0

0.5

(f)

Fig. 4. Isovorticity lines with TOLΩ = TOLI = 0.01 at (a) t = 5; (b) t = 10; (c)t = 15; (d) t = 20; (e) t = 30; (f) t = 60

0 5 10 15−0.5

0

0.5

Fig. 5. Lower and upper eddy at Re = 800: iso-lines with vorticity ω = 0

0 5 10 15−0.5

0

0.5

Fig. 6. Steady horizontal velocity profile at Re = 800

14

The results of the simulation at Re = 800 (Figures 4(a)-4(f)) initially show atransient process, involving a sequence of vortex shedding along the upper andlower walls. Afterwards, in accordance with the literature [28,16], the vorticalfeatures are convectively transported outside the domain and the flow evolvestowards a steady state with two recirculating regions (Figures 5-6).

0 50 100 150 2000

1.5

3x 10

4

time

Nel

e

TOLΩ

= TOLI = 0.04

TOLΩ

= TOLI = 0.02

TOLΩ

= TOLI = 0.01

TOLΩ

= TOLI = 0.005

Fig. 7. Re = 800: Nele vs. time at differ-ent space and time tolerances

0 20 40 60 800

0.2

0.4

0.6

0.8

1

time

∆t

TOLΩ

= TOLI = 0.04

TOLΩ

= TOLI = 0.02

TOLΩ

= TOLI = 0.01

TOLΩ

= TOLI = 0.005

Fig. 8. Re = 800: ∆t[n] vs. time at dif-ferent space and time tolerances

We have performed some simulations with different space and time tolerancesat Re = 800 and with initial timestep size ∆t[n] = 5e − 2. Figures 7 and8 respectively show the evolution of number of elements and timestep-lengthduring the simulation with different tolerances. At the beginning of the simula-tions, the mesh is subject to strong refinement near the inlet, where the vortexshedding takes place. In the following instants the mesh is refined in order tofollow the moving of transient flow features inside the domain. Finally, whenthe steady state is reached, the mesh is hardly coarsened until a minimumnumber of elements. (Figures 9(a)-9(f)). The timestep size has a progressiveincrease from the beginning to the end of the simulations (Figure 8).

Varying the tolerances TOLΩ and TOLI produces different effects on thenumber of elements and ∆t[n], respectively. Imposing a TOLΩ more restrictivemeans the space error has to be smaller and smaller; as a consequence of that,a decrease of space tolerances always coincides with an increase of the numberof elements. The behavior of timestep-length is quite different. If the timetolerance TOLI decreases, we observe a proportional reduction of the timestepsize during the transient flow; afterwards, when the flow gets the steady state,the time error indicator reduces very quickly and, therefore, ∆t[n] reaches themaximum allowed value (∆t ' 1).

Tables 1 and 2 report the lower and upper eddy recirculation lengths computedwith different values for TOLΩ and TOLI and also the final number of elementswhen the flow reaches the steady state. The values of Table 1 are obtained bycomplete simulations of the transient flow until the convergence to the steadystate. The values of Table 2 are computed starting from the steady solution

15

0 5 10 15−0.5

0

0.5

(a)

0 5 10 15−0.5

0

0.5

(b)

0 5 10 15−0.5

0

0.5

(c)

0 5 10 15−0.5

0

0.5

(d)

0 5 10 15−0.5

0

0.5

(e)

0 5 10 15−0.5

0

0.5

(f)

Fig. 9. Mesh with TOLΩ = TOLI = 0.01 at (a) t = 5; (b) t = 10; (c) t = 15; (d)t = 20; (e) t = 30; (f) t = 700

TOLΩ =TOLI

Lower eddylength

Upper eddystart

Upper eddystop

Upper eddylength

Elements

0.04 5.9271 5.0513 10.1204 5.0691 1178

0.02 5.9046 4.6978 10.3891 5.6913 1817

0.01 6.0526 4.8194 10.5011 5.6817 2861

0.005 6.0817 4.8482 10.4908 5.6426 7373

Table 1Dimensionless eddy lengths at Re = 800 and at different space and time tolerances.

previously computed with a less restrictive tolerances TOLΩ and TOLI . Thefirst two columns of this Table respectively report the starting and the finaltolerances.

