Continuous data assimilation applied to a velocity-vorticity

formulation of the 2D Navier-Stokes equations

Matthew Gardner∗ Adam Larios † Leo G. Rebholz‡ Duygu Vargun §

Camille Zerfas ¶

Abstract

We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulationof the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity,and nudging applied to the velocity only. We prove that under a typical finite element spatialdiscretization and backward Euler temporal discretization, application of CDA preserves theunconditional long-time stability property of the velocity-vorticity method and provides optimallong-time accuracy. These properties hold if nudging is applied only to the velocity, and ifnudging is also applied to the vorticity then the optimal long-time accuracy is achieved morerapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an applicationproblem of channel flow past a flat plate.

1 Introduction

Performing accurate simulations of complex fluid flows that match real-world observations or ex-periments typically requires highly precise knowledge of the initial data. However, such data isoften known in very sparsely-distributed locations, which is the case in, e.g., weather observation,ocean monitoring, etc. Thus, accurate, deterministic simulations based on initial data are oftenimpractical. Data assimilation is a collection of methods that works around this difficulty by incor-porating incoming data into the simulation to increase accuracy, hence data assimilation techniquesare highly desirable to incorporate into simulations. However, the underlying physical equationsoften suffer from stability issues which can reduce the accuracy gained by using data assimilation.While there are many ways to stabilize numerical simulations, it is far from obvious how to adaptdata assimilation techniques to combine them with cutting-edge stabilization methods. Thereforeit becomes worthwhile to seek new ways to incorporate data assimilation into stabilized schemes.In this article, we propose and analyze a new approach to this problem which combines continuousdata assimilation with velocity-vorticity stabilization.

Since Kalman’s seminal paper [42] in 1960, a wide variety of data assimilation algorithms havearisen (see, e.g., [14, 43, 47, 51]). In [6], Azouani, Olson, and Titi proposed a new algorithm known

∗Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634; email: mdgardn@g.clemson.edu†Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588; email: alarios@unl.edu, par-

tially supported by NSF Grants DMS 1716801 and CMMI 1953346.‡Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634; email: rebholz@clemson.edu,

partially supported by NSF Grant DMS 2011490.§Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634; email: dvargun@clemson.edu.¶Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634; email: czerfas@g.clemson.edu.

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as continuous data assimilation (CDA), also referred to as the AOT algorithm. Their approachrevived the so-called “nudging” methods of the 1970’s (see, e.g., [5, 36]), but with the additionof a spatial interpolation operator. This seemingly minor change had profound impacts, and theauthors of [6] were able to prove that using only sparse observations, the CDA algorithm appliedto the 2D Navier-Stokes equations converges to the correct solution exponentially fast in time,independent of the choice initial data. This stimulated a large amount of recent research on theCDA algorithm; see, e.g., [3, 4, 7, 8, 10, 11, 12, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30,31, 32, 38, 40, 41, 44, 45, 46, 52, 53, 54, 60, 61]. The recent paper [15] showed that CDA can beeffectively used for weather prediction, showing that it can indeed be a powerful tool on practicallarge scale problems. Convergence of discretizations of CDA models was studied in [29, 37, 45, 61], and found results similar to those at the continuous level. Our interest in the CDA algorithmarises from its adaptability to a wide range of nonlinear problems, as well as its small computationalcost and straight-forward implementation. These qualities make it an ideal candidate for combiningdata assimilation with stabilization techniques; in particular, with the recently developed velocity-vorticity stabilization, described below.

Flows of incompressible, viscous Newtonian fluids are modeled by the Navier-Stokes equations(NSE), which take the form

ut − ν∆u+ (u · ∇)u+∇p =f,

∇ · u =0,(1.1) NSE

together with suitable boundary and initial conditions. Here, u denotes a velocity vector field, p ispressure, f is external (given) force, and ν > 0 represents the kinematic viscosity which is inverselyproportional to the Reynolds number. Solving the NSE is important in many applications, howeverit is well known that doing so can be quite difficult, especially for small ν. Many different toolshave been used for more accurate numerical simulations of the NSE, for example using NSE formu-lations tailored to particular application problems [13, 33, 49, 57] or discretization and stabilizationmethods [39, 48, 59], and more recently using observed data to improve simulation [6, 9, 45, 62, 63].

We consider in this paper discretizations of a continuous data assimilation (CDA) enhancementapplied to the following velocity-vorticity (VV) formulation of the 2D NSE:

ut − ν∆u+ ω × u+∇P = f,

∇ · u = 0,

ωt − ν∆ω + (u · ∇)ω = rot f.

(1.2) VV

Here, ω represents the (scalar) vorticity, P := p + 12 |u|

2 is the Bernoulli pressure, and rot is the

2D curl operation: rot(f1f2

):= ∂f1

∂y −∂f2∂x . In the NSE, the velocity and vorticity are coupled via

the relationship ω = rotu (or equivalently, the Biot-Savart Law). However, the VV formulationtypically does not enforce this relationship, so u and ω are only coupled via the evolution equationsin (1.2), and the relationship ω = rotu is recovered a posteriori, so that at the continuous level,(1.2) is formally equivalent to (1.1). However, in practice, discretizations of VV can behave quitedifferently from typical discretizations of NSE, providing better stability as well as accuracy (espe-cially for vorticity) for vortex dominated or strongly rotating flows, see [2, 50, 56, 58] and referencestherein. A very interesting property of (1.2) was recently shown in [35], where it was proven thatthe system (1.2) when discretized with standard finite elements and a decoupling backward Euleror BDF2 temporal discretization was unconditionally long-time stable in both L2 and H1 norms

2

for both velocity and vorticity; no such analogous result is known for velocity-pressure discretiza-tions/schemes. Hence the scheme itself is stabilizing, even though it is still formally consistentwith the NSE. The recent work in [2] showed that these unconditionally long-time stable schemesalso provide optimal vorticity accuracy, yielding a vorticity solution that is one full order of spatialaccuracy better than for an analogous velocity-pressure scheme.

We consider herein CDA applied to (1.2), which yields a model of the form

vt − ν∆v + w × v +∇q + µ1IH(v − u) = f,

∇ · v = 0,

wt − ν∆w + (v · ∇)w + µ2IH(w − ω) = rot f,

(1.3) VVda

where IH is an appropriate interpolation operator, IH(u) and IH(ω) are assumed known frommeasurements, and µ1, µ2 ≥ 0 are nudging parameters. If µ2 = 0, then vorticity is not nudged andIH(ω) need not be assumed known. Due to the success of (1.2) in recent papers [2, 35, 56] andthat of CDA in the works mentioned above, combining these ideas and studying (1.2) is a naturalnext step to see whether CDA will provide optimal long-time accuracy for the VV schemes alreadyknown to be unconditionally long-time stable. Herein, we do find that CDA provides convergenceof (1.3), with any initial condition, to the true NSE solution (up to optimal discretization error)and moreover that CDA preserves the long-time stability.

This paper is organized as follows. In Section 2, we introduce the necessary notation andpreliminaries needed in the analysis. In Section 3, we propose and analyze a fully discrete schemefor (1.3), and show that for nudging velocity and vorticity together and nudging just velocity,algorithms are long-time stable in L2 and H1 norms and long-time optimally accurate in L2 velocityand vorticity (under the usual CDA assumptions on the coarse mesh and nudging parameter). InSection 4, we illustrate the theory with numerical tests, and finally draw conclusions in section 5.

2 Notation and Preliminariesprelim

We now provide notation and mathematical preliminaries to allow for a smooth analysis to follow.We consider the domain Ω ⊂ R2 to be the 2π-periodic box, with the L2(Ω) norm and inner productdenoted by ‖ · ‖ and (·, ·) respectively, while all other norms will be appropriately labeled.

