Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow

time-marching for time-dependent incompressible fluid flow

Haijin Wang^† Yunxian Liu^‡ Qiang Zhang^§ Chi-Wang Shu^¶ June 15, 2017

Abstract

The main purpose of this paper is to study the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with multi-step implicit-explicit (IMEX) time discretization schemes, for solving time-dependent incompressible fluid flows. We will give theoretical analysis for the Oseen equation, and assess the perfor- mance of the schemes for incompressible Navier-Stokes equations numerically. For the Oseen equation, using first order IMEX time discretization as an example, we show that the IMEX-LDG scheme is unconditionally stable forQ^k elements on cartesian meshes, in the sense that the time-stepτ is only required to bebounded from aboveby a pos- itive constant independent of the spatial mesh sizeh. Furthermore, by the aid of the Stokes projection and an elaborate energy analysis, we obtain theL^∞(L²) optimal error estimates for both the velocity and the stress (gradient of velocity), in both space and time. By theinf-sup argument, we also obtain theL^∞(L²) optimal error estimates for the pressure. Numerical experiments are given to validate our main results.

Keywords. local discontinuous Galerkin method, implicit-explicit scheme, incompress- ible flow, Oseen equation, Navier-Stokes, stability, error estimate.

AMS. 65M12, 65M15, 65M60

1 Introduction

In this paper we study a fully discrete local discontinuous Galerkin (LDG) scheme coupled with multi-step implicit-explicit (IMEX) time discretization, for solving the following Oseen

†College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu Province, P. R. China. E-mail: [email protected]. Research sponsored by NUPTSF grant NY215067, Natural Science Foundation of Jiangsu Province grant BK20160877 and NSFC grant 11601241.

‡School of Mathematics, Shandong University, Jinan 250100, Shandong Province, P. R. China. E-mail:

[email protected]. Research sponsored by NSFC grant 11471194.

§Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P. R. China. E- mail: [email protected]. Research supported by NSFC grants 11271187, 11571290 and 11671199.

¶Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A. E-mail:

[email protected]. Research supported by DOE grant DE-FG02-08ER25863 and NSF grant DMS- 1418750.

1

equation 





∂u

∂t −ν∆u+∇ ·(γ⊗u) +∇p=f,

∇ ·u= 0, u(x,0) =u⁰(x)

(1.1) in Ω ⊂ R². Here x = (x, y), ν is the kinematic viscosity, u = (u₁, u₂) is the velocity, γ = (γ₁, γ₂) is a given convective velocity field, p is the pressure, and f = (f₁, f₂) is the source term. Since p is uniquely defined up to an additive constant, we also assume that R

Ωpdxdy = 0. For the simplicity of analysis, we consider periodic boundary conditions, and assume f = 0 in this paper, and without loss of generality, we assume γ₁ and γ₂ are positive.

The LDG method was introduced by Cockburn and Shu [9] for convection-diffusion problems, motivated by the work of Bassi and Rebay [2] for compressible Navier-Stokes equations. As an extension of discontinuous Galerkin (DG) schemes for hyperbolic conser- vation laws [10], the LDG scheme can easily handle meshes with hanging nodes, elements of general shapes and local spaces of different types, thus it is flexible for hp-adaptivity.

Besides, the LDG methods enforce the conservation laws locally and in a conservative way, and it performs well for problems with shocks, steep gradients, or boundary layers. Owing to these advantages, LDG methods have been designed to solve incompressible fluid flows, such as the Stokes system [7], the Oseen equation [6] and Navier-Stokes equations [8]. How- ever, all these works are for the steady problems. As far as the authors know, only a few workshave been done on LDG methods for time-dependent incompressible flows. Recently, Wang et al. [18] adopted the characteristic local discontinuous Galerkin methods for time- dependent impressible Navier-Stokes problems, they used the characteristic method to deal with the time derivative term and the nonlinear convection term together, and discretized the remaining terms by the LDG spatial discretization. The unconditional stability of the first order scheme was presented, but the error analysis was absent.

In [15–17], several Runge-Kutta type implicit-explicit (IMEX) time marching schemes [1]

coupled with the LDG spatial discretization for solving one and two dimensional convection- diffusion problems have been studied, where the corresponding IMEX-LDG schemes have shown good stability and accuracy. These schemes have also been applied to the drift- diffusion model of semiconductor devices in [14]. The advantages of IMEX schemes lie in that: it can not only overcome the severe time step restriction, compared with the pure explicit time discretization schemes; but also obtain an elliptic-type algebraic system, which is easy to solve efficiently by many standard iterative methods. In this paper, we would like to study the stability and accuracy of the IMEX-LDG schemes for solving the time- dependent Oseen equations, where the convection term is treated explicitly, and both the viscosity term and the pressure term are treated implicitly. Since the LDG discretization for the Stokes system results in a differential algebraic equation (DAE) [13] with index 2, Runge-Kutta time discretization might reduce the order of accuracy in time for pressure, hence we consider multi-step IMEX time discretizations, which are developed in [12].