The separation and reattachment points are defined as points on the boundary

16

startingTOLΩ =TOLI

finalTOLΩ =TOLI

Lowereddylength

Uppereddystart

Uppereddystop

Uppereddylength

Elements

0.01 0.0025 6.0964 4.8487 10.4935 5.6448 7791

0.005 0.00125 6.0943 4.8538 10.4805 5.6267 15424

0.0025 0.000625 6.0962 4.8550 10.4808 5.6258 44285

Table 2Dimensionless eddy lengths at Re = 800 and at different space and time tolerances.

0 1 2 3 4 5

x 104

5.9

5.95

6

6.05

6.1

6.15

Nele

L r/D

TOLΩ

= TOLI = 6.25e−4

TOLΩ

= TOLI = 4e−2

TOLΩ

= TOLI = 2e−2

TOLΩ

= TOLI = 2e−2

TOLΩ

= TOLI = 1e−2

TOLΩ

= TOLI = 5e−3

TOLΩ

= TOLI = 2.5e−3

Fig. 10. Lower eddy recirculation vs.Nele

0 1 2 3 4 5

x 104

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

Nele

Ur/D

TOLΩ

= TOLI = 4e−2

TOLΩ

= TOLI = 2e−2

TOLΩ

= TOLI = 1e−2

TOLΩ

= TOLI = 5e−3

TOLΩ

= TOLI = 2.5e−3

TOLΩ

= TOLI = 1.25e−3

TOLΩ

= TOLI = 6.25e−4

Fig. 11. Upper eddy recirculation vs.Nele

where the vorticity ω vanishes. (Figure 5). These values are obtained by linearinterpolation between two boundary grid points where the vorticity changessign. The eddy length for the upper eddy is defined as the difference betweenthe x coordinates of its reattachment and separation points. The lower eddylength is defined as the x coordinate of its reattachment point.

The computation of recirculation lengths is affected by different sources oferror. The computation of vorticity as curl of the velocity ∇× u is affectedby an amplification of the computational error due to the computation ofderivatives of the numerical solution and, of course, this error is larger oncoarser grids.

In our opinion the differences among the six simulations are mainly due to thisamplification effect more than to a lack of accuracy of the whole numericalsolution. In fact, the simulations show the transient flow converging to a steadystate for all the tolerance values and that means the numerical error does notgrow too much to give unsteady solutions as noticed in [28,16].

Anyway Figure 10 and 11 show the lower and upper eddy recirculation lengthsreaching an asymptotic value with the increasing of the number of elementsdue to the decreasing of the tolerances and, therefore, over a certain numberof elements, the solution is independent on the mesh. This approach suggests

17

Lower eddylength

Upper eddystart

Upper eddystop

Upper eddylength

Computed results*

Present study 6.0962 4.8550 10.4808 5.6258

Gartling [27] 6.1 4.85 10.48 5.63

Barton [4] 5.755 4.57 5.24 5.76

Kim and Moin [36] 6.0 - - 5.75

Guj and Stella [30] 6.025 4.85 10.15 5.30

Gresho et al [28] 6.10 4.86 10.49 5.63

Keskar and Lyn [35] 6.095 4.855 10.48 5.625

Grigoriev and Dargush [29] 6.09 4.85 10.47 5.62

Rogers and Kwak [44] 5.74 - - 5.535

Erturk [23] 5.917 4.738 10.2765 5.5385

Sohn [46] 5.8 - - 4.63

Lee and Mateescu [38] 6.0 4.80 10.30 5.50

Eperimental results

Lee and Mateescu [38] † 6.45 5.15 10.25 5.1

Armaly et al [1] ‡ 7.0 10.0 5.7 4.3

Table 3Comparison of computed results and experimental measurements of dimensionlesseddy lengths (*ER = 2 and Re = 800; †ER = 2 and Re = 805; ‡ER = 1.94 andRe = 800)

an efficient strategy to reach an accurate and mesh independent solution. Infact for each tolerance the adaptive method allows us to get an optimal meshfor a required accuracy and reducing the tolerances we increase the accuracyup to get a mesh independent solution.