For simplicity, we use herein periodic boundary conditions for velocity and vorticity. Extensionto full nonhomogeneous Dirichlet conditions can be performed by following analysis in [50], al-though for no-slip velocity together with the more physically consistent natural vorticity boundarycondition studied in [55, 56] more work would be needed to handle the boundary integrals. Wedenote the natural corresponding function spaces for velocity, pressure, and vorticity by

X := H1#(Ω)2 =

v ∈ H1

loc(R)2, v is 2π-periodic in each direction,

∫Ωv dx = 0

,

Q := L2#(Ω) =

q ∈ L2

loc(R), q is 2π-periodic in each direction,

∫Ωq dx = 0

,

W := H1#(Ω) =

v ∈ H1

loc(R), v is 2π-periodic in each direction,

∫Ωv dx = 0

.

3

In X (and W ), we have the Poincare inequality: there exists a constant CP depending only onΩ such that for any φ ∈ X (or W ),

‖φ‖ ≤ CP ‖∇φ‖.

We define the skew-symmetric trilinear operator b∗ : X ×W ×W → R to use for the nonlinearterm in the vorticity equation, by

b∗(u, ω, χ) :=1

2((u · ∇ω, χ)− (u · ∇χ, ω)) .

The following lemma is proven in [45], and is useful in our analysis.

geoseries Lemma 2.1. Suppose constants r and B satisfy r > 1, B ≥ 0. Then if the sequence of real numbersan satisfies

ran+1 ≤ an +B,

we have that

an+1 ≤ a0

(1

r

)n+1

+B

r − 1.

2.1 Discretization preliminaries

Denote by τh a regular, conforming triangulation of the domain Ω, and let Xh ⊂ X, Qh ⊂ Qbe velocity-pressure spaces that satisfy the inf-sup condition. We will assume the use of Xh =X ∩Pk(τh) and Qh = Q∩Pk−1(τh) Taylor-Hood or Scott-Vogelius elements (on appropriate meshesand/or polynomial degrees, see [34] and references therein). The discrete vorticity space is definedas Wh := W ∩ Pk(τh). Define the discretely divergence free subspace by

Vh := vh ∈ Xh | (∇ · vh, qh) = 0 ∀ qh ∈ Qh.

We will assume the mesh is sufficiently regular so that the inverse inequality holds in Xh: Thereexists a constant C such that

‖∇χh‖ ≤ Ch−1‖χh‖ ∀ χh ∈ Xh.

The discrete Laplacian operator is defined as: For φ ∈ H1(Ω)2, ∆hφ ∈ Xh satisfies

(∆hφ, vh) = −(∇φ,∇vh) ∀vh ∈ Xh. (2.2) laplace

The definition for ∆h is written the same way when applied in Wh, since this is simply the abovedefinition restricted to a single component.

The discrete Stokes operator Ah is defined as: For φ ∈ H1(Ω)2, find Ahφ ∈ Vh such that for allvh ∈ Vh,

(Ahφ, vh) = −(∇φ,∇vh). (2.3) stokes

By the definition of discrete Laplace and Stokes operators, we have the Poincare inequalities

‖∇χh‖ ≤ CP ‖∆hχh‖ ∀χh ∈ Xh, (2.4) Pforlap

‖∇φh‖ ≤ CP ‖Ahφh‖ ∀φh ∈ Vh. (2.5) Pforsto

4

We recall the following discrete Agmon inequalities and discrete Lp bounds [35, 39]:

‖vh‖L∞ ≤ C‖vh‖1/2‖Ahvh‖1/2 ∀vh ∈ Vh, (2.6) agmonlap

‖vh‖L∞ ≤ C‖vh‖1/2‖∆hvh‖1/2 ∀vh ∈ Xh, (2.7) agmonstk

‖∇vh‖L3 ≤ C‖vh‖1/3‖∆hvh‖2/3 ∀vh ∈ Xh. (2.8) l3ineq

We note that all bounds above for Xh trivially hold in Wh, since Wh functions can be consideredas components of functions in Xh.

A function space for measurement data interpolation is also needed. Hence we require anotherregular conforming mesh τH , and define XH = Pr(τH)2 and WH = Pr(τH) for some polynomialdegree r. We require that the coarse mesh interpolation operator IH used for data assimilationsatisfies the following bounds: for any w ∈ H1(Ω)d,

‖IH(w)− w‖ ≤ CH‖∇w‖, (2.9) interp1

‖IH(w)‖ ≤ C‖w‖. (2.10) interp2

These are key properties for the interpolation operator that allow for both mathematical theoryas well as providing guidance on how small H should be (i.e. how many measurement points areneeded). We note the same IH operator is used for vector functions and scalar functions, with itbeing applied component-wise for vector functions.

3 Analysis of a CDA-VV schemeanalysisDA

We consider now a discretization of (1.3) that uses a finite element spatial discretization andbackward Euler temporal discretization. The backward Euler discretization is chosen only forsimplicity of analysis; all results extend to the analogous BDF2 scheme following analysis in [2, 35].One difference of our scheme below compared to other discretizations of CDA is that IH is alsoapplied to the test functions in the nudging terms. This was first proposed by the authors in[61], and allows for a simpler stability analysis as well as to the use of special types of efficientinterpolation operators.

VVNLbe Algorithm 3.1. Given v0h ∈ Vh and w0

h ∈ Wh, find (vn+1h , wn+1

h , Pn+1h ) ∈ (Xh,Wh, Qh) for n =

0, 1, 2, ..., satisfying

1

∆t

(vn+1h − vnh , χh

)+ (wnh × vn+1

h , χh)− (Pn+1h ,∇ · χh) + ν(∇vn+1

h ,∇χh)

+µ1(IH(vn+1h − un+1), IH(χh)) = (fn+1, χh), (3.1) nlda5

(∇ · vn+1h , rh) = 0, (3.2) nlda6

1

∆t

(wn+1h − wnh , ψh

)+ b∗(vn+1

h , wn+1h , ψh) + ν(∇wn+1

h ,∇ψh)

+µ2(IH(wn+1h − rotun+1), IH(ψh)) = (rot fn+1, ψh), (3.3) nlda7

for all (χh, ψh, rh) ∈ (Xh,Wh, Qh), where IH(un+1), IH(rotun+1) are assumed known for all n ≥ 1.

We begin our analysis with long-time stability estimates, followed by long-time accuracy.

5

3.1 Stability analysis of Algorithm 3.1

In this subsection, we prove that Algorithm 3.1 is unconditionally long-time L2 and H1 stable forboth velocity and vorticity. This property was proven for the scheme without nudging in [35], andso these results show that CDA preserves this important property that is (seemingly) unique toVV schemes of this form.

L2stabboth Lemma 3.4 (L2 stability of velocity and vorticity ). Let f ∈ L∞(0,∞;L2) and u ∈ L∞(0,∞;H1).Then, for any ∆t > 0, any integer n > 0, and nudging parameters µ1, µ2 ≥ 0, velocity and vorticitysolutions to Algorithm 3.1 satisfy

‖vnh‖2 ≤α−n‖v0h‖2 +

CC2P

ν(ν−1‖f‖2L∞(0,∞;H−1) + µ1‖u‖2L∞(0,∞;L2)) =: C1, (3.5) L2stabvel

‖ωnh‖2 ≤α−n‖ω0h‖2 +

CC2P

ν(ν−1‖f‖2L∞(0,∞;L2) + µ2‖ rotu‖2L∞(0,∞;L2)) =: C2, (3.6) L2stabvortmu

where α = 1 + νC−2P ∆t.

Proof. Begin by choosing χh = 2∆tvn+1h in (3.1), which vanishes the nonlinear and pressure terms,

and leaves

‖vn+1h ‖2 − ‖vnh‖2 + ‖vn+1

h − vnh‖2 + 2∆tν‖∇vn+1h ‖2 + 2∆tµ1‖IH(vn+1

h )‖2

= 2∆t(fn+1, vn+1h ) + 2∆tµ1(IH(un+1), IH(vn+1

h )).