In our LDG schemes, we adopt the equal order polynomials for the velocity, the stress (gradient of velocity) and the pressure. We will prove that for the Oseen equations, the corresponding IMEX-LDG schemes are unconditionally stable forQk elements on cartesian meshes, in the sense that the time-step τ is only required to be bounded from above by a positive constant independent of the spatial mesh size h. Furthermore, by the aid of the

so-calledStokes projection and an elaborate energy analysis, we obtain theL^∞(L²) optimal error estimates for both the velocity and the stress, in both space and time. We also obtain the optimal error estimates for the pressure in theL^∞(L²) norm, by theinf-sup argument and an optimal estimate for the time difference of velocity. These results, especially the optimal error estimates for pressure, constitute the main highlight of this paper. Since the proof is quite technical, we only take the first order scheme (i.e, forward Euler in the explicit discretization and backward Euler in the implicit discretization) as an example to show the idea.

Another important contribution of this paper lies in that, it is the first time that the so called Stokes projection associated with the LDG spatial discretization is studied and adopted to obtain the optimal error estimate, which will consequentially provide a powerful tool in the analysis for Navier-Stokes problems. Furthermore, it is worth mentioning that, the numerical flux we consider for the viscosity term and the pressure term are the classical

“alternating” fluxes without any extra penalty terms, this is different from [3, 6, 18], where penalty terms are added to enhance the stability of the scheme. Finally we point out that, our long-term goal is to study IMEX-LDG methods for the incompressible Navier-Stokes equations, the analysis of the Oseen problem is just an intermediate step.

The paper is organized as follows. In Section 2 we present the semi-discrete LDG schemes for the Oseen equation, and give some properties of the LDG schemes. We will focus on the study of the Stokes projection in Section 3, which plays an important role in obtaining the optimal error estimates. Sections 4 and 5 are devoted to the stability and error analysis of the first order fully-discrete IMEX-LDG methods, respectively. In Section 6 we will present numerical results to verify our results. The concluding remarks and a few technical proofs are given in Section 7 and the Appendix, respectively.

2 The semi-discrete LDG scheme and its properties

Let σ=√

ν∇u=√

ν(∇u₁,∇u₂)^⊤, then (1.1) can be rewritten as the following equivalent first-order differential system









∂u

∂t −√

ν∇ ·σ+∇ ·(γ⊗u) +∇p=f, σ=√

ν∇u,

∇ ·u= 0, u(x,0) =u⁰(x).

(2.1)

Here the notations are the same as those in [7]. The gradient of the vector u is a matrix, with (∇u)_ij =∂_ju_i. The divergence of the matrixσ is a vector, with (∇ ·σ)_i =Pd

j=1∂_jσ_ij. γ⊗uis a matrix whose (i, j)th component isγ_iu_j.

We will define the semi-discrete LDG scheme based on equation (2.1). To this end, we would like to give the finite element space and some notations firstly.

2.1 Discontinuous finite element space

Let Ω_h ={K} be a quasi-uniform partition of the domain Ω with rectangular element K, whereh = max

K h_K, withh_K being the diameter of element K. We denote Γ_h as the set of

all element interfaces. Associated with this mesh, we define the discontinuous finite element space

Σ_h=

σ∈L²(Ω)^2×2 :σ|K ∈Q_k(K)^2×2,∀K ∈Ω_h , V_h=

v∈L²(Ω)²:v|K ∈Q_k(K)²,∀K∈Ω_h , Q_h=

q∈L²(Ω) :q|K ∈Q_k(K),∀K ∈Ω_h,and Z

Ω

qdxdy= 0 ,

which is contained in the following (mesh-dependent) broken Sobolev space Σ×V ×Q Σ =n

σ∈L²(Ω)^2×2 :σ|K ∈H¹(K)^2×2,∀K∈Ω_ho , V =n

v∈L²(Ω)²:v|K ∈H¹(K)²,∀K ∈Ω_ho , Q=n

q ∈L²(Ω) :q|K ∈H¹(K),∀K ∈Ω_hand Z

Ω

qdxdy= 0o ,

respectively. In this paper Q_k= P_k⊗P_kdenotes the space of tensor product of polynomials of degree at most k.

2.2 Notations

We use a fixed vector β = (1,1)^⊤ to uniquely define the left and right elements K_L and K_R which share the same element interface e. Namely, β·n_KL|e > 0 and β·n_KR|e <0, respectively, where n_K is the outward normal of K. Along each side e, there are two traces for any function p, denoted by p⁺ = (p|KR)|e and p⁻ = (p|KL)|e, respectively, and we denote the jump by [[p]] = p⁺−p⁻ for scalar functions, [[v]] = ([[v₁]],[[v₂]])^⊤ for vector- valued functions, and [[r]] = ([[rij]])2×2 for matrix-valued functions. Besides, we use ∂_K⁻ = {e ⊂ ∂K,β·n_K|e < 0} and ∂_K⁺ = {e ⊂ ∂K,β·n_K|e > 0} to denote the inflow and outflow sides of elementK, respectively. For non-periodic boundary conditions, we denote

∂Ω⁻ = {e⊂ ∂Ω,β·n_e < 0} and ∂Ω⁺ ={e ⊂ ∂Ω,β·n_e > 0} as the inflow and outflow boundaries, respectively.