Table 3 reports the recirculation length for the lower and upper eddies (ob-tained with TOLΩ = TOLI = 6.25e−4), compared with literature results. Weobserve a substantial good agreement of present numerical predictions withboth numerical and experimental results. In particular there is a very goodaccordance between the present study and the results obtained by Gartling,Gresho et al., Keskar and Lyn, Grigoriev and Dargush.

18

H

H/2

$

%-

-

-

-

-

-

-

u

D

L

l

Fig. 12. Square cylinder geometry

5.2 Square cylinder

Now we consider the 2D laminar flow around a square cylinder centered insidea channel. This is a classical test case, that has been deeply investigated byBreuer et al. [11]. The Reynolds number is based on the cylinder diameterD and the maximum inlet velocity umax. The blockage ratio B = D/H isfixed al 1/8. In order to reduce the influence of inflow and outflow boundaryconditions, the length of the channel is set to L/D = 50 and an inflow lengthof l = L/4 is chosen. No-slip boundary conditions are applied on the channelwalls and on the cylinder surface. The following parabolic velocity profile isprescribed at the channel inlet:

ux = − 1

16y2 +

1

2y

uy = 0.

At the outflow Neumann conditions (20) are applied. These conditions ensurethat vortices can approach and pass the outflow boundary without significantdisturbances or reflections into the inner domain.

A large range of Reynolds numbers: 1 ≤ Re ≤ 300 is investigated. At firstwe compute the solution for Re = 1 starting from a stagnant flow inside thedomain. Afterwards, we use the obtained flowfield as initial condition for theother simulations.

In the following we discuss the solutions obtained at different Reynolds num-bers and with different space and time tolerances. In particular we have per-formed the simulations at Re = 1, 60, 100, 125, 150, 175, 250, 280 withTOLΩ = TOLI = 0.02 and at Re = 30, 40, 50, 100, 200 with TOLΩ =TOLI = 0.01, 0.02, 0.04, 0.08. Finally, we present a comparison with the liter-ature results regarding integral flow parameters, such as recirculation length,

19

Strouhal number and drag coefficient.

8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

(a)

8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

(b)

8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

(c)

Fig. 13. Isovorticity lines around the square cylinder for different Reynolds numbers(a) Re = 1; (b) Re = 30; (c) Re = 100

In accordance with Breur et al. [11], the computed flow is steady for 1 ≤ Re <60 and at Re ≥ 60 it becomes unsteady. At Re = 1 the flow past the cylinderpersists without separation (Figure 13(a)). Increasing the Reynolds number,the wake comprises a steady recirculation region of two symmetric vortices,

20

as shown in Figure 13(b) at Re = 30, whose length grows as Re increases.At Re > 60 the flow becomes periodic and the von Karman street develops(Figure 13(c)).

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

Fig. 14. Initial uniform mesh: 7184 elements.

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

Fig. 15. Mesh after the first iterations at Re = 100: 3514 elements.

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

Fig. 16. Final mesh at Re = 100: 7919 elements.

The simulations start from an uniform mesh, that is subject to an immediateadaptation. That causes a concentration of grid points on the cylinder wallsand, successively, along the wake (Figures 14-16).

0 50 100 150 2003000

4000

5000

6000

7000

8000

9000

time

Nel

e

Fig. 17. Re = 100: Nele vs. time

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

time

∆t

Fig. 18. Re = 100: ∆t[n] vs. time

21

10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

Re

L r/D

TOLΩ

= TOLI = 0.01

TOLΩ

= TOLI = 0.02

TOLΩ

= TOLI = 0.04

TOLΩ

= TOLI = 0.08

−0.065 + 0.0554Re

Fig. 19. Recirculation lenght vs.Reynolds number at different tolerances

0 0.5 1 1.5 2 2.5 3

x 104

2.68

2.685

2.69

2.695

2.7

2.705

2.71

2.715

2.72

2.725

Nele

L r/D

TOLΩ

= TOLI = 8e−2

TOLΩ

= TOLI = 4e−2

TOLΩ

= TOLI = 2e−2

TOLΩ

= TOLI = 1e−2

TOLΩ

= TOLI = 5e−3

Fig. 20. Re = 50: recirculation lenght vsNele

When the steady state is got (Re < 60) or when the flow becomes periodic(Re > 60), the mesh stays about unchanged and the timestep size is nearlyconstant, for the given tolerances TOLΩ and TOLI (Figures 17-18).

