Finite elements for the Navier-Stokes problem with outflow condition

Finite elements for the Navier-Stokes problem with outflow condition Daniel Arndt, Malte Braack, and Gert Lube

Abstract. This work is devoted to the Directional Do-Nothing (DDN) condition as an outflow boundary condition for the incompressible Navier-Stokes equation. In contrast to the Classical Do-Nothing (CDN) condition, we have stability, existence of weak solutions and, in the case of small data, also uniqueness. We derive an a priori error estimate for this outflow condition for finite element discretizations with inf-sup stable pairs. Stabilization terms account for dominant convection and the divergence free constraint. Numerical examples demonstrate the stability of the method.

1

Introduction

The classical do-nothing condition is very often prescribed at outflow boundaries for fluid dynamical problems. However, in the case of the Navier-Stokes equations in a domain Ω ⊂ Rd , d ∈ {2, 3}, not even existence of weak solutions can be shown, see [10]. The reason is that this boundary condition does not exhibit any control about inflow across such boundaries, see [4]. This has also severe impact onto the stability of numerical algorithms for flows at higher Reynolds numbers. Denoting the velocity field by u and the pressure by p, the directional do-nothing (DDN) boundary condition ν∇u · n − pn − β(u · n)− u = 0

at S1

(1)

on S1 ⊆ ∂Ω with normal vector n and a parameter β ≥ 0 is one possibility to circumvent this disadvantage. Here, (u · n)− (x) = min(0, u(x) · n(x)) denotes the negative part of the flux across the boundary at x ∈ ∂Ω. In particular, existence of weak solutions is proved in [4], and in several applications the stability is enhanced compared to the classical do-nothing condition (β = 0), see e.g. [2,12]. In the case of pure outflow, i.e. if u·n ≥ 0 on S1 , this condition is identical to the classical do-nothing (CDN). In particular, it reproduces Poiseuille flow for a laminar flow in a tube with parabolic inflow. D. Arndt · G. Lube University of G¨ ottingen, Germany e-mail: [email protected]; [email protected] M. Braack Mathematical Seminar, University of Kiel, Germany e-mail: [email protected]

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Arndt, Braack, Lube

The outflow condition (1) has similarities with the convective boundary condition in [6,5], but no reference solution nor Stokes solution on a larger domain is needed. We also like to refer to the recent work [8] where a different open boundary condition is proposed which makes use of a smoothed step function and overcomes backflow instabilities as well.

2

Variational formulation

The variational spaces for velocity and pressure are given by V := {u ∈ H 1 (Ω)d | u = 0 a.e. on S0 }, Q := L2 (Ω), respectively. The norm in L2 (ω) (and in L2 (ω)d ) for ω ⊆ Ω is denoted by || · ||ω . For ω = Ω we surpress the index. The H −1 (Ω)-norm is denoted by || · ||−1 . In order to formulate the Navier-Stokes system in variational form we consider the decomposition Z 1 1 (w · n)u · φ ds, ((w · ∇)u, φ) = (((w · ∇)u, φ) − (u, (w · ∇)φ))) + 2 2 ∂Ω of the convective term for divergence free vector fields w and use the notation c(w; u, φ) := 21 ((w · ∇)u, φ) − 21 (u, (w · ∇)φ) Z 1 + 2 (w · n) − β(w · n)− u · φ ds. S1

Lemma 1. The nonlinear convective term can be expressed by Z c(w; u, φ) = ((w·∇)u, φ) + 12 (div w u, φ) − β (w · n)− u · φ ds S1

for all w, u, φ ∈ V . Proof. This identity follows easily by integration by parts. The semi-linear form for the Navier-Stokes system with DDN condition reads A(w; u, p; φ, χ) := c(w; u, φ) + (ν∇u, ∇φ) − (p, div φ) + (div u, χ). We seek (u, p) ∈ V × Q s.t. (∂t u, φ) + A(u; u, p; φ, χ) = hf , φi

∀φ ∈ V , ∀χ ∈ Q.

(2)

Diagonal testing with the solution, φ := u, χ := p, results in Z 1 1 1 ∂t ||u||2 + ν||∇u||2 + (u · n)+ + − β (u · n)− |u|2 ds 2 2 2 S1 ≤ ||f ||−1 ||∇u||. For β ≥ 1/2 the arising boundary integral is non-negative. This property is needed to show existence of weak solutions, see techniques in [4]. Moreover, the solution is unique in the case of small data, see [3].