In this paper, we will use the standard norms and semi-norms in the Sobolev space for scalar functions. For any given domainD, we denote bykwkD theL² norm ofwon D. For any integers≥0, letH^s(D) represent the space equipped with the normk·ks,D, in which the function itself and the derivatives up to the s-th order are all in L²(D). IfD= Ω, we omit the subscript Ω for convenience. We define similar norms for vector-valued functionu and matrix-valued functionσaskuks,D = (P_d

i=1ku_ik²s,D)^1/2, andkσks,D= (P_d

i,j=1kσ_ijk²s,D)^1/2, respectively. We also define kukΓh = (P

e∈Γhkuk²e)^1/2 and kσkΓh = (P

e∈Γhkσk²e)^1/2. We denote k∇vk= (P

K∈Ωhk∇vk²K)^1/2 forv∈V_h. In addition, we would like to write

σ :r= Xd i,j=1

σ_ijr_ij, v·σ·n= Xd i,j=1

v_iσ_ijn_j =σ: (v⊗n).

The symbol C is used as a generic constant, and εis used to denote an arbitrary small constant, bothCandεmay have different values in different occurrences. The symbolµ >0

is used to represent an inverse constant which appears in the following inverse inequalities k∇vk ≤µh⁻¹kvk, kvkΓh ≤p

µh⁻¹kvk, (2.2)

krkΓh ≤p

µh⁻¹krk, (2.3)

for any functions v∈V_h and r∈Σ_h.

2.3 The semi-discrete LDG scheme

The LDG scheme for the Oseen equation (2.1) is to find the approximation (σ_h,u_h, p_h) ∈ Σ_h×V_h×Q_h, such that the following variation forms hold for any element K ∈Ω_h and any test function (r,v, w) ∈Σ_h×V_h×Q_h,

Z

K

∂u_h

∂t ·vdxdy=HK(u_h,v) +LK(σ_h,v) +PK(p_h,v), (2.4a) Z

K

σ_h:rdxdy=K^K(u_h, r), (2.4b)

0 =Q^K(u_h, w). (2.4c)

Here

H^K(u_h,v) = Z

K

(γ⊗u_h) :∇vdxdy− Z

∂K

v·γ⊗ud_h,γ·n_Kds, (2.5a) LK(σ_h,v) = −√

ν Z

K

σ_h :∇vdxdy− Z

∂K

v·σc_h·n_Kds

, (2.5b)

PK(p_h,v) = Z

K

p_h∇ ·vdxdy− Z

∂K

b

p_hv·n_Kds, (2.5c)

KK(u_h, r) = −√ ν

Z

K

u_h· ∇ ·rdxdy− Z

∂K

d

u_h,σ·r·n_Kds

, (2.5d)

QK(u_h, w) = Z

K

u_h· ∇wdxdy− Z

∂Kud_h,p·n_Kwds. (2.5e) The “hat” terms are the so-called numerical flux, which are taken as the “alternating”

numerical flux

d

u_h,σ=u⁻_h, σc_h=σ⁺_h, d

u_h,p=u⁻_h, pb_h =p⁺_h, (2.6) for the viscosity and the pressure terms. For the convection term, we can simply take the upwind numerical flux or the central flux. Since the convection velocity is assumed to be positive, we simply take the upwind flux

d

u_h,γ=u⁻_h. (2.7)

The initial condition u_h(x,0) can be taken as any approximation of the initial solution u⁰(x), for example, the Stokes projection Π_hu⁰(x), which is to be defined in Section 3.

Denote (u,v) =P

K

R

Ku·vdxdy and (σ, r) =P

K

R

Kσ:rdxdy. Summing (2.4) over all elements K, we get the global form of the semi-discrete LDG scheme:

(u_ht,v) =H(u_h,v) +L(σ_h,v) +P(p_h,v), (2.8a)

(σ_h, r) =K(u_h, r), (2.8b)

0 =Q(u_h, w). (2.8c)

Here Ξ(·,·) = P

KΞ_K(·,·) for Ξ = H,L,P,K,Q. We have now defined the semi-discrete LDG scheme for the Oseen equation (2.1).

Remark 2.1. Integrating by parts gives rise to the following equivalent forms of (2.5):

H^K(u_h,v) = − Z

K

(v⊗γ) :∇u_hdxdy+ Z

∂_K⁻

v·γ⊗[[u_h]]·n_Kds, (2.9a) LK(σ_h,v) =√

ν

"Z

K

v·(∇ ·σ_h) dxdy+ Z

∂_K⁺

v·[[σ_h]]·n_Kds

#

, (2.9b)

PK(p_h,v) = − Z

K

v· ∇p_hdxdy− Z

∂_K⁺

[[p_h]]v·n_Kds, (2.9c) K^K(u_h, r) =√

ν

"Z

K

r:∇u_hdxdy− Z

∂⁻_K

[[u_h]]·r·n_Kds

#

, (2.9d)

QK(u_h, w) = − Z

K

w∇ ·u_hdxdy+ Z

∂_K⁻

[[u_h]]·n_Kwds. (2.9e)

2.4 Properties of the LDG spatial discretization

In this subsection, we will give several lemmas to illustrate a few properties of the LDG spatial discretization. All the properties are trivial generalizations of the two-dimensional scalar case [17]. We refer the readers to [17] for the proof.