We compute integral parameters from the flowfields and we compare themwith literature results. The length of the closed near-wake Lr is obtained byinterpolation between two grid points past the cylinder, along the center ofthe domain (y/D = 4), where the horizontal velocity changes sign.

Figure 19 reports Lr as function of Re for different value of space and timetolerances. It increases with the Reynolds number and the values are fitted bythe following curve proposed by Breur et al.:

Lr/D = −0.065 + 0.0554 Re, for 5 < Re < 60.

Figure 20 shows that imposing more restrictive space tolerances and, therefore,increasing the number of the elements, the solution reaches an asymptoticvalue. Anyway the difference between the lowest and the largest values is verysmall.

These results show at low Reynolds number (Re = 10, 20, 30, 40, 50, 55),the solution is nearly independent on the mesh; in fact Lr is almost the samefor all the Reynolds numbers. For Re < 60 the timestep-length, and then theTOLI , does not influence the final solution because the flowfield is steady.

The Strouhal number is obtained by evaluation of the flowfield time period Θ:

St =D

Θu.

Simulations performed with different space and time tolerances show that, at

22

TOLΩ TOLI Strouhalnumber

TOLΩ TOLI Strouhalnumber

0.08 0.08 0.122 0.08 0.08 0.122

0.04 0.04 0.128 0.08 0.04 0.128

0.02 0.02 0.138 0.08 0.03 0.133

0.01 0.01 0.138 0.08 0.02 0.134

Breuer et al (LBA) [11] ≈ 0.134 0.08 0.015 0.136

Breuer et al (FVM) [11] ≈ 0.139 0.08 0.01 0.137

Table 4Comparison of computed Strouhal numbers at different space and time toleranceswith data from literature at Re = 200.

50 100 150 200 250 3000.08

0.09

0.1

0.11

0.12

0.13

0.14

Re

St

Breuer et al. (FVM)

Breuer et al. (LBA)

Galletti et al.

Current results

Mukhopadhyay et al.

Fig. 21. Strouhal number vs. Reynoldsnumber with TOLΩ = TOLI = 0.02.

0 50 100 150 200 250 3001.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

Re

Cd

Breuer et al. (FVM)

Breuer et al. (LBA)

Current results

Fig. 22. Drag coefficient vs. Reynoldsnumber with TOLΩ = TOLI = 0.02.

Re = 200, the choices of TOLΩ and TOLI really affect the accuracy. In fact,the Strouhal number meaningfully changes with space and time tolerances.Table 4 reports the computed Strouhal numbers for different tolerances. Forspace and time tolerances small enough we have a good agreement with themost reliable results. Our numerical results start to be less dependent fromthe tolerances imposed for TOLI = TOLΩ ≤ 0.02 and we consider reliablethese results.

The 4-6 columns of Table 4 show that, for this problem, the error caused by thetime discretization has a larger impact in comparison with the error caused bythe space discretization. In fact, reducing only TOLI , with a fixed not muchrestrictive TOLΩ (0.08), the solution converges to the results of Breuer et al.[11].

Figure 21 reports the Strouhal number versus the Reynolds number; we canobserve that, at first, the St increases with Re until a local maximum atRe ≈ 150 and, successively, it decreases.

23

The drag coefficient Cd is one of the most important characteristic quantitiesof the flow around a cylinder. In the range of small Reynolds numbers thecontributions of the viscous and pressure forces to the total drag are of thesame order of magnitude, while increasing Re, the viscous forces become negli-gible. Figure 22 reports the computed time-averaged drag coefficient Cd vs Rewith TOLΩ = TOLI = 0.02. The Cd(Re) curve presents a local minimum atRe ≈ 150. Both the St and the Cd has a good accordance with the literatureresults [11,26].

