1. Introduction
This paper is concerned with the finite element discretizations of the unsteady NavierStokes equations equationparentequation
(1.1a)  
(1.1b)  
(1.1c)  
(1.1d) 
where () is a bounded domain with Lipschitz and polyhedral boundary ; the velocity and pressure are two unknowns; is the constant kinematic viscosity; represents the external body force at ; and denotes the initial velocity.
Physically, there are several balance laws implied in the NavierStokes equations such as the balances of kinetic energy, linear momentum and angular momentum (EMA) [14, 7]. In some cases (e.g., vanishing external force and viscosity), these quantities are conservative under some appropriate assumptions [7]. Since at least Arakawa devised an energy and enstrophy conservation method for twodimensional NSEs in [2], scientists have recognised that these conservation (balance) laws are important references for unlocking efficient and stable numerical methods for NSEs [15, 39, 38, 1, 14, 36]. A violation of these laws may bring unexpected instability.
In this paper, we consider infsup stable mixed methods for NSEs. The crucial points for these balance laws are the discretization of the nonlinear term and the finite elements one uses. The most common formulation for this term might be the socalled convective formulation (CONV). However, it was shown in [7] that this formulation was not EMAconserving unless exactly divergencefree elements were used. We note that most classical elements are not divergencefree [24]. Also in [7], Charnyi et al. proposed an EMAconserving formulation (”EMAC”) for the common elements. The numerical experiments in [7, 29, 28, 37] demonstrated that this formulation was always among the best discretizations of the nonlinear term. In addition, the EMAC formulation was also shown to preserve the 2D enstrophy, helicity and total vorticity under certain appropriate assumptions. In the paper [36], Olshanskii and Rebholz proved that the Gronwall constant in the EMAC error estimates does not explicitly depend on the Reynolds number, which is contrast to other formulations such as SKEW. We note that, although the EMAC formulation has many fascinating properties, it is not easy to find a fully satisfactory linearization scheme for it. In [8], Charnyi et al. proposed two linearization schemes for this method: a skewsymmetric linearization and the Newton linearization. The former did not conserve momentum and angular momentum, which gave a poor performance for some problems, the latter conserved momentum and angular momentum but was not energystable. The objective of this paper is to design an alternative formulation that is very suited to Picard linearization.
The idea for this paper goes back to a class of locally discontinuous Galerkin (DG) methods for NSEs, cf. [10]. By replacing one of the velocity with its a divergencefree approximation in the nonlinear term, a class of energystable DG methods were developed in [10]. Similar techniques were also applied in [21, 26] and to coupled flowtransport problems [34, 19, 24]. Meanwhile, seeking an exactly divergencefree approximation to some discretely divergencefree velocity is also an objective of the pressurerobust reconstruction methods [32, 27, 31], where a large class of divergencefree reconstruction operators were proposed, almost covered all the classical conforming elements. In this paper, we will use these operators to reconstruct CONV for conforming elements.
One motivation for our methods is that, the CONV formulation not only has the easiest form but also usually shows good performance before the scheme blows up due to the energy instability (e.g., see the comparison experiments in [7]). In our opinion, it has the potential to simulate a problem accurately and the divergencefree reconstruction is one of the keys to unlock this potential. Here we focus on unsteady NSEs and give some theoretical analysis of the reconstructed CONV formulation. In what follows, we shall refer to it simply as “modified CONV” sometimes. We prove that modified CONV conserves kinetic energy, linear momentum, total vorticity, 2D enstrophy and helicity under the same assumptions as EMAC [7]. Further, we also prove that the Picard linearization conserves them as well. Compared to EMAC, our formulation does not conserve angular momentum theoretically. However, the numerical experiments show that modified CONV preserves this quantity well. For the semidiscrete scheme, we prove that the Gronwall constant of the error bounds does not depend on the Reynolds number explicitly under the assumption . In this aspect a related concept is called (Re)semirobustness [40, 22], which is a type of robustness that the constants (including Gronwall constant) in velocity error estimates do not explicitly depend on the inverse of the viscosity. For Resemirobust methods/analysis, we refer to [42, 41, 40, 6, 11, 12, 16].
The rest of this paper is organized as follows. In Section 2 we state the reconstructed scheme and discuss the conservative properties and error estimates of the semidiscrete scheme. Section 3 is devoted to investigating the conservative properties of Picard linearization, matching the CrankNicolson time stepping method. Section 4 studies some numerical experiments.