Lemma 2.1. For any u∈V and v ∈ V_h, there exists C_γ =kγk independent of u and v such that

H(v,v)≤0. (2.10)

|H(u,v)| ≤C_γ(k∇uk+p

µh⁻¹k[[u]]kΓh)kvk, (2.11)

|H(u,v)| ≤C_γkuk(k∇vk+p

µh⁻¹k[[v]]kΓh). (2.12) Lemma 2.2. For anyp∈Qand v ∈V_h, we have

|P(p,v)| ≤ kpk(k∇vk+p

µh⁻¹k[[v]]kΓh). (2.13) Lemma 2.3. For any(r,v, w)∈Σ_h×V_h×Q_h, we have

L(r,v) +K(v, r) = 0, (2.14)

P(w,v) +Q(v, w) = 0. (2.15)

The next lemma illustrates an important relationship between the gradient and the element interface jump of the numerical solution with the numerical solution of the gradient, which plays a key role in the stability and error analysis. Along the similar line as the proof in [17], we can obtain the proof of this lemma, hence it is omitted here.

Lemma 2.4. Assume the pair of functions (u_h, σ_h) ∈V_h×Σ_h satisfy (2.8b) for arbitrary r∈Σ_h, then

k∇u_hk+p

µh⁻¹k[[u_h]]kΓh ≤ C_µ

√νkσ_hk, (2.16)

where C_µ is a positive constant only depending on the inverse constant µ.

2.5 The inf-sup condition

We would like to begin this subsection by defining a projection P_h following [5]: for any ρ∈H₀¹(Ω)²,P_hρ|K ∈Q²_k(K) satisfies

Z

K

(P_hρ−ρ)· ∇vdxdy= 0, ∀v ∈Q_k(K), Z

e

(P_hρ−ρ)·n_evds= 0, ∀v ∈P_k(e),∀e∈∂_K⁻.

(2.17)

There holds the following approximation property [5]

kρ−P_hρk+h|ρ−P_hρ|¹+h^1/2kρ−P_hρk^Γh ≤Chkρk¹. (2.18) By (2.18) and the triangle inequality, we can easily get

kP_hρk¹≤Ckρk¹. (2.19)

To get the estimate for the pressure, we will resort to the standard inf-sup argument in the analysis for Stokes problems. Since this argument will be used several times in this paper, we present the key results in the following lemma.

Lemma 2.5. For arbitraryq ∈L²₀(Ω) .

={v∈L²(Ω)|:R

Ωv= 0}, there existsw^⋆=w^⋆(q)∈ H₀¹(Ω)², such that

− Z

Ω

q∇ ·w^⋆dxdy≥α₁kqk², kw^⋆k1≤α₂kqk, (2.20) for some positive constants α₁, α₂ independent of q. And

kqk²≤C

K(Phw^⋆, r) +Q(Phw^⋆, q) +krk²

, (2.21)

for arbitrary r∈Σ_h, where Ph is defined in (2.17).

Proof. The conclusion (2.20) is cited from [11]. Hence we only show (2.21).

On one hand, we have

Q(P_hw^⋆, q) =Q(w^⋆, q) =− Z

Ω

q∇ ·w^⋆dxdy≥α1kqk² (2.22)

by the definition of P_h and Q, (2.9e) and (2.20), where we have used the fact that w^⋆ ∈ H₀¹(Ω)² is continuous across the element interface.

On the other hand, for arbitraryr∈Σ_h, it follows from (2.9d) and the continuity ofw^⋆ that

K(w^⋆, r) =√ ν

Z

Ω∇w^⋆ :rdxdy.

So

K(P_hw^⋆, r) =K(w^⋆, r)− K(w^⋆−P_hw^⋆, r) =√ ν

Z

Ω∇w^⋆ :rdxdy

−√

ν X

K∈Ωh

"Z

K∇(w^⋆−P_hw^⋆) :rdxdy− Z

∂⁻_K

[[w^⋆−P_hw^⋆]]·r·n_Kds

# ,

where (2.9d) is used in the second line. Thus, by the aid of the Cauchy-Schwarz inequality and the trace inverse inequality (2.3), we get

|K(P_hw^⋆, r)| ≤√

νkw^⋆k1krk+√

ν(|w^⋆−P_hw^⋆|1+p

µh⁻¹kw^⋆−P_hw^⋆kΓh)krk. Then by the approximation property (2.18) we get

|K(P_hw^⋆, r)| ≤Ckw^⋆k1krk ≤Cα₂kqkkrk ≤ α₁

2 kqk²+C²α²₂ 2α1 krk², where we have used (2.20) and the Young’s inequality. So

K(P_hw^⋆, r)≥ −α₁

2 kqk²−C²α²₂

2α₁ krk². (2.23)

Hence we obtain (2.21) immediately by adding up (2.22) and (2.23). This completes the proof of this lemma.

3 The Stokes projection

In the later error analysis, theStokes projection plays a salient role in obtaining theL^∞(L²) optimal error estimates. The advantage of the Stokes projection lies in that the impact of pressure errors can be eliminated when estimating the velocity error. However, as far as we know, the Stokes projection associated with the LDG scheme has not been studied, and the approximation properties of the Stokes projection are not clear. Hence it is worth taking the effort to study it rigorously.

In this section, we will give the definition of the Stokes projection firstly, and then show the existence and uniqueness of the projection, the optimal approximation properties will be given and proven at the end of this section. To prove the optimal approximation properties, we need to resort to a series of projections which will be defined in Subsection 3.4. For the readers who are eager to enter into the stability and error analysis of the schemes, they can skip Subsections 3.4 and 3.5 for the time being.