Finite elements for the Navier-Stokes problem with outflow condition

3

3

Finite element discretization

For the discretization of (2) in space we use inf-sup stable finite elements of order k for uh , for instance the classical Taylor-Hood element Q2 /Q1 on quadrilaterals (for d = 2). Due to the fact that we use a divergencefree projection in the analysis below, we require for d = 3 on hexahedrons Q3 /Q2 elements, see [9]. It is well-known that the convective terms and the divergence-free should be stabilized in order to obtain more accurate discrete solutions with enhanced divergence properties and less over- and undershoots. We use a combination of div-div stabilization and local projection (LPS) of the convective terms X Sh (w; u, φ) := γM (div u, div φ)M + αM (κM [(wM ·∇)u], κM [(wM ·∇)φ])M M ∈Mh

with local fluctuation operator κM : L2 (M ) → L2 (M ) on patches M , and piecewise constant approximation wM of w. We allow for the one-level (M ∈ Th ) or the two-level (M ∈ T2h ) variant, but the common requirements according to [11] should be satisfied. The stabilization parameter γM for the divergence stabilization is patch-wise constant in the following range: 0 < γ0 hmax ≤ γM ≤ γmax ,

(3)

with positive constants γ0 , γmax > 0 and the maximal mesh size hmax = max{hT : T ∈ Th }. The LPS parameter α must be non-negative (may vanish) and may depend on uM but is bounded (α0 ≥ 0): 0 ≤ αM ≤ α0 |uM |−2 .

(4)

Similar to the work [1] we assume for the a priori estimate the following local approximation property: ||u − uM ||L∞ (M ) ≤ C||u||L∞ (M ) .

(5)

The semi-discrete system consists in seeking uh ∈ V h , ph ∈ Qh s.t. (∂t uh , φ) + A(uh ; uh , ph ; φ, χ) + Sh (uh ; uh , φ) = hf , φi

(6)

for all φ ∈ V h and all χ ∈ Qh .

4

A priori estimate

For the a priori estimate we split the error eu := u − uh into interpolation error η u := u − ih u and projection error ξ u := ih u − uh . Here, ih : V → V h is a divergence-free projection. We use the following norm in V : Z 2 2 2 1 1 |||u|||uh = ν||∇u|| + 2 (uh · n)+ + ( 2 − β)(uh · n)− |u| dσ S1

+Sh (uh ; u, u).

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Arndt, Braack, Lube

A bound on the interpolation error η u is well-known, see [9]. Therefore we focus on the projection error. Theorem 2. Under the previous assumptions (3)-(5), enough regularity of the continuous solution u, p, and β > 1/2 it holds for the projection error: ||ξ u ||2L∞ (0,T ;L2 (Ω)) +

Z 0

T

|||ξ u (t)|||2uh dt ≤ C

Z

T

eCG (t−τ )

0

X

φM (τ )dτ

M

with the Gronwall constant CG := c(1+|u|L∞ (0,T ;W 1,∞ (Ω)) + h||u||2L∞ (0,T ;W 1,∞ (Ω)) + (1+ν −1 )||u||L∞ (S1 ) ), and the quantity φM depending on u and on the interpolation errors η u , ηp : φM :=||∂t η u ||2M + (c1 + c3 )||∇η u ||2M + c2 ||η u ||2M + c3 ||κM (∇u)||2M + c4 ||ηp ||2M , and coefficients c1 , . . . , c4 : −1 c1 = ν + γM , c2 = h−2 ||u||2L∞ (M ) , c3 = αM |u|2M , c4 = (ν + γM )−1 . M +ν

This bound is similar to the one published in [1] (for S1 = ∅). The difference is the additional term ν −1 ||u||L∞ (S1 ) in the Gronwall constant. Proof. In the first step we subtract (2) and (6) and perform diagonal testing. Due to the additive splitting of the error, eu = η u + ξ u , and after reordering terms we arrive at 1 ∂t ||ξ u ||2 + |||ξ u |||2uh = −(∂t η u , ξ u ) − ν(∇η u , ∇ξ u ) + (div ξ u , ηp )−(div η u , ξp ) 2 −c(u; u, ξ u ) + c(uh ; ih u, ξ u ) + Sh (uh ; ih u, ξ u ) Using Lemma 1 we obtain −c(u; u, ξ u ) + c(uh ; ih u, ξ u ) = −(u · ∇u, ξ u ) + (uh · ∇ih u, ξ u ) + 21 (div uh ih u, ξ u ) Z +β (u · n)− u − (uh · n)− ih u · ξ u ds S1