Throughout the paper we use with or without subscript, to denote a generic positive constant. The norm (seminorm) of the Sobolev space or () is denoted by (, respectively). The standard inner product of or is denoted by . When (, ), with the convention that the index (, , respectively) is omitted.
2. A modified convective formulation
Let be shaperegular partitions of [9] and with the diameter of element . Let , and denotes a pair of infsup stable finite element spaces on . Here we do not give them a concrete definition first.
A continuous variational form for eq. 1.1 is: equationparentequation
(2.1a)  
(2.1b) 
where
and
for any and . Here the trilinear form is the socalled convective formulation (CONV).
A semidiscrete analog of eq. 2.1 reads: equationparentequation
(2.2a)  
(2.2b) 
Let be the velocity solution of eq. 2.1 or eq. 2.2. Introduce the kinetic energy:
The following identity is widely used in energystability analysis:
(2.3) 
Equation 2.3 gives a skewsymmetric structure for and in when is divergencefree. Based on this fact, one could further prove that the continuous velocity is energystable in eq. 2.1. In the discrete level, however, the divergence constraint is usually relaxed for most of the classical elements such as the BernardiRaugel elements [4] and the TaylorHood elements [18, 5]. Then the skewsymmetric structure loses. The discretely divergencefree subspace of is defined by
(2.4) 
To fix the lack of skewsymmetry, one commonly used method is to modify the CONV formulation into the following SKEW formulation:
(2.5) 
However, Olshanskii and Rebholz [36] showed that the SKEW formulation might give a poor accuracy for long time simulations with higher Reynolds number. In this paper, we consider another skewsymmetrization way for CONV, where we shall use the divergencefree projection operators.
We define
with
the unit external normal vector on
. Denote by a generic divergencefree projection operator, which may be different for different space pairs. Introduce a corresponding finite element space such that . Here we assume that has the following properties:Assumption 1.
➀ for all ; ➁ there exists a constant independent of such that for all .
For most cases, the inequality in Assumption 1 ➁ can be obtained by a combination of the approximation property of , the inverse inequality and triangle inequality.
Introduce the modified convective formulation (modified CONV)
(2.6) 
We consider the reconstructed scheme: equationparentequation
(2.7a)  
(2.7b) 
The following lemma is very essential for our analysis.
Lemma 1.
For any we have
if

;

or or on .
Proof.
Lemma 1 can be regarded as a special case of the skewsymmetric discontinuous Galerkin formulation (see, e.g., [13, Lemma 6.39]). Here for completeness we reprove this special case. To make the proof process more clear, we will derive it in the component form. Let denotes the component form of any . We have
(2.8) 
The integration by parts on each element gives
(2.9)  
where denotes the th component of the unit external normal vector of , . A combination of the continuity and boundary conditions of and implies that
(2.10) 
and the first condition in Lemma 1 gives
(2.11)  
Then Lemma 1 follows immediately from eqs. 2.11, 2.10, 2.9 and 2.8. ∎
By Lemma 1 and Assumption 1 one can obtain
(2.12) 
2.1. Conservation laws
In this subsection, we focus on the conservative properties of the modified method eq. 2.7. First, let us give the definitions of momentum, angular momentum, helicity, enstrophy and vorticity. Denote by the exact velocity in eq. 1.1 or the discrete velocity in eq. 2.7. Let when represents the continuous solution; let be the solution of some finite element discretization of the NavierStokes vorticity equation (see eq. 2.14 and eq. 2.15 below) when is the discrete velocity. Note that the operator curl in two dimensions denotes a scalar operator [18]. Then we define
Remark 1.
In what follows we shall discuss some conservative properties for the case (the Euler equations) sometimes. For an Euler equation, the boundary conditions should be altered in eq. 1.1. Here we replace the noslip boundary condition () with the nopenetration boundary condition () for it. In this case the method eq. 2.7 should only strongly enforce the nopenetration boundary condition also. By abuse of notation, we shall use eq. 1.1 and eq. 2.7 to denote the Euler equation and the corresponding finite element methods as well, respectively.
Theorem 2.1.
Under Assumption 1 ➀, the modified method eq. 2.7 properly balances the kinetic energy:
(2.13) 
Proof.
Equation 2.12 implies that . Then eq. 2.13 follows immediately from taking in eq. 2.7a. ∎
Remark 2.