3.1 The definition

Let u ∈ V satisfy ∇ ·u = 0, σ = √ν∇u and p ∈ Q, we define their Stokes projection (Π_hσ,Πhu,Π_hp)∈Σ_h×V_h×Q_h as follows: for arbitrary (r,v, w)∈Σ_h×V_h×Q_h, there hold

L(Π_hσ,v) +P(Π_hp,v) =L(σ,v) +P(p,v), (3.1a) (Π_hσ, r) =K(Π_hu, r), (3.1b) Q(Πhu, w) =Q(u, w). (3.1c) In addition, to ensure the uniqueness of the projection, we let

Z

Ω

(Π_hu−u)·e_idxdy= 0, for i= 1,2, (3.1d) wheree₁ = (1,0)^⊤ and e₂= (0,1)^⊤, and

Z

Ω

(Π_hp−p) dxdy= 0. (3.1e)

Remark 3.1. The condition (3.1e) holds automatically since the averages of bothpand Π_hp are 0.

3.2 Existence and uniqueness

Lemma 3.1. The projection defined in (3.1) exists uniquely.

Proof. To show that the Stokes projection exists uniquely, it is enough to show that, if (σ,u, p) = (0,0,0), then the only possible solution of (Π_hσ,Πhu,Π_hp) defined in (3.1) equals (0,0,0). To this end, we consider

0 =L(Π_hσ,v) +P(Π_hp,v), (3.2a)

(Π_hσ, r) =K(Π_hu, r), (3.2b)

0 =Q(Πhu, w). (3.2c)

Taking (r,v, w) = (Π_hσ,Π_hu,Π_hp) in (3.2) and adding them together leads tokΠ_hσk² = 0 by Lemma 2.3. Hence Π_hσ= 0, thus it follows from (3.2b) that

0 =K(Πhu, r) =−√

ν X

K∈Ωh

Z

K

Πhu· ∇ ·rdxdy− Z

∂K

(Πhu)⁻·r·n_Kds

. (3.3) By takingr =Π_hu⊗β, whereβ = (1,1)^⊤, we get

X

K∈Ωh

Z

K

Π_hu· ∇ ·(Π_hu⊗β) dxdy− Z

∂K

(Π_hu⊗β) : ((Π_hu)⁻⊗n_K) ds

= 0. (3.4) Noting that

Z

K

Πhu· ∇ ·(Πhu⊗β) dxdy= 1 2

Z

∂K

(Πhu⊗β) : (Πhu⊗n_K) ds.

Thus (3.4) becomes X

K∈Ωh

"

1 2

Z

∂⁻_K

((Π_hu)⁺⊗β) : ((Π_hu)⁺⊗n_K) ds−1 2

Z

∂_K⁺

((Π_hu)⁻⊗β) : ((Π_hu)⁻⊗n_K) ds

− Z

∂_K⁻

((Πhu)⁺⊗β) : ((Πhu)⁻⊗n_K) ds

#

= 0, which is equivalent to

X

e∈Γh

1 2

Z

e

[(Π_hu)⁺·(Π_hu)⁺−2(Π_hu)⁺·(Π_hu)⁻+ (Π_hu)⁻·(Π_hu)⁻](β·n_e) ds

= X

e∈Γh

1 2

Z

e

[[Π_hu]]·[[Π_hu]](β·n_e) = 0,

by the periodic boundary condition, here β ·n_e < 0 for all e ∈ Γ_h. It implies Πhu is continuous across each element interface. Hence, from (3.3) and integrating by parts, we have

X

K∈Ωh

"Z

K∇Π_hu:rdxdy− Z

∂_K⁻

[[Π_hu]]·r·n_Kds

#

= X

K∈Ωh

Z

K∇Π_hu:rdxdy= 0. (3.5) Choosing r =∇Π_hu, we get∇Π_hu= 0 in each element K, which implies that Π_hu is a constant. Then according to the condition (3.1d) we deduce thatΠhu=0.

Along the similar line as above, we can obtain Π_hp= 0. Thus we have completed the proof of this lemma.

3.3 The approximation property

To obtain the optimal approximation properties of the Stokes projection (3.1), we will resort to the following adjoint Stokes problem similar as that considered in [3, 7],

−√

ν∇ ·s+∇q =λ, in Ω, (3.6a)

s=√

ν∇z, in Ω, (3.6b)

∇ ·z = 0, in Ω (3.6c)

for any givenλ∈L²(Ω)², with the periodic boundary condition and R

Ωq = 0. In addition, we assume it satisfies the following elliptic regularity assumption

νkzk2+√

νksk1+kqk1 ≤C_∗kλk. (3.7) The approximation property of the Stokes projection is stated as follows.

Lemma 3.2. Assume u∈H^k+2(Ω)²,σ =√

ν∇u∈H^k+1(Ω)^2×2 and p∈H^k+2(Ω)satisfy- ingR

Ωp= 0, then there exists a bounding constant C depending on the regularity of(u, σ, p) and the elliptic regularity constant C_∗ defined in (3.7), such that

ku−Π_huk+kσ−Π_hσk+kp−Π_hpk ≤Ch^k+1. (3.8) We will prove this lemma in Subsection 3.5.