The first right hand side term is bounded using the dual norm and Young’s inequality via

2∆t(fn+1, vn+1h ) ≤ C∆tν−1‖fn+1‖2−1 + ∆tν‖∇vn+1

h ‖2,

and for the interpolation term, we use Cauchy-Schwarz, the interpolation property (2.10) andYoung’s inequality to get

2∆tµ1(IH(un+1), IH(vn+1h )) ≤2∆tµ1‖IH(un+1)‖‖IH(vn+1

h )‖≤C∆tµ1‖un+1‖‖IH(vn+1

h )‖≤C∆tµ1‖un+1‖2 + ∆tµ1‖IH(vn+1

h )‖2.

Combining the above estimates and dropping ‖vn+1h − vnh‖2 and ‖IH(vn+1

h )‖2 from the left handside produces the bound

‖vn+1h ‖2 + ∆tν‖∇vn+1

h ‖2 ≤ ‖vnh‖2 + C∆tν−1‖fn+1‖2−1 + C∆tµ1 ‖un+1‖2,

and thanks to the Poincare inequality, we obtain

(1 + C−2P ∆tν)‖vn+1

h ‖2 ≤ ‖vnh‖2 + C∆t(ν−1‖fn+1‖2−1 + µ1‖un+1‖2

).

Defining α = 1 + C−2P ∆tν > 1, and then applying Lemma 2.1 reveals the L2 stability bound (3.5)

for the velocity solution of Algorithm 3.1.Applying similar analysis to the above will produce the stated L2 vorticity bound.

6

h1stanVVNLbothbe Lemma 3.7 (H1 stability of velocity and vorticity ). Let f ∈ L∞(0,∞;H1) and u ∈ L∞(0,∞;H1).Then, for any ∆t > 0, any integer n > 0, and nudging parameters µ1, µ2 ≥ 0, velocity and vorticitysolutions to Algorithm 3.1 satisfy

‖∇vnh‖2 ≤α−n‖∇v0h‖2 +

CC2P

ν2

(‖f‖2L∞(0,∞;L2) + C4

2C21ν−2 + µ2

1‖u‖2L∞(0,∞;L2) + µ21C

21

)=: C1,

(3.8) H1stabvel

‖∇wnh‖2 ≤α−n‖∇w0h‖2 +

C2PC

ν2

(| rot fn+1‖2 + ν−4C6

1C22 + ν−2C4

1C22 + µ2

2‖ rotun+1‖2 + µ22C

22

),

(3.9) H1stabvortmu

where α = 1 + νC−2P ∆t.

Proof. After testing the velocity equation (3.1) with χh = 2∆tAhvn+1h , we obtain

‖∇vn+1h ‖2 − ‖∇vnh‖2 + ‖∇(vn+1

h − vnh)‖2 + 2∆tν‖Ahvn+1h ‖2

≤ 2∆t(fn+1, Ahvn+1h ) + 2∆t|(wnh × vn+1

h , Ahvn+1h )|+ 2∆tµ1(IH(un+1 − vn+1

h ), IH(Ahvn+1h )).

We now bound the right hand side terms. First, the forcing term is bounded by Cauchy-Schwarzand Young’s inequalities via

2∆t(fn+1, Ahvn+1h ) ≤ C∆tν−1‖fn+1‖2 +

ν

4∆t‖Ahvn+1‖2. (3.10) fw

Then, for the nonlinear terms, we again apply Holder, discrete Agmon (2.7) and generalized Younginequalities, and the result of Lemma 3.4 to get

2∆t|(wnh × vn+1h , Ahv

n+1h )| ≤2∆t‖wnh‖‖vn+1

h ‖L∞‖Ahvn+1h ‖

≤C∆t‖wnh‖‖vn+1h ‖1/2‖Ahvn+1

h ‖3/2

≤C∆tν−3‖wnh‖4‖vn+1h ‖2 +

ν

8∆t‖Ahvn+1

h ‖2

≤CC42C

21∆tν−3 +

ν

8∆t‖Ahvn+1

h ‖2.

Lastly, the interpolation term is bounded using Cauchy-Schwarz and interpolation property 2.10,followed by Young’s inequality and the result of Lemma 3.4 to obtain

2∆tµ1(IH(un+1 − vn+1h ), IH(Ahv

n+1h ))

≤ 2∆tµ1

(|(IHun+1, IHAhv

n+1h )|+ |(IHvn+1

h , IHAhvn+1h )|

)≤ C∆tµ1‖IHun+1‖‖IHAhvn+1

h ‖+ C∆tµ1‖IHvn+1h ‖‖IHAhvn+1

h ‖

≤ C∆tµ21ν−1‖un+1‖2 + C∆tµ2

1ν−1C2

1 +ν

8∆t‖Ahvn+1

h ‖2. (3.11) intpstab

Combining all these bounds for right hand side terms and dropping nonnegative term ‖∇(vn+1h −

vnh)‖2 on left hand side give us that

‖∇vn+1h ‖2 + ∆tν‖Ahvn+1

h ‖2 ≤ ‖∇vnh‖2 + C∆tν−1‖fn+1‖2 + ν−3∆tCC42C

21 + ∆tµ2

1ν−1C‖un+1‖2

+ C∆tµ21ν−1C2

1 .

7

By the Poincare inequality (2.5), we now get

α‖∇vn+1h ‖2 ≤ ‖∇vnh‖2 + C∆t

(ν−1‖fn+1‖2 + ν−3C4

2C21 + µ2

1ν−1‖un+1‖2 + µ2

1ν−1C2

1

),

where α = 1 + νC−2P ∆t. Finally, we apply Lemma 2.1 and reveal (3.8).

For the vorticity estimate, choose ψh = 2∆t∆hwn+1h in (3.3) to get

‖∇wn+1h ‖2 − ‖∇wnh‖2 + ‖∇(wn+1

h − wnh)‖2 + 2∆tν‖∆hwn+1h ‖2

≤ 2∆t|(rot fn+1,∆hwn+1h )|+ 2∆t|b∗(vn+1

h , wn+1h ,∆hw

n+1h )|+ 2∆tµ2(IH(rotun+1 − wn+1

h ), IH(∆hwn+1h )).

From here, the proof follows the same strategy as the H1 velocity proof above, except the nonlinearterm is handled slightly differently. We use the discrete Agmon inequality (2.7), the discrete Sobolevinequality (2.8), the result of Lemma 3.4, the H1 stability bound for vorticity (3.8) proven above,and the generalized Young’s inequality, as follows.

2∆t|b∗(vn+1h , wn+1

h ,∆hwn+1h )|

≤ 2∆t

(|(vn+1

h · ∇wn+1h ,∆hw

n+1h )|+ 1

2|((∇ · vn+1

h )wn+1h ,∆hw

n+1h )|

)≤ 2∆t‖vn+1

h ‖L6‖∇wn+1h ‖L3‖∆hw

n+1h ‖+ ∆t‖∇vn+1

h ‖‖wn+1h ‖L∞‖∆hw

n+1h ‖

≤ C∆tC1‖wn+1h ‖1/3‖∆hw

n+1h ‖5/3 + C∆tC1‖wn+1

h ‖1/2‖∆hwn+1h ‖3/2

≤ C∆tC1C1/32 ‖∆hw

n+1h ‖5/3 + C∆tC1C

1/22 ‖∆hw

n+1h ‖3/2

≤ C∆tν−5C61C

22 + C∆tν−3C4

1C22 +

ν

3∆t‖∆hw

n+1h ‖2.

Now proceeding as in the velocity H1 bound will produce the H1 vorticity stability bound (3.9).