= −(eu · ∇u, ξ u ) − (uh · ∇η u , ξ u ) + 21 (div uh ih u, ξ u ) Z +β (u · n)− u − (uh · n)− ih u · ξ u ds. S1

Integration by parts a second time yields Z (uh · ∇η u , ξ u ) = −(η u , uh · ∇ξ u ) − (div uh η u , ξ u ) +

(uh · n)η u · ξ u ds. S1


5

We obtain the identity 1 ∂t ||ξ u ||2 + |||ξ u |||2uh = R + T, 2 with volume integrals R := −(∂t η u , ξ u ) − ν(∇η u , ∇ξ u ) + (div ξ u , ηp ) − (div η u , ξp ) −(eu · ∇u, ξ u ) + 21 (div uh ih u, ξ u ) + (η u , uh · ∇ξ u ) +(div uh η u , ξ u ) + Sh (uh ; ih u, ξ u ), and boundary integrals Z T := β(u · n)− u − β(uh · n)− ih u − (uh · n)η u · ξ u ds. S1

For R we may use the result of [1]: R≤

X 1 1 ||∂t η u ||2 + CG ||ξ u ||2 + |||ξ u |||2uh + 2 φM . 2 4 M

The techniques in [1] do not require any further integration by parts. Therefore, the approach for the Dirichlet case without any outflow condition also applies to bound R in our case. The remaining terms of T will be bounded in the sequel. A basic calculus yield T = T1 + T2 with Z T1 := −

{(uh · n)+ + (1 − β)(uh · n)− }η u · ξ u ds, S1

Z {(u · n)− − (uh · n)− }u · ξ u ds.

T2 := β S1

Since the triple-norm includes control on the boundary fluxes, T1 is bounded by ||| · |||uh provided β > 21 : Z T1 = − {(uh · n)+ + (1 − β)(uh · n)− }η u · ξ u ds S1 |β − 1| ≤ max 2, |||η u |||uh |||ξ u |||uh . β − 21 T2 can be bounded by the trace theorem in L1 -norm and the product rule with arbitrary > 0: Z T2 = β {(u · n)− − (uh · n)− }u · ξ u ds S1

≤ cS ||u||L∞ (S1 ) |||eu ||ξ u |||W 1,1 (Ω) ≤ cS ||u||L∞ (S1 ) (|||eu ||ξ u |||L1 (Ω) + ||∇(|eu ||ξ u |)||L1 (Ω) ) ≤ cS ||u||L∞ (S1 ) (1 + ν −1 )(||ξ u ||2 + ||η u ||2 ) + (|||ξ u |||2uh + |||η u |||2uh ).

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Arndt, Braack, Lube

Hence, the sum of the two terms T1 and T2 can now be bounded by T ≤ CG (||ξ u ||2 + ||η u ||2 ) + ( + cβ −1 )|||η u |||2uh + |||ξ u |||2uh , with a still arbitrary parameter > 0. In combination with the upper bound for R we arrive for = 14 at X ∂t ||ξ u ||2 + |||ξ u |||2uh ≤ ||∂t η u ||2 + 4 φM + CG (||ξ u ||2 + ||η u ||2 ) + c0β |||η u |||2uh . M

Application of the Gronwall lemma yields the assertion.

t u

Remark 3. If the solution u is sufficiently smooth, the previous Theorem can be used to derive an upper bound of the projection error in terms of powers of hM by using 2 2 2 φM (τ ) ≤ Ch2k M ||∂t u(τ )||H 2 (M ) + (c5 + c6 )||u(τ )||H k+1 (M ) + c4 ||p(τ )||H k (M ) with c4 = (ν + γM )−1 , c5 = 1 + ν + γmax + α0 , and c6 = h2M ν −1 ||u||2L∞ (M ) . Remark 4. The term c6 in the previous Remark can be avoided by using a different bound in the proof of Theorem 2, see Dallmann [7] (page 44). eG = CG + ||u||2 ∞ . However, this leads to a larger Gronwall constant C L (Ω)