3.4 Useful projections

To prove Lemma 3.2, we would like to introduce another series of projections (π_h,π_h, π_h) : Σ×V ×Q → Σ_h ×V_h ×Q_h, corresponding to matrix-valued functions, vector-valued functions and scalar functions, respectively. To define these projections conveniently, we would like to represent any rectangular element as K = I_i ×J_j, where I_i = (x_i−¹

2, x_i+¹

2) and J_j = (y_j−¹

2, y_j+¹

2). In what follows, we will present these useful projections and their properties.

First we define two projections π⁺_h and π_h⁻ from H¹(Ω_h) =

v ∈ L²(Ω) : v|K ∈ H¹(K),∀K ∈Ω_h onto W_h =

v∈L²(Ω) :v|^K ∈Q_k(K),∀K ∈Ω_h , following [19]:

Z

K

(π_h⁺w−w)vdxdy= 0, ∀v∈Q_k−1(K), Z

Jj

(π_h⁺w−w)_i−¹

2,yvdy= 0, ∀v∈P_k−1(J_j), Z

Ii

(π_h⁺w−w)_x,j−¹

2vdx= 0, ∀v∈P_k−1(I_i), π⁺_hw(x⁺

i−¹₂, y⁺

j−¹₂) =w(x_i−¹

2, y_j−¹

2), ∀i∀j, andZ

K

(π_h⁻w−w)vdxdy= 0, ∀v∈Q_k−1(K), Z

Jj

(π_h⁻w−w)_i+¹

2,yvdy= 0, ∀v∈P_k−1(J_j), Z

Ii

(π_h⁻w−w)_x,j+¹

2vdx= 0, ∀v∈P_k−1(I_i), π⁻_hw(x⁻

i+¹₂, y⁻

j+¹₂) =w(x_i+¹

2, y_j+¹

2), ∀i∀j.

Next we are going to define the projections (π_h,π_h, π_h), based on the projections defined above and the projectionP_h defined in (2.17).

1. For matrix-valued function σ = (σ₁,σ₂) ∈ Σ, where σ₁ and σ₂ are its two column vectors, we define

π_hσ= (P_hσ₁,P_hσ₂). (3.9) 2. For vector-valued function u= (u₁, u₂)^⊤∈V, we define

π_hu= (π_h⁻u₁, π⁻_hu₂)^⊤. (3.10) 3. For scalar function p∈Q, we define

π_hp=π⁺_hp. (3.11)

The following properties of these projections hold:

1. For arbitraryv∈V_h, we have

L(σ−π_hσ,v) = 0. (3.12) 2. For arbitraryσ∈H^s(Ω)^2×2,u∈H^s(Ω)² andp∈H^s(Ω) withs≥1, by the standard scaling argument [4], along the similar analysis as in [5], we can obtain the following approximation properties

kσ−π_hσk+h|σ−π_hσ|¹+h^1/2kσ−π_hσk^Γh≤Ch^min{k+1,s}kσk^s, (3.13a) ku−π_huk+h|u−π_hu|¹+h^1/2ku−π_huk^Γh≤Ch^min{k+1,s}kuk^s, (3.13b) kp−π_hpk+h|p−π_hp|1+h^1/2kp−π_hpkΓh≤Ch^min{k+1,s}kpks. (3.13c)

3. From [5], we can also get the following superconvergence properties: suppose u ∈ H^k+2(Ω)² and p∈H^k+2(Ω), then

|P(p−π_hp,v)| ≤Ch^k+1kpkk+2kvk, (3.14a)

|K(u−π_hu, r)| ≤C√

νh^k+1kukk+2krk, (3.14b)

|Q(u−π_hu, w)| ≤Ch^k+1kukk+2kwk. (3.14c) 3.5 Proof of Lemma 3.2

For the simplicity of notations, we denote byη= (η

σ,η_u, η_p) = (Π_hσ−σ,Π_hu−u,Π_hp−p).

We begin the proof with the following relationship

0 =L(η_σ,v) +P(η_p,v), (3.15a)

(ησ, r) =K(η_u, r), (3.15b)

0 =Q(ηu, w), (3.15c)

for any test function (r,v, w) ∈Σ_h×V_h×Q_h.

Based on the projections defined in Subsection 3.4, we split the error η = (η

σ,η_u, η_p) into two parts, namely

η_σ =σ−π_hσ+π_hη_σ, ηu=u−πhu+πhηu, η_p =p−π_hp+π_hη_p. (3.16) Thus, to prove Lemma 3.2, the remaining work is to estimate (π_hη

σ,π_hη_u, π_hη_p) in a sharp way, since (3.13) hold. To do that, we will proceed in three steps.

Step 1. First we give the estimate for kπ_hη_σk. By (3.15) and the division ofη (3.16), we have

−L(π_hη

σ,v)− P(π_hη_p,v) =L(σ−π_hσ,v) +P(p−π_hp,v), (3.17a) (π_hη

σ, r)− K(π_hηu, r) = −(σ−π_hσ, r) +K(u−π_hu, r), (3.17b)

−Q(π_hη_u, w) =Q(u−π_hu, w). (3.17c) Noting the error’s orthogonal property (3.12), taking (r,v, w) = (π_hη_σ,π_hη_u, π_hηp) in (3.17), and adding the three equalities together, we obtain

kπ_hη

σk²= −(σ−π_hσ, π_hη

σ) +P(p−π_hp,π_hηu) +K(u−π_hu, π_hη

σ) +Q(u−π_hu, π_hη_p), where the left hand side is owing to Lemma 2.3. By applying the Cauchy-Schwarz inequality and the approximation property (3.13a) to the first term on the right hand side, and the superconvergence properties (3.14) to the remaining three terms, we obtain

kπ_hη

σk² ≤Ch^k+1(kπ_hη

σk+kπ_hη_uk+kπ_hη_pk)