3.2 Long-time accuracy of Algorithm 3.1

We now consider the difference between the solutions of (3.1) - (3.3) to the NSE solution. We willshow that the algorithm solution converges to the true solution, up to an optimal O(∆t + hk+1)discretization error, independent of the initial condition, provided a restriction on the coarse meshwidth and nudging parameters. We will give two results, the first for µ2 > 0 and the second forµ2 = 0; while they both provide optimal long-time accuracy, when µ2 > 0 the convergence to thetrue solution occurs more rapidly in time.

In our theory below for long-time accuracy of Algorithm 3.1, we assume the use of Scott-Vogelius elements. This is done for simplicity, as for non-divergence-free elements like Taylor-Hoodelements, similar optimal results can be obtained (although with some additional terms and differentconstants) but require more technical details; see, e.g., [29].

nlthmL2 Theorem 3.12 (Long-time L2 accuracy of Algorithm 3.1 with µ1 > 0 and µ2 > 0). Let true solu-tions u ∈ L∞(0,∞;Hk+2(Ω)), p ∈ L∞(0,∞;Hk(Ω)) where k ≥ 1 and ut, utt ∈ L∞(0,∞;H1), andwe assume properties of the domain permits optimal L2 and H1 accuracy of the discrete Stokesprojection in Vh and discrete H1

0 projection into Wh. Then, assume that time step ∆t is sufficientlysmall, and that µ1 and µ2 satisfy

max

1, Cν−1(‖ωn+1‖2L∞ + ‖ηn+1

w ‖2L∞)≤ µ1 ≤

Cν

H2,

8

max

1, Cν−1(‖un+1‖2L3 + ‖ηn+1

v ‖2L3

)≤ µ2 ≤

Cν

H2,

where H is chosen so that this inequality holds. Then, for any time tn, n = 0, 1, 2, ..., we have forsolutions of Algorithm 3.1 using Scott-Vogelius elements,

‖vnh − un‖2 + ‖ωnh − rotun‖2 ≤ (1 + λ∆t)−n(‖v0h − u0‖2 + ‖ω0

h − rotu0‖2) + Cλ−1R,

whereR :=

(µ−1

1 ∆t2 + µ−12 ∆t2 + ν−1h2k+2 + µ1h

2k+2 + µ2h2k+2

),

and λ = minµ14 +

νC−2P4 , µ24 +

νC−2P4

with C independent of ∆t, h and H.

Proof. The true NSE solution satisfies the VV system

1

∆t(un+1 − un) + ωn × un+1 +∇Pn+1 − ν∆un+1 = fn+1 −∆tutt(t

∗) + (ωn − ωn+1)× un+1,

∇ · un+1 = 0,

1

∆t(ωn+1 − ωn) + un+1 · ∇ωn+1 − ν∆ωn+1 = rot fn+1 −∆tωtt(t

∗∗),

where un is the velocity at time tn, Pn the Bernoulli pressure, ωn := rotun, and t∗, t∗∗ ∈ [tn, tn+1].Note that by Taylor expansion, we can write ωn − ωn+1 = −∆tωt(s

∗) where s∗ ∈ [tn, tn+1].The difference equations are obtained by subtracting the solutions to Algorithm 3.1 from the

NSE solutions by defining the differences between velocity and vorticity as env := un − vnh andenw := ωn−wnh , respectively. Next, we will decompose the error into a term that lies in the discretespace Vh and one outside. To do so, add and subtract the discrete Stokes projection of un, denotedsnh, to env and let ηnv := snh − un, φnh,v := vnh − snh. Then env = φnh,v + ηnv and φnh,v ∈ Vh. In a similar

manner, by taking the H10 projection of rotun into Wh, we obtain enw = φnh,w + ηnw with φnh,w ∈ Vh.

For velocity, since (∇ηn+1v ,∇φn+1

h,v ) = 0, the difference equation becomes

1

2∆t[‖φn+1

h,v ‖2 − ‖φnh,v‖2 + ‖φn+1

h,v − φnh,v‖2] + ν‖∇φn+1

h,v ‖2 + µ1‖φn+1

h,v ‖2

= −∆t(utt(t∗), φn+1

h,v )− 1

∆t(ηn+1v − ηnv , φn+1

h,v ) + ∆t(ωtt(s∗), φn+1

h,v )− (enω × vn+1h , φn+1

h,v )

− (ωn+1 × ηn+1v , φn+1

h,v )− 2µ1(IH(φn+1h,v )− φn+1

h,v , φn+1h,v )− µ1‖IHφn+1

h,v − φn+1h,v ‖

2

− µ1(IHηn+1v , IHφ

n+1h,v ), (3.13) veldiff

and similarly for vorticity, we have

1

2∆t[‖φn+1

h,w ‖2 − ‖φnh,w‖2 + ‖φn+1

h,w − φnh,w‖2] + ν‖∇φn+1

h,w ‖2 + µ2‖φn+1

h,w ‖2

= −∆t(ωtt(t∗∗), φn+1

h,w )− 1

∆t(ηn+1w − ηnw, φn+1

h,w ) + b∗(en+1v , ηn+1

w , φn+1h,w )

+ b∗(un+1, ηn+1w , φn+1

h,w ) + b∗(en+1v , ωn+1, φn+1

h,w )− 2µ2(IH(φn+1h,w )− φn+1

h,w , φn+1h,w )

− µ2‖IHφn+1h,w − φ

n+1h,w ‖

2 − µ2(IHηn+1w , IHφ

n+1h,w ), (3.14) vortdiffmu

9

where in (3.13) we have added and subtracted φn+1h,v to write it in the form found above using

µ1(IHen+1v , IHχh) = µ1(IHφ

n+1h,v , IHχh) + µ1(IHη

n+1v , IHχh)

= µ1(IHφn+1h,v − φ

n+1h,v + φn+1

h,v , IHχh − φn+1h,v + φn+1

h,v ) + µ1(IHηn+1v , IHχh)

= µ1‖φn+1h,v ‖

2 + µ1(IHφn+1h,v − φ

n+1h,v , φ

n+1h,v ) + µ1(φn+1

h,v , IHχh − φn+1h,v )

+ µ1(IHφn+1h,v − φ

n+1h,v , IHχh − φ

n+1h,v ) + µ1(IHη

n+1v , χh),

and similarly for (3.14).Next, we bound the terms on right hand side of difference equations, starting with the velocity

difference equation (3.13). The first three right hand side terms are bounded using Cauchy-Schwarzand Young’s inequalities, via

∆t(utt(t∗), φn+1

h,v ) ≤ ∆t‖utt‖L∞(0,∞;L2(Ω))‖φn+1h,v ‖

≤ C∆t2µ−11 ‖utt‖

2L∞(0,∞;L2(Ω)) +

µ1

20‖φn+1

h,v ‖2,

∆t(ωtt(s∗), φn+1

h,v ) ≤ C∆t‖ωtt‖L∞(0,∞;L2(Ω))‖un+1‖L∞‖φn+1h,v ‖

≤ C∆t2µ−11 ‖ωtt‖

2L∞(0,∞;L2(Ω))‖u

n+1‖2L∞ +µ1

20‖φn+1

h,v ‖2,

1

∆t(ηn+1v − ηnv , φn+1

h,v ) = (ηv,t(s∗), φn+1

h,v )

≤ ‖ηv,t(s∗)‖‖φn+1h,v ‖

≤ Cµ−11 ‖ηv,t(s

∗)‖2 +µ1

20‖φn+1

h,v ‖2,

where s∗ ∈ [tn, tn+1].For nonlinear terms in (3.13), first we add and subtract en+1

w in the first component, and un+1

in second component to get

(enw × vn+1h , φn+1

h,v ) = ((enw − en+1w )× en+1

v , φn+1h,v ) + (en+1

w × en+1v , φn+1

h,v )

+ ((enw − en+1w )× un+1, φn+1

h,v ) + (en+1w × un+1, φn+1

h,v )

= ((enw − en+1w )× ηn+1

v , φn+1h,v ) + (en+1

w × ηn+1v , φn+1

h,v )

+ ((enw − en+1w )× un+1, φn+1

h,v ) + (en+1w × un+1, φn+1

h,v ).