5

Numerical results

We want to support the above analysis by numerical examples that show the desired convergence results in space. In particular we like to see that the error does not blow up in time or space, even if there is inflow at the boundary S1 . The considered domain is given by Ω := (0, 2π) × (−π, π). We use the directional do-nothing (DDN) at S1 := {(2π, y) : −π ≤ y ≤ π} with the parameter β = 1, and Dirichlet boundary at S0 := ∂Ω \ S1 . Let χ : R → [0, 1] defined as χ(y) = 1 if y < 0, and χ(y) = 0 for y ≥ 0. The exact solution in analytical form and the corresponding right hand side are given by u(x, y) = (sin(y) cos(t)2 , 0)T , p(x, y) = − 21 χ(y) sin(y)2 cos(t)4 , f (x, y) = (− sin(2t) sin(y) + cos(t)2 sin(y)ν, −χ(y) cos(y) sin(y) cos(t)4 )T . We investigate the convergence behavior for the classical Taylor-Hood pair Q2 /Q1 . Since we are not interested in the error due to time discretization we set ∆t = 10−4 and evaluate the error at T = 10−2 . In Figure 1 we depict the L2 -errors with respect to velocity and pressure, ||u − uh || and ||p − ph ||, in dependence of a uniform mesh size h for various viscosities ν. We compare with (γ = 1) and without (γ = 0) div-div stabilization. For the velocity error in L2 we observe convergence of third order in the


10

10

L2 (p)

L2 (u)

10 2

γ = 1, γ = 1, γ = 1, γ = 0, γ = 0, γ = 0, h3

0

ν ν ν ν ν ν

= 100 = 10−2 = 10−4 = 100 = 10−2 = 10−4

7

10

0

γ = 1, γ = 1, γ = 1, γ = 0, γ = 0, γ = 0, h2

ν ν ν ν ν ν

= 100 = 10−2 = 10−4 = 100 = 10−2 = 10−4

-2

10 10 -4 -1 10

10 0

h

-2

10 -1

10 0

h

Fig. 1. L2 -errors of u and p for Taylor-Hood (Q2 /Q1 ) elements.

case γ = 1. Without div-div stabilization the convergence order of ||u − uh || is reduced. For the pressure, second order convergence can be observed which is in line with our analysis. The pressure error does essentially not depend on any of the parameters. In Figure 2 we show the errors |u − uh |1 , and ||div (u − uh )||. Both quantities show quadratic convergence, i.e. at optimal rate, if div-div stabilization is used. For the velocity energy error ||u − uh || and the H 1 (Ω) error the results deviate from the optimal rate of convergence (h3 resp. h2 ) if no div-div stabilization is used. However, the biggest impact of the stabilization can be seen for the divergence error ||div (u − uh )||. For sufficiently small viscosity the error stays nearly constant if no div-div stabilization is used. Optimal convergence rates can be recovered if div-div stabilization is used. With respect to the LPS stabilization we did not observe any significant influence for the considered norms. Compared to results in [1] for Dirichlet boundary conditions the div-div stabilization seems to play a more important role.

References 1. D. Arndt, H. Dallmann, and G. Lube, Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem, Numer. Meth. Partial Diff. Eq. 31:4 (2015), 1224–1250. 2. Y. Bazilevs, J. Gohean, T. Hughes, R. Moser, and Y. Zhang, Patientspecific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device, Comp. Meth. Appl. Mech. Eng. 198:45 (2009), 3534–3550. 3. M. Braack, Outflow condition for the Navier–Stokes equations with skewsymmetric formulation of the convective term, BAIL 2015, Lecture Notes Comput. Science and Engineer. (to appear 2016).

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L2 (div u)

10

0

γ = 1, γ = 1, γ = 1, γ = 0, γ = 0, γ = 0, h2

ν ν ν ν ν ν

= 100 = 10−2 = 10−4 = 100 = 10−2 = 10−4

H 1 (u)

10 0

γ = 1, γ = 1, γ = 1, γ = 0, γ = 0, γ = 0, h2

ν ν ν ν ν ν

= 100 = 10−2 = 10−4 = 100 = 10−2 = 10−4

10 -2 10 -2 10 -1

10 0

h

10 -1

10 0

h

Fig. 2. Errors ||div (u − uh )|| and |u − uh |1 for Taylor-Hood (Q2 /Q1 ) elements.

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