6 Conclusions

We have presented a space-time adaptive method and we have applied it to twotest cases. The performed simulations show the solution becomes independenton the mesh and on the timestep-length over defined values of the space andtime tolerances. Moreover, the adaptive strategy allows to control the errordue to both space and time discretization and to have an optimization of thegrid points distribution and timestep size.

The most accurate solution of the backward facing step at Re = 800 is com-puted by means of 44285 elements (TOLΩ = 6.25e-4), but we have obtainedsatisfactory results already with 7791 elements. Gresho et al. [28] have simu-lated the same solution through a time-dependent finite element method with245760 grid points, Erturk [23] and Lee and Mateescu [38] by means of a finitedifference method with 429351 and 25000 points, respectively.

The Reynolds number deeply affects an adaptive method. Increasing Re thescales are always smaller and, therefore, we need finer grid mesh to solve theflowfield carefully. For this reason the current solutions of the confined flowpast a square cylinder are computed with different grids. The most accuratesolution (TOLΩ = 0.01) is obtained at Re = 100 by means of 16820 elementsand at Re = 200 the required elements are 71703.

Breuer et al. [11] use two different computational methods: lattice-Boltzmannautomata (LBA) and finite-volume method (FVM). The finest grid for LBA isan equidistant grid with 640000 lattice nodes, while the most accurate solutionobtained by FVM is computed by a non-equidistant stretched grid with 190400control volumes.

In order to verify the effectiveness of the adaptive technique, we have per-formed some simulations for the square cylinder at different Reynolds num-bers (Re = 100, 200) through the same code, but with fixed timestep-lengthand uniform grids. For each simulation, we have chosen a larger number ofelements and a smaller ∆t[n] than those used in the adaptive simulations with

24

TOLΩ = TOLI = 0.02 (corresponding to satisfactory accuracy). At Re = 100we have used an uniform grid with 14368 elements, whereas the final adaptedgrid has 7919 elements; at Re = 200 the elements are 57472 for the uniformgrid and 17238 for the final adapted one. For both Re = 100 and Re = 200,the timestep-length is fixed at 1e-2, whereas in the simulations performed bythe adaptive algorithm its minimum value is 5e-2. The solutions computedwithout the adaptive algorithm do not have the expected periodic behaviour,but display a steady flowfield. This means that the discretization error is toolarge and that the small scales of the flow are dumped giving an inaccuratesolution.

Although the computational methods we use are quite different from the meth-ods used in the quoted works, we can observe a general optimization of thedegrees of freedom and a meaningful computational resources saving. More-over, the simulations performed with uniform grids and fixed timestep-lengthshow that we need many more degrees of freedom to obtain the same accu-racy reached by the adaptive method. That proves the space-time adaptivetechnique we have presented is an effective tool to obtain accurate and eco-nomical finite element solutions of the incompressible unsteady Navier-Stokesproblems.

7 Acknowledgement

The first author was supported by Regione Piemonte via the project “Air-ToLyMi”: Modeling and simulating sustainable mobility strategies. A studyof three real test cases: Turin, Lyon, Milan. (CIPE grant 2006). The secondauthor was supported by “Progetto Lagrange - Fondazione C.R.T.”

References

[1] Armaly BF, Durst F, Pereira JCF, Schonung BJ. Experimental and theoreticalinvestigation of backward-facing step flow. J Fluid Mech 1983;127:473-96.

[2] Babuska I, Rheinboldt WC. Error estimates for adaptive finite element method.SIAM J Numer Anal 1978;15(4):736-54.

[3] Balay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG,Curfman McInnes L, Smith BF, Zhang H. PETSc Users Manual. ANL-95/11 -Revision 2.1.5, Argonne National Laboratory; 2004.

[4] Barton IE. The entrance effect of laminar flow over a backward-facing stepgeometry. Int J Numer Methods Fluids 1998;28:841-57.