The all resulting terms are bounded by Holder’s and Young’s inequalities to obtain

((enw − en+1w )× ηn+1

v , φn+1h,v ) ≤ C‖φnh,w − φn+1

h,w ‖‖ηn+1v ‖L3‖φn+1

h,v ‖L6 + C‖ηnw − ηn+1w ‖L∞‖ηn+1

v ‖‖φn+1h,v ‖

≤ Cν−1‖φnh,w − φn+1h,w ‖

2‖ηn+1v ‖2L3 +

ν

8‖∇φn+1

h,v ‖2

+ Cµ−11 ‖η

nw − ηn+1

w ‖2L∞‖ηn+1v ‖2 +

µ1

20‖φn+1

h,v ‖2,

10

(en+1w × ηn+1

v , φn+1h,v ) ≤ C‖φn+1

h,w ‖‖ηn+1v ‖L3‖φn+1

h,v ‖L6 + C‖ηn+1w ‖‖ηn+1

v ‖L∞‖φn+1h,v ‖

≤ Cν−1‖φn+1h,w ‖

2‖ηn+1v ‖2L3 +

ν

8‖∇φn+1

h,v ‖2

+ Cµ−11 ‖η

n+1w ‖2‖ηn+1

v ‖2L∞ +µ1

20‖φn+1

h,v ‖2,

((enw − en+1w )× un+1, φn+1

h,v ) ≤ C‖φnh,w − φn+1h,w ‖‖u

n+1‖L3‖φn+1h,v ‖L6 + C‖ηnw − ηn+1

w ‖‖un+1‖L∞‖φn+1h,v ‖

≤ Cν−1‖φnh,w − φn+1h,w ‖

2‖un+1‖2L3 +ν

8‖∇φn+1

h,v ‖2

+ Cµ−11 ‖η

nw − ηn+1

w ‖2‖un+1‖2L∞ +µ1

20‖φn+1

h,v ‖2,

(en+1w × un+1, φn+1

h,v ) ≤ C‖φn+1h,w ‖‖u

n+1‖L3‖φn+1h,v ‖L6 + C‖ηn+1

w ‖‖un+1‖L∞‖φn+1h,v ‖

≤ Cν−1‖φn+1h,w ‖

2‖un+1‖2L3 +ν

8‖∇φn+1

h,v ‖2

+ Cµ−11 ‖η

n+1w ‖2‖un+1‖2L∞ +

µ1

20‖φn+1

h,v ‖2.

Then, for last nonlinear term, we apply Holder’s, Poincare’s and Young’s inequalities and get

(ωn+1 × ηn+1v , φn+1

h,v ) ≤ ‖ωn+1‖L∞‖ηn+1v ‖‖φn+1

h,v ‖

≤ Cν−1‖ωn+1‖2L∞‖ηn+1v ‖2 +

ν

8‖∇φn+1

h,v ‖2.

Next, the first interpolation term on the right hand side of (3.13) will be bounded with Cauchy-Schwarz inequality and (2.9) to obtain

µ1(IH(φn+1h,v )− φn+1

h,v , φn+1h,v ) ≤ µ1‖IH(φn+1

h,v )− φn+1h,v ‖‖φ

n+1h,v ‖

≤ µ1CH‖∇φn+1h,v ‖‖φ

n+1h,v ‖

≤ Cµ1H2‖∇φn+1

h,v ‖2 +

µ1

20‖φn+1

h,v ‖2.

For the second interpolation term, we apply inequality (2.10), which yields

µ1‖IHφn+1h,v − φ

n+1h,v ‖

2 ≤ µ1H2‖∇φn+1

h,v ‖2.

Finally, the last interpolation term will be bounded using Cauchy-Schwarz, (2.10), and Young’sinequality to get the bound

µ1(IHηn+1v , IHφ

n+1h,v ) ≤ Cµ1‖ηn+1

v ‖2 +µ1

20‖φn+1

h,v ‖2.

We now move on to the vorticity difference equation, (3.14). All the linear terms are majorized ina similar manner as in the velocity case, and so we show below the bounds only for the nonlinearterms. Due to the use of Scott-Vogelius elements, the skew-symmetric form reduces to the usualconvective form, so b∗(u, v, w) = (u ·∇v, w), with ‖∇·u‖ = 0. To bound the first nonlinear term on

11

the right hand side of (3.14), we begin by breaking up the velocity error term, then apply Holder’sand Young’s inequalities, yielding

b∗(en+1v , ηn+1

w , φn+1h,w ) = (en+1

v · ∇ηn+1w , φn+1

h,w )

= (φn+1h,v · ∇η

n+1w , φn+1

h,w ) + (ηn+1v · ∇ηn+1

w , φn+1h,w )

= (φn+1h,v · ∇φ

n+1h,w , η

n+1w ) + (ηn+1

v · ∇φn+1h,w , η

n+1w )

≤ C‖φn+1h,v ‖‖∇φ

n+1h,w ‖‖η

n+1w ‖L∞ + C‖ηn+1

v ‖‖∇φn+1h,w ‖‖η

n+1w ‖L∞

≤ Cν−1‖φn+1h,v ‖

2‖ηn+1w ‖2L∞ +

ν

10‖∇φn+1

h,w ‖2 + Cν−1‖ηn+1

v ‖2‖ηn+1w ‖2L∞

+ν

10‖∇φn+1

h,w ‖2.

For the second nonlinear term, we use Holder’s, Poincare’s and Young’s inequalities, which gives

b∗(un+1, ηn+1w , φn+1

h,w ) = (un+1 · ∇ηn+1w , φn+1

h,w )

= (un+1 · ∇φn+1h,w , η

n+1w )

≤ C‖un+1‖L∞‖∇φn+1h,w ‖‖η

n+1w ‖

≤ Cν−1‖un+1‖2L∞‖ηn+1w ‖2 +

ν

10‖∇φn+1

h,w ‖2.

For the last nonlinear term, we begin by breaking up the velocity error term, then apply Holder’sand Young’s inequalities to get

b∗(en+1v , ωn+1, φn+1

h,w ) = (en+1v · ∇ωn+1, φn+1

h,w )

= (φn+1h,v · ∇ω

n+1, φn+1h,w ) + (ηn+1

v · ∇ωn+1, φn+1h,w )

= (φn+1h,v · ∇φ

n+1h,w , ω

n+1) + (ηn+1v · ∇φn+1

h,w , ωn+1)

≤ C‖φn+1h,v ‖‖∇φ

n+1h,w ‖‖ω

n+1‖L∞ + C‖ηn+1v ‖‖∇φn+1

h,w ‖‖ωn+1‖L∞

≤ Cν−1‖φn+1h,v ‖

2‖ωn+1‖2L∞ +ν

10‖∇φn+1

h,w ‖2 + Cν−1‖ηn+1

v ‖2‖ωn+1‖2L∞

+ν

10‖∇φn+1

h,w ‖2.