When and , one can similarly obtain . In this case, kinetic energy is conserved by our method.
To analysis the conservative properties of the other quantities, we need some extra assumptions, which are similar to the EMAC analysis [7].
Assumption 2.
The exact solution , the discrete solution and the source term are only supported on a subdomain such that there exists restriction for with in and on . Here represents the th usual basis of .
Theorem 2.2.
Under Assumption 1 ➀ and Assumption 2, the modified method eq. 2.7 conserves momentum (for with zero linear momentum), helicity (for and ), 2D enstrophy (for and ), and total vorticity.
We divide the proof of the above theorem into several subsections.
2.1.1. Momentum
Remark 3.
For the analysis of angular momentum, note that . Then substituting into eq. 2.7, one could similarly obtain that
Here we apply the fact that and . Assuming that has zero angular momentum, i.e., , the above equation gives that since in general. Thus angular momentum is not preserved by our methods.
2.1.2. Helicity
Let be the curl of the exact solution : . Note that . In two dimensions, represents a scalar and we use the symbol to denote it. Taking the curl operator on eq. 1.1a gives the following NavierStokes vorticity equations:
(2.14) 
For the discrete case, we apply the strategy in [35, 7]. Here we do not analyze the quantity for the discrete solution . We consider a sightly altered virticity which is a solution of some direct discretization of eq. 2.14. As it was said in [7], ‘this discrete vorticity still depends on the computed velocity , but more implicitly, through the equation coefficients’. We further assume vorticity also vanishes on and near the boundary due to Assumption 2. The corresponding finite element methods to eq. 2.14 reads: Find and the Lagrange multifier such that
(2.15)  
for any and , where is the solution of eq. 2.7. Note that eq. 2.15 implies that for all and further .
2.1.3. 2D Enstrophy
In two dimensions the vorticity satisfies
The corresponding finite element scheme reads:
(2.18) 
Similarly, we have . Thus, taking and for the case we have
(2.19) 
Thus we prove the conservative property of the 2D enstrophy.
2.1.4. Vorticity
Note that under Assumption 2. Set in eq. 2.15 and we obtain
(2.22) 
Equation 2.12 implies that
(2.23) 
and
(2.24) 
Substituting eq. 2.23 and eq. 2.24 into eq. 2.22 gives that
which implies that the total vorticity is conserved.
2.2. Semidiscrete error analysis
Denote by the Stokes projection which satisfies that
(2.25) 
Assumption 3.
The Stokes projection satisfies the following estimate
(2.26) 
To satisfy the estimate eq. 2.26, some extra conditions might be required. For example, the domain is convex. For the concrete analysis of eq. 2.26, we refer the readers to the paper [17]. Let
and
The following error equation will be used in error analysis.
(2.27)  
The above equation can be further rewritten as
(2.28)  
where we use the fact that .
Theorem 2.3 (Semidiscrete estimate).
Let solve eq. 2.7 and solve eq. 2.1 with , and Under Assumption 1 and Assumption 3 it holds
(2.29)  
with
and
independent of
Proof.
Set in eq. 2.28 and we arrive at
(2.30)  
Applying the CauchySchwarz inequality and the weighted Young’s inequality one obtains
(2.31) 
(2.32) 
To analyze the nonlinear terms, here we use a similar splitting way as [40, 36]. We split the difference of the two nonlinear terms as
(2.33) 
Further,
(2.34)  
Then the CauchySchwarz inequality, Equation 2.26 and the Young’s inequality give that
(2.35) 
(2.36) 
and
(2.37) 
where in the last inequality we also use Assumption 1. Substituting eqs. 2.37, 2.36 and 2.35 into eqs. 2.34 and 2.33 provides
which, together with eqs. 2.32 and 2.31, gives that
(2.38) 
Then integrating over and using the Gronwall inequality and Hölder inequality we finally obtain
(2.39)  
Finally, the estimate eq. 2.29 follows immediately from eq. 2.39 and the triangle inequality. ∎
3. The Picard linearization scheme
In practice, a commonly used linearization way is replacing one of the velocity solutions of the nonlinear term with last time step solutions. In this section we prove that this way preserves all the conservative properties from the semidiscrete version when matching the CrankNicolson time discretizations. The linearized CrankNicolson scheme is that
(3.1)  
for all , where
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