25

[5] Bergman A, Bernardi C, Mighazli Z. A posteriori analysis of the finite elementdiscretization of some parabolic equations. Math Comp 2004;74:1117-38.

[6] Bernardi C, Verfurth R. A posteriori analysis of the fully discretized time-dependent Stokes equations. M2AN 2004;38:437-55.

[7] Berrone S. Adaptative discretization of the stationary and incompressibleNavier-Stokes equations by stabilized Finite Element Methods. ComputMethods Appl Mech Engrg 2001;34:4435-55.

[8] Berrone S. Robustness in a posteriori error analysis for FEM flow models.Numer Math 2002;91:389-422.

[9] Berrone S. Robust a posteriori error estimates for finite element discretizationsof the heat equation with discontinuos coefficients. Mathematical Modelling andNumerical Analysis 2006;6:991-1021.

[10] Berrone S. Skipping transition conditions in a posteriori error estimates for finiteelement discretizations of parabolic equations. Preprint n. 5/2008, Dipartimentodi Matematica, Politecnico di Torino.

[11] Breuer M, Bernsdorf J, Zeiser T, Durst F. Accurate computations of the laminarflow past a square cylinder based on two different methods: lattice-Boltzmannand finite-volume. Intl J Heat Fluid Flow 2000;21:186-96.

[12] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York-Berlin: Springer-Verlag; 1991.

[13] Canuto C, Hussaini H, Quarteroni A, Zang T. Spectral Methods. Evolution toComplex Geometries and Applications to Fluid Dynamics. Berlin-Heidelberg-New York: Springer Verlag; 2007.

[14] Cao J. Application of a posteriori error estimation to finite element simulationof incompressible Navier-Stokes flow. Comput Fluids 2005;34:972-90.

[15] Cao J. Application of a posteriori error estimation to finite element simulationof compressible Navier-Stokes flow. Comput Fluids 2005;34:991-1024.

[16] Chan DC, Mittal R. Large-eddy simulation of a backward facing step flowusing a least-sqares spectral element method. Center for Turbulence Research,Proceeding for the summer program; 1996.

[17] Ciarlet PG. The finite element method for elliptic problems. Amsterdam: North-Holland; 1978.

[18] Ciarlet PG, Lions J-L. Handbook of Numerical Analysis, Volume IX.Amsterdam: North Holland; 1990.

[19] Dautray R, Lions J-L. Mathematical Analysis and Numerical Methods forScience and Technology, Volume 5 and 6. New York-Berlin: Springer-Verlag;1990.

26

[20] Dahmen W, Urban K, Vorloeper J. Adaptive Wavelet Methods – Basic Conceptsand Applications to the Stokes Problem. In: Wavelet Analysis, D.-X. Zhou (ed.),New Jersey: World Scientific; 2002. p.39-80.

[21] Deville MO, Fischer PF, Mund EH. Higher Order Methods for IncompressibleFluid Flow. Cambridge: Cambridge University Press; 2002.

[22] Dorfler W. A convergent adaptive algorithm for Poisson’s equation. SIAM JNumer Anal 1996;33(3):1106-24.

[23] Ertutk E. Numerical solutions of 2-D steady incompressible flow over abackward-facing step, Part I: High Reynolds number solutions. Comput Fluids2008;37:633-55.

[24] Franca LP, Frey SL, Hughes TJR. Stabilized finite element methods: I.Application to the advective-diffusion model. Comput Methods Appl MechEngrg 1992;95:253-76.

[25] Franca LP, Frey SL. Stabilized finite element methods: II. The incompressibleNavier-Stokes equations. Comput Methods Appl Mech Engrg 1992;99:209-33.

[26] Galletti B, Bruneau CH, Zannetti L, Iollo A. Low-order modelling of laminarflow regimes past a confined square cylinder. J Fluid Mech 2004;503:161-70.

[27] Gartling DK. A test problem for outflow boundary conditions - flow over abackward-facing step. Int J Numer Methods Fluids 1990;11:953-67.