≤ε(kπ_hη_σk²+kπ_hη_uk²+kπ_hηpk²) +Ch^2k+2, (3.18) for arbitrary ε > 0, where the Young’s inequality is used in the last inequality. Thus by choosingε small enough, we get

kπ_hη

σk²≤Ch^2k+2+ε(kπ_hη_uk²+kπ_hη_pk²). (3.19)

Step 2. We would like to estimate kπ_hηpk in this step. Owing to Lemma 2.5, for π_hηp ∈ L²₀(Ω), there exists v^⋆∈H₀¹(Ω)² such thatkv^⋆k1≤α₂kπ_hη_pk and

kπ_hηpk² ≤C[K(P_hv^⋆, π_hη_σ) +Q(P_hv^⋆, π_hηp

| {z }

Z

) +kπ_hη_σk²], (3.20)

where we chooser=π_hη

σin (2.21). By using Lemma 2.3, (3.17a) and the error’s orthogonal property (3.12) successively, we get

Z = − L(π_hη_σ,P_hv^⋆)− P(π_hηp,P_hv^⋆) =P(p−π_hp,P_hv^⋆). (3.21) Thus, by using the superconvergence property (3.14a), (2.19), (2.20) and the Young’s in- equality, we have

Z ≤Ch^k+1kP_hv^⋆k ≤Ch^k+1kv^⋆k1 ≤Cα₂h^k+1kπ_hη_pk ≤εkπ_hη_pk²+Ch^2k+2. (3.22) Consequently, we get

kπ_hη_pk² ≤Ch^2k+2+Ckπ_hη_σk² ≤Ch^2k+2+ε(kπhηuk²+kπ_hη_pk²), (3.23) by using the estimate (3.19). Hence choosing εsmall enough gives rise to

kπ_hη_pk² ≤Ch^2k+2+εkπ_hη_uk². (3.24) Step 3. In this step we will estimate kπ_hη_uk. For arbitrary λ ∈ L²(Ω)², let z, s, q be defined in the adjoint Stokes problem (3.6), then from (3.6) we have

(π_hη_u,λ) = (π_hη_u,−√

ν∇ ·s+∇q) =−L(s,π_hη_u)− P(q,π_hη_u)

=L(π_hs−s,πhηu)− L(π_hs,πhηu) +P(π_hq−q,πhηu)− P(π_hq,πhηu)

=K(πhηu, π_hs) +P(π_hq−q,πhηu) +Q(πhηu, π_hq)

= (η_σ, π_hs) +K(πhu−u, π_hs) +P(π_hq−q,π_hη_u) +Q(πhu−u, π_hq)

=V₁+V₂+V₃+V₄, (3.25)

where we have used (2.9b), (2.9c) and the continuity ofsand q in the first line, the error’s orthogonal property (3.12) and Lemma 2.3 in the third line, and the properties (3.17b), (3.17c) in the fourth line.

Next we turn to estimate each term on the right hand side of (3.25). Firstly, by using the Cauchy-Schwarz inequality, the triangle inequality, the approximation property (3.13a) and the elliptic regularity (3.7) we get

|V₁| ≤ kη_σkkπ_hsk ≤C(kσ−π_hσk+kπ_hη_σk)ksk1 ≤ CC_∗

√ν (h^k+1+kπ_hη_σk)kλk. (3.26) Secondly, due to the superconvergence property (3.14) we have

|V₂+V₄| ≤Ch^k+1(√

νkπ_hsk+kπ_hqk)≤Ch^k+1(√

νksk1+kqk1)≤CC_∗h^k+1kλk. (3.27)

Thirdly, from (3.17b) we can derive that k∇π_hη_uk+p

µh⁻¹k[[πhη_u]]k^Γh≤ C_µ

√ν(h^k+1+kπ_hη_σk). (3.28) The proof of this relationship is similar to but a little more complicated than the proof for Lemma 2.4, so we only give the idea of the proof and omit the details to save space.

According to (3.17b), we choose suitable test functions r (refer to [17]), then we can get (3.28) by using the Cauchy-Schwarz inequality for the terms (π_hη_σ, r) and (σ−π_hσ, r), and the superconvergence property (3.14b) for the termK(u−Π_hu, r).

Hence, a simple application of Lemma 2.2, the approximation property (3.13c), (3.28) and the elliptic regularity (3.7) yields

|V₃| ≤Ckπ_hq−qk(k∇π_hη_uk+p

µh⁻¹k[[π_hη_u]]kΓh)

≤Chkqk1(h^k+1+kπ_hη_σk)≤ CC∗Cµ

√ν h(h^k+1+kπ_hη_σk)kλk. (3.29) As a consequence, by taking λ=πhηu, we can derive

kπ_hη_uk ≤C(h/√

ν+ 1/√

ν+ 1)(h^k+1+kπ_hη

σk)≤Ch^k+1+εkπ_hη_uk, (3.30) whereεis an arbitrary small value. Thus, if we take εsmall enough we can get

kπ_hη_uk ≤Ch^k+1. (3.31)

As a result, we get

kπ_hη_pk ≤Ch^k+1, and kπ_hη

σk ≤Ch^k+1 (3.32)

from (3.24) and (3.19). Consequently, (3.8) follows by the triangle inequality, this completes

the proof of Lemma 3.2.

4 The fully discrete schemes and their stability analysis

4.1 The fully discrete schemes

In this subsection we would like to present the fully-discrete LDG schemes coupled with three specific multi-step IMEX time-marching methods up to the third order, which have been considered in [16] for one-dimensional convection-diffusion equations.

Let {tⁿ=nτ}^Mn=0 be the uniform partition of the time interval [0, T], with time step τ such that M τ =T. Givenuⁿ_h, we would like to find the numerical solution (u_h, p_h) at the next time leveltⁿ⁺¹ by the multi-step IMEX time-marching schemes which were developed in [12].

First order:

(uⁿ⁺¹_h ,v) = (uⁿ_h,v) +τ[H(uⁿ_h,v) +L(σⁿ⁺¹_h ,v) +P(pⁿ⁺¹_h ,v)], (4.1a)

(σⁿ⁺¹_h , r) =K(uⁿ⁺¹_h , r), (4.1b)

0 =Q(uⁿ⁺¹_h , w), ∀n≥0. (4.1c)

Second order:

(uⁿ⁺¹_h ,v) = (uⁿ_h,v) +3

2τH(uⁿ_h,v)−1

2τH(uⁿ⁻¹_h ,v) +3

4τL(σⁿ⁺¹_h ,v) +1

4τL(σⁿ⁻¹_h ,v) +3

4τP(pⁿ⁺¹_h ,v) +1

4τP(pⁿ⁻¹_h ,v), ∀n≥1, (4.2a) (σⁿ_h, r) =K(uⁿ_h, r), ∀n≥0, (4.2b)

0 =Q(uⁿ_h, w), ∀n≥0. (4.2c)

Third order:

(uⁿ⁺¹_h ,v) = (uⁿ_h,v) + 23

12τH(uⁿ_h,v)−4

3τH(uⁿ⁻¹_h ,v) + 5

12τH(uⁿ⁻²_h ,v) +2

3τL(σⁿ⁺¹_h ,v) + 5

12τL(σⁿ⁻¹_h ,v)− 1

12τL(σⁿ⁻³_h ,v) +2

3τP(pⁿ⁺¹_h ,v) + 5

12τP(pⁿ⁻¹_h ,v)− 1

12τP(pⁿ⁻³_h ,v), ∀n≥3, (4.3a)

(σⁿ_h, r) =K(uⁿ_h, r), ∀n≥0, (4.3b)

0 =Q(uⁿ_h, w), ∀n≥0. (4.3c)

Remark 4.1. Schemes (4.2) and (4.3), developed in [12], use standard Adams-Bashforth extrapolation for the explicit part, and a kind of modified Adams-Moulton interpolation which requires the coefficient at time step tⁿ⁺¹ to dominate the sum of the absolute values of the other coefficients. This extra requirement results in a stretched stencil. We consider this kind of IMEX schemes to ensure the unconditional stability.

4.2 Stability

In this subsection, we take the first order scheme (4.1) as an example, to investigate the sta- bility of the IMEX-LDG methods for solving Oseen equations. The unconditional stability will be obtained for the velocityu_h, the stressσ_h =√

ν∇u_h and the pressurep_h. Although the stability results for the second and the third order schemes are a little different from the stability for the first order scheme, they can be obtained analogously. So we omit the proofs for higher order schemes to save space, we refer the readers to [16] for more details.

Theorem 4.1. There exists a positive constant τ₀ independent of h, such that if τ ≤ τ₀, then the solutions of scheme (4.1) satisfy

kuⁿ_hk ≤ ku⁰_hk, (4.4)

kσⁿ_hk ≤e^C0nτkσ⁰_hk, (4.5)

kpⁿ_hk ≤C

ku⁰_hk+1

τku¹_h−u⁰_hk+e^C0nτkσ⁰_hk

, (4.6)

where C₀ is a positive constant depending on 1/√

ν,C is a positive constant independent of h and τ.

Proof. Step 1: prove (4.4). Taking v = uⁿ⁺¹_h , r = τ σⁿ⁺¹_h , w = τ pⁿ⁺¹_h in (4.1a), (4.1b) and (4.1c), respectively, and adding them together, we get

1

2kuⁿ⁺¹_h k²+ ¹₂kuⁿ⁺¹_h −uⁿ_hk²−¹₂kuⁿ_hk²+τkσⁿ⁺¹_h k²

| {z }

LHS

=τH(uⁿ_h,uⁿ⁺¹_h )

| {z }

RHS

, (4.7)

where theLHS is obtained by applying Lemma 2.3 and the equality (a−b)a= ¹₂[a²+ (a− b)²−b²] for arbitrarya, b. To bound the RHS, we write it as

RHS=τH(uⁿ⁺¹_h ,uⁿ⁺¹_h )−τH(uⁿ⁺¹_h −uⁿ_h,uⁿ⁺¹_h ).

Hence, owing to Lemmas 2.1 and 2.4 we have

RHS≤Cγτkuⁿ⁺¹_h −uⁿ_hk(k∇uⁿ⁺¹_h k+p

µh⁻¹k[[uⁿ⁺¹_