Replacing the right hand sides of (3.13) and (3.14) with the computed bounds and droppingnonnegative terms with ‖φn+1

h,v − φnh,v‖2 yields the bound

1

2∆t

(‖φn+1

h,v ‖2 + ‖φn+1

h,w ‖2 − ‖φnh,v‖2 − ‖φnh,w‖2

)+

(1

2∆t− Cν−1(‖ηn+1

v ‖2L3 − ‖un+1‖2L3)

)‖φn+1

h,w − φnh,w‖2

+ν

4‖∇φn+1

h,v ‖2 +

(ν4− Cµ1H

2)‖∇φn+1

h,v ‖2 +

ν

4‖∇φn+1

h,w ‖2 +

(ν4− Cµ2H

2)‖∇φn+1

h,w ‖2

+µ1

4‖φn+1

h,v ‖2 +

(µ1

4− Cν−1(‖ωn+1‖2L∞ + ‖ηn+1

w ‖2L∞))‖φn+1

h,v ‖2

+µ2

4‖φn+1

h,w ‖2 +

(µ2

4− Cν−1(‖un+1‖2L3 + ‖ηn+1

v ‖2L3))‖φn+1

h,w ‖2

≤ C∆t2(µ−1

1 ‖utt‖2L∞(0,∞;L2) + µ−1

1 ‖ωtt‖2L∞(0,∞;L2)‖u

n+1‖2L∞ + µ−12 ‖ωtt‖

2L∞(0,∞;L2)

)+ Cµ−1

1

(‖ηv,t‖2L∞(0,∞;L2) + ‖ηn+1

w − ηnw‖2L∞‖ηn+1v ‖2 + ‖ηn+1

w ‖2‖ηn+1v ‖2L∞ + ‖ηn+1

w − ηnw‖2‖un+1‖2L∞

+ ‖ηn+1w ‖2‖un+1‖2L∞

)+ Cµ−1

2 ‖ηw,t‖2L∞(0,∞;L2) + Cν−1

(‖ωn+1‖2L∞‖ηv‖2 + ‖ηn+1

v ‖2‖ηn+1w ‖2L∞

+ ‖ηn+1v ‖2‖ωn+1‖2L∞ + ‖ηn+1

w ‖2‖un+1‖2L∞)

+ Cµ1‖ηn+1v ‖2 + Cµ2‖ηn+1

w ‖2.

12

Using the assumptions on H and the nudging parameters, the time step restriction, and smoothnessof the true solution, this reduces to

1

2∆t

(‖φn+1

h,v ‖2 + ‖φn+1

h,w ‖2 − ‖φnh,v‖2 − ‖φnh,w‖2

)+ν

4‖∇φn+1

h,v ‖2 +

ν

4‖∇φn+1

h,w ‖2 +

µ1

4‖φn+1

h,v ‖2 +

µ2

4‖φn+1

h,w ‖2

≤ C∆t2(µ−1

1 + µ−11 + µ−1

2

)+ Cµ−1

2 ‖ηw,t‖2L∞(0,∞;L2)

+ Cµ−11

(‖ηv,t‖2L∞(0,∞;L2) + ‖ηn+1

w − ηnw‖2L∞‖ηn+1v ‖2 + ‖ηn+1

w ‖2‖ηn+1v ‖2L∞ + ‖ηn+1

w − ηnw‖2 + ‖ηn+1w ‖2

)+ Cν−1

(‖ηv‖2 + ‖ηn+1

v ‖2‖ηn+1w ‖2L∞ + ‖ηn+1

v ‖2 + ‖ηn+1w ‖2

)+ Cµ1‖ηn+1

v ‖2 + Cµ2‖ηn+1w ‖2.

Now define

λ1 :=µ1

4+νC−2

P

4,

λ2 :=µ2

4+νC−2

P

4.

Using this in the inequality after applying Poincare’s inequality and multiplying each side by 2∆t,we get

(1 + ∆tλ1)‖φn+1h,v ‖

2 + (1 + ∆tλ2)‖φn+1h,w ‖

2 ≤ C∆t(µ−1

1 ∆t2 + ν−1∆t2 + ν−1h2k+2 + µ1h2k+2 + µ2h

2k+2)

+ ‖φnh,v‖2 + ‖φnh,w‖2.

Then, with R :=(µ−1

1 ∆t2 + ν−1∆t2 + ν−1h2k+2 + µ1h2k+2 + µ2h

2k+2)

and λ := min λ1, λ2, weobtain the bound

(1 + λ∆t)(‖φn+1

h,v ‖2 + ‖φn+1

h,w ‖2)≤ C∆tR+ ‖φnh,v‖2 + ‖φnh,w‖2.

By Lemma 2.1, this implies

‖φn+1h,v ‖

2 + ‖φn+1h,w ‖

2 ≤ Cλ−1R+ (1 + λ∆t)−(n+1)(‖φ0h,v‖2 + ‖φ0

h,w‖2).

Lastly, applying triangle inequality completes the proof.

Theorem 3.15 (Long-time L2 accuracy of Algorithm 3.1 with µ1 > 0, µ2 = 0 ). Let true solutionu ∈ L∞(0,∞;Hk+2(Ω)), p ∈ L∞(0,∞;Hk(Ω)) where k ≥ 1 and ut, utt,∈ L∞(0,∞;H1). Then,assume that time step ∆t is sufficiently small, µ2 = 0, and that µ1 satisfies

max

1, Cν−1(‖un+1‖L∞ + ‖ηn+1v ‖L∞), Cν−1(‖ωn+1‖L∞ + ‖ηn+1

w ‖L∞)≤ µ1 ≤

Cν

H2,

where H is chosen so that this inequality holds. Then for any time tn, n = 0, 1, 2, ..., solutions ofof Algorithm 3.1 using Scott-Vogelius element satisfy

‖vnh − un‖2 + ‖ωnh − rotun‖2 ≤ (1 + λ∆t)−n(‖v0h − u0‖2 + ‖ω0

h − rotu0‖2) + Cλ−1R, (3.16) l2accboth

whereR :=

(µ−1

1 ∆t2 + ν−1∆t2 + ν−1h2k+2 + µ1h2k+2

),

and λ =C−2

P ν4 with C independent of ∆t, h and H .

13

Remark 3.17. Algorithm 3.1 converges to the true solutions up to optimal discretization errorin both cases µ1, µ2 > 0 and µ1 > 0, µ2 = 0. The key difference between two cases is that whenµ2 = 0, the convergence in time to reach optimal accuracy is much slower since λ does not scalewith the nudging parameters. This phenomena is illustrated in our numerical tests.

Proof. We follow the same steps with the proof of Theorem 3.12. The difference equation forvelocity is already the same with (3.13), and just two nonlinear terms in the velocity differenceequation are bounded with differently in this case. By Holder, Poincare and Young’s inequalities,we get the bounds

(en+1w × ηn+1

v , φn+1h,v ) ≤ C‖φn+1

h,w ‖‖ηn+1v ‖L3‖φn+1

h,v ‖L6 + C‖ηn+1w ‖‖ηn+1

v ‖L∞‖φn+1h,v ‖

≤ Cµ−11 ‖∇φ

n+1h,w ‖

2‖ηn+1v ‖2L∞ +

µ1

16‖φn+1

h,v ‖2

+ Cµ−11 ‖η

n+1w ‖2‖ηn+1

v ‖2L∞ +µ1

20‖φn+1

h,v ‖2,

(en+1w × un+1, φn+1

h,v ) ≤ C‖φn+1h,w ‖‖u

n+1‖L3‖φn+1h,v ‖L6 + C‖ηn+1

w ‖‖un+1‖L∞‖φn+1h,v ‖

≤ Cµ−11 ‖∇φ

n+1h,w ‖

2‖un+1‖2L∞ +µ1

16‖φn+1

h,v ‖2

+ Cµ−11 ‖η

n+1w ‖2‖un+1‖2L∞ +

µ1

20‖φn+1

h,v ‖2.

All terms on the right hand side of vorticity difference equation for Theorem 3.12 are boundedidentically. Proceeding as in the previous proof, we arrive at

1

2∆t

(‖φn+1

h,v ‖2 + ‖φn+1

h,w ‖2 − ‖φnh,v‖2 − ‖φnh,w‖2

)+

(1

2∆t− Cν−1(‖ηn+1

v ‖2L3 − ‖un+1‖2L3)

)‖φn+1

h,w − φnh,w‖2

+µ1

4‖φn+1

h,v ‖2 +

(µ1

4− Cν−1(‖ωn+1‖2L∞ + ‖ηn+1

w ‖2L∞))‖φn+1

h,v ‖2 +

ν

4‖∇φn+1

h,v ‖2

+(ν

4− Cµ1H

2)‖∇φn+1

h,v ‖2 +

ν

4‖∇φn+1

h,w ‖2 +

(ν4− Cµ−1

1 (‖un+1‖L∞ + ‖ηn+1v ‖L∞)

)‖∇φn+1

h,w ‖2

≤ C∆t2(µ−1

1 ‖utt‖2L∞(0,∞;L2) + µ−1

1 ‖ωtt‖2L∞(0,∞;L2)‖u

n+1‖2L∞ + µ−12 ‖ωtt‖

2L∞(0,∞;L2)

)+ Cµ−1

1

(‖ηv,t‖2L∞(0,∞;L2) + ‖ηn+1

w − ηnw‖2L∞‖ηn+1v ‖2 + ‖ηn+1

w ‖2‖ηn+1v ‖2L∞ + ‖ηn+1

w − ηnw‖2‖un+1‖2L∞+ ‖ηn+1

w ‖2‖un+1‖2L∞)

+ Cν−1(‖ωn+1‖2L∞‖ηv‖2 + ‖ηn+1

v ‖2‖ηn+1w ‖2L∞ + ‖ηn+1

v ‖2‖ωn+1‖2L∞+ ‖ηn+1

w ‖2‖un+1‖2L∞)

+ Cµ1‖ηn+1v ‖2.

Provided ∆t is sufficiently small and the restriction

max

1, Cν−1(‖un+1‖L∞ + ‖ηn+1v ‖L∞), Cν−1(‖ωn+1‖L∞ + ‖ηn+1

w ‖L∞)≤ µ1 ≤

Cν

H2,

holds, then applying Poincare inequality to the terms on left hand side and using

λ1 :=µ1

4+C−2P ν

4,

λ2 :=C−2P ν

4,

14

and assumptions on the true solution, we obtain

(1 + ∆tλ1)‖φn+1h,v ‖

2 + (1 + ∆tλ2)‖φn+1h,w ‖

2 ≤ C∆t(µ−1

1 ∆t2 + ν−1∆t2 + ν−1h2k+2 + µ1h2k+2

)+ ‖φnh,v‖2 + ‖φnh,w‖2.

From here, the proof is finished in the same way as the previous theorem.

3.3 Second order temporal discretization

We now present results for a second order analogue of the first order algorithm studied above.

VVNLbdf2 Algorithm 3.2. Find (vn+1h , wn+1

h , qn+1h ) ∈ (Xh,Wh, Qh) for n = 0, 1, 2, ..., satisfying

1

2∆t

(3vn+1h − 4vnh + vn−1

h , χh)

+ ((2wnh − wn−1h )× vn+1

h , χh)− (Pn+1h ,∇ · χh) + ν(∇vn+1

h ,∇χh)

+µ1(IH(vn+1h − un+1), IHχh) = (fn+1, χh),

(3.18) nlda5a

(∇ · vn+1h , rh) = 0, (3.19) nlda6a

1

2∆t

(3wn+1

h − 4wnh − vn−1h , ψh

)+ (vn+1

h · ∇wn+1h , ψh) + ν(∇wn+1

h ,∇ψh)

+µ2(IH(wn+1h − rotun+1), IH(ψh)) = (rot fn+1, ψh),

(3.20) nlda7a

for all (χh, ψh, rh) ∈ (Xh,Wh, Qh), with v0 ∈ X and IH(un+1), IH(rotun+1) given.

Stability and convergence results follow in the same manner as the first order scheme resultsabove, using G-stability theory as in [1, 2, 45].

nlthmbdf2L2 Theorem 3.21 (Long-time stability and accuracy of Algorithm 3.2 with µ1 > 0 and µ2 > 0). Forany time step ∆t > 0, and any time tn, n = 0, 1, 2, ..., we have that solutions of Algorithm 3.2 satisfy

‖vnh‖+ ‖wnh‖+ ‖∇vnh‖+ ‖∇wnh‖ ≤ C,

with C independent of n, ∆t, h, H.Furthermore, if we suppose the true solution u ∈ L∞(0,∞;Hk+2(Ω)), p ∈ L∞(0,∞;Hk(Ω))

where k ≥ 1 and ut, uttt,∈ L∞(0,∞;H1), that time step ∆t is sufficiently small, Scott-Vogeliuselements are used, and that µ1 and µ2 satisfy C(u) ≤ µ1, µ2 ≤ Cν

H2 , we have the bound

‖vnh − un‖2 + ‖ωnh − rotun‖2 ≤(1 + λ∆t)−n(‖v0

h − u0‖2 + ‖ω0h − rotu0‖2 + ‖v1

h − u1‖2 + ‖ω1h − rotu1‖2) + Cλ−1R,

whereR :=

(µ−1

1 ∆t4 + µ−12 ∆t4 + ν−1h2k+2 + µ1h

2k+2 + µ2h2k+2

),

and λ = minµ14 +

νC−2P4 , µ24 +

νC−2P4

with C independent of ∆t, h and H.

15

4 Numerical Experiments

In this section, we illustrate the above theory with two numerical tests, both using Algorithm 3.2.Our first test is for convergence rates on a problem with analytical solution, and the second test isfor flow past a flat plate. For both tests, we report results only for (P2, P1) Taylor-Hood elementsfor velocity and pressure, and P2 for vorticity; however we also tried Scott-Vogelius elements onbarycenter refined meshes that produced similar numbers of degrees of freedom, and results werevery similar to those of Taylor-Hood. The coarse velocity and vorticity spaces XH and WH aredefined to be piecewise constants on the same mesh used for the computations. The interpolationoperator IH was taken to be the L2 projection operator onto XH (or WH), which is known tosatisfy (2.9)-(2.10) [18].

4.1 Experiment 1: convergence rate test

For our first test, we investigate the theory above for Algorithm 3.2. Here we use the analyticsolution

u =

[cos(π(y − t))sin(π(x+ t))

], p = (1 + t2) sin(x+ y),

on the unit square domain Ω = (0, 1)2 with kinematic viscosity ν = 1.0, and use and the NSE todetermine f and boundary conditions. We take the final time T = 1, and choose initials conditionsfor Algorithm 3.2’s velocity and vorticity to be 0. For the discretization, (P2, P1) Taylor-Hoodelements are used for velocity and pressure, P2 for vorticity, and a time step size of ∆t = 0.001.From Section 3, we expect third order spatial convergence rate in the L2 norm for large enoughtimes. Results are presented below for two cases, µ2 > 0 and µ2 = 0.

4.1.1 Results for µ1 > 0 and µ2 > 0

To test this case, we first calculated spatial convergence rates at the final time T = 1 with the L2

error, using successively refined uniform meshes and µ1 = µ2 = 100. Errors and rates are shown intable 1, and show clear third order spatial convergence of both velocity and vorticity. Deteriorationof the rates for the smallest h is expected since the time step ∆t is fixed while the spatial meshwidth decreases.

h ‖ev(T )‖ rate ‖ew(T )‖ rate

1/4 2.62008e-03 - 7.70647e-03 -1/8 3.20467e-04 3.0314 9.68456e-04 2.99231/16 3.97307e-05 3.0146 1.20888e-04 3.00411/32 4.94529e-06 3.0061 1.50809e-05 3.00291/64 6.19332e-07 2.9973 1.99325e-06 2.91951/128 8.13141e-08 2.9247 3.15236e-07 2.5855

Table 1: Shown above are L2 velocity and vorticity errors and convergence rates on varying meshwidths, at the final time T = 1, using Algorithm 3.2 with µ1 = µ2 = 100. rates1

We next consider convergence to the true solution exponentially in time (up to discretizationerror). Here we take h = 1/32, and compute solutions using µ1 = µ2 = µ, with µ = 1, 10, 100, 1000.Results are shown in figure 1, as L2 error versus time for velocity and vorticity. We observe

16

exponential convergence in time of both velocity and vorticity, up to about 10−5, which is consistentwith the choices of h and ∆t and the accuracy of the method. We note that as µ is increased,convergence is faster in time, which is consistent with our theory for the case of µ1 > 0 and µ2 > 0.

Figure 1: Shown above are L2 velocity and vorticity errors for Algorithm 3.2 with µ1 = µ2 = µ,with varying µ > 0. covplot1

4.1.2 Results for µ1 > 0 and µ2 = 0

We now consider the same tests as above, but with µ2 = 0. This is an important case, since itis not always practical to obtain accurate vorticity measurement data. Just as in the first case,we first calculated spatial convergence rates at the final time T = 1 for the L2 error, on the samesuccessively refined uniform meshes, but now with µ1 = 100 and µ2 = 0. Errors and rates areshown in table 1, and show clear third order spatial convergence of both velocity and vorticity.Deterioration of the rates for the smallest h is expected since the time step ∆t was fixed at 0.001,although the vorticity errors are slightly worse than for the case of µ2 = 100 shown in table 1, andthe deterioration of the rates occurs a bit earlier. Hence we observe essentially the same velocityerrors and rates compared to the case of µ2 = 100, and slightly worse vorticity error but still withoptimal L2 accuracy.

h ‖ev(T )‖ rate ‖ew(T )‖ rate

1/4 2.62003e-03 - 7.79431e-03 -1/8 3.20466e-04 3.0313 9.70492e-04 3.00561/16 3.97175e-05 3.0123 1.20897e-04 3.00491/32 4.94501e-06 3.0057 1.50883e-05 3.00231/64 6.17406e-07 3.0017 2.08215e-06 2.85731/128 8.11244e-08 2.9280 9.37122e-07 1.1518

Table 2: Shown above are L2 velocity and vorticity errors and convergence rates on varying meshwidths, at the final time T = 1, using Algorithm 3.2 with µ1 = 100 and µ2 = 100. rates

To test exponential convergence in time for the case of µ2 = 0, we again take h = 1/32, andcompute solutions using µ1 = 1, 10, 100, 1000. Results are shown in figure 2, as L2 error versustime for velocity and vorticity. Although we do observe exponential convergence in time of bothvelocity and vorticity, up to about 10−5 which is the same accuracy reached when µ2 = 100 above.

17

An important difference here compared to when µ2 = 100 is that the convergence of vorticity to thetrue solution is independent of µ1, and the convergence of velocity is slower for larger choices of µ1.This reduced dependence of the convergence on the nudging parameters when µ2 = 0 is consistentwith our theory. Hence without vorticity nudging, long-time optimal accuracy is still achieved, butit takes longer in time to get there.

Figure 2: Shown above are L2 velocity and vorticity errors (from left to right) for Algorithm 3.2with varying µ1 and µ2 = 0. covplot

4.2 Experiment 2: Flow past a normal flat plate

To test Algorithm 3.2 on a more practical problem, we consider flow past a flat plate with Re = 50.The domain of this problem is [−7, 20]× [−10, 10] rectangular channel with a 0.125× 1 plate fixedten units into the channel from the left, vertically centered. The inflow velocity is uin = 〈0, 1〉,no-slip velocity and the corresponding natural vorticity boundary condition from [56] are used onthe walls and plate, and homogeneous Neumann conditions are enforced weakly at the outflow. Asetup for the domain is shown in figure 3. There is no external forcing applied, f = 0. The viscosityis taken to be ν = 1/50 which is inversely proportional to Re, based on the height of the plate.The end time for the test is T = 80. A DNS was run until for 160 time units (from t=-80 to t=80),and for t > 0 measurement data for the VV-DA simulation was sampled from the DNS.

Figure 3: Setup for the flow past a normal flat plate. fig:plate_setup

We computed solutions using a Delaunay generated triangular meshes that provided 27, 373 total

18

degree of freedom with (P 2, P 1, P 2) velocity-pressure-vorticity elements, and time step ∆t = 0.02.We first compared convergence in time to the DNS solution, for two cases: µ1 = µ2 > 0 andµ1 > −0, µ2 = 0. Plots of L2 velocity and vorticity error for both of these cases are shown in figure4, with varying nudging parameters. There is a clear advantage seen in the plots for the simulationswith µ2 > 0: when vorticity is nudged in addition to velocity, convergence to the true solution ismuch faster in time. The convergence when µ2 = 0 appears to still be occurring, but is much slowerand even by t = 80 the L2 vorticity error is barely smaller than O(1). We note that just like in theanalytical test problem, when µ2 = 0 the vorticity convergence in time is independent of µ1.

Figure 4: L2 velocity and vorticity errors (from left to right) for Algorithm 3.2 with µ1 = µ2 = µ >0(top) and µ1 = µ > 0, µ2 = 0 (bottom) errorplot

To further illustrate the convergence of the DNS, we show contour plots of the DNS solution, theVVDA solution, and their difference, in figures 5-8. For these simulations, we used µ1 = µ2 = 10in figures 5-6, but used µ2 = 0 for figures 7-8. As expected due to the plots in figure 4, whenµ1 = µ2 = 10 we observe rapid convergence in the plots for VV-DA velocity and vorticity to theDNS velocity and vorticity, and by t = 1 the contour plots are visually indistinguishable. Thisis not the case, however, when µ2 = 0. In this case, while the velocity plots do agree with DNSvelocities by t = 1 (in the eyeball norm), the vorticity error remains observable at t = 10 andeven at t = 20 there are some small difference. The contour plots of the errors at early times for

19

vorticity show the largest errors occur near vortex centers, indicating that the VV-DA method isnot accurately predicting the strength of the vortices.

Figure 5: Contour plots of velocity for DNS (left), VV-DA with µ1 = µ2 = 10 (center), and theirdifference (right), for times t = 0, 0.1, 1, 10, 20, 80 (top to bottom). both_vel

20

Figure 6: Contour plots of vorticity for DNS (left), VV-DA with µ1 = µ2 = 10 (center), and theirdifference (right), for times t = 0, 0.1, 1, 10, 20, 80 (top to bottom). both_vort

21

Figure 7: Contour plots of velocity for DNS (left), VV-DA with µ1 = 10, µ2 = 0 (center), andtheir difference (right), for times t = 0, 0.1, 1, 10, 20, 80 (top to bottom). justvel_vel

22

Figure 8: Contour plots of vorticity for DNS (left), VV-DA with µ1 = 10, µ2 = 0 (center), andtheir difference (right), for times t = 0, 0.1, 1, 10, 20, 80 (top to bottom). justvel_vort

5 Conclusions

We have analyzed a VV scheme for NSE enhanced with CDA, using linearized backward Euler orBDF2 in time and finite elements in space. We proved that applying CDA preserves the uncondi-tional stability properties of the scheme, and also yields optimal long-time accuracy if both velocityand vorticity are nudged, or velocity-only. If only velocity is nudged, then the convergence in timeto the true solution is slower, but still exponentially fast in time. Numerical tests illustrate thetheory, including the difference between nudging only velocity or also nudging vorticity.

For future directions, since nudging vorticity is difficult in practice due to accurate measurement

23

data not typically being available, one may try to obtain better results for the velocity-only-nudgingby penalizing the difference between wh and rot vh in the vorticity equation. That is, by settingµ2 = 0 and adding the term γ(w − rotu) to the vorticity equation (1.3), it may be possible toanalytically prove a convergence result resembling Theorem 3.12. Determining whether this ispossible, and if so then for what values of γ, and whether it works in practice (i.e. how large areassociated constants), would need a detailed further study which the authors plan to undertake.

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