[28] Gresho PM, Gartling DK, Cliffe KA, Garrat TJ, Spense A, Winters KH,Goodrich JW, Torczynski TR. Is the steady viscous incompressible 2D flowover a backward-facing step at Re=800 stable? Int J Numer Methods Fluids1993;17:501-41.

[29] Grigoriev MM, Dargush GF. A poly-region boundary element method forincompressible viscous fluid flows. Int J Numer Methods Eng 1999;46:1127-58.

[30] Guj G, Stella F. Numerical solutions of high-Re recirculating flows vorticity-velocity form. Int J Numer Methods Fluids 1998;8:405-16.

[31] Harari I, Hughes TJR. What are C and h?: Inequalities for the analysisand design of finite element methods. Comput Methods Appl Mech Engrg1992;97:157-92.

[32] Hughes TJR, Franca LP, Balestra M. A new finite element formulation forcomputational fluid dynamics: V. Circumventing the Babuska-Brezzi condition:A stable Petrov-Galerkin formulation of the Stokes problem accommodatingequal-order interpolations. Comput Methods Appl Mech Engrg 1986;59:85-99.

[33] Johnson C. Adaptive finite element methods for diffusion and convectionproblems. Comput Methods Appl Mech Engrg 1990;82:301-22.

[34] Karniadakis GE, Sherwin SJ. Spectral/hp Element Methods for CFD. Oxford:Oxford University Press; 2005.

27

[35] Keskar J, Lyn DA. Computations of a laminar backward-facing step flow atRe = 800 with spectral domain decomposition method. Int J Numer MethodsFluids 1999;29:411-27.

[36] Kim J, Moin P. Application of a fractional-step method to incompressibleNavier-Stokes equations. J Comput Phys 1985;59:308-23.

[37] Kossaczky I. A recusrive approach to local mesh refinement in two and threedimensions. J Comput Appl Math 1994;55:275-88.

[38] Lee T, Mateescu D. Experimental and numerical investigation of 2D backward-facing step flow. J Fluids Struct 1998;12:703-16.

[39] Medina J, Picasso M, Rappaz J. Error estimates and adaptive finite elementsfor nonlinear diffusion-convection problems. Math Models and Math in ApplSci 1996;6:689-712.

[40] Mitchell W. A comparison of adaptive refinemnet tecniques for elliptic problems.ACM Trans Math Soft 1989;15:326-47.

[41] Morin P, Nochetto RH, Siebert KG. Convergence of adaptive finite elementmethods. SIAM Rev. 2002;44(4):631-58.

[42] Picasso M. Adaptive finite elements for a linear parabolic problem. ComputMethods Appl Mech Engrg 1998;167:223-37.

[43] Prudhomme S, Oden JT. A posteriori error estimattion and error controlfor finite element approximations of time-dependent Navier-Stokes equations.Finite Elem. Anal. Des. 1999;33:247-62.

[44] Rogers SE, Kwak D. An upwind differencing scheme for the incompressibleNavier-Stokes equations. Appl Numer Math 1991;8:43-64.

[45] Schmidt A, Siebert KG. Design of adaptive finite element software. The finite

element toolbox ALBERTA. Lecture Notes in Computational Science andEngineering 42. Springer; 2005.

[46] Sohn J. Evaluation of FIDAP on some classical laminar and turbulentbenchmarks. Internat J Numer Methods Fluids 1988;8:1469-90.

[47] Temam R. Navier-Stokes Equations and Nonlinear Functional Analysis. No. 66in CBMS-NSF Series in Applied Mathematics, Philadelphia: SIAM; 1995.

[48] Temam R. Some developments on the Navier-Stokes equations in the secondhalf of the 20th century. in Developpment des Mathematiques au cours de laseconde moitie du XXeme siecle, Basel: J.P. Pier ed, Birkhauser Verlag; 2000.

[49] Verfurth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Chichester-New York: John Wiley & Sons; 1996.

[50] Verfurth R. A posteriori error estimates for finite element discretization of theheat equations, Calcolo 2003;40:195-212.

[51] Verfurth R. Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J Numer Anal 2005;43:1783-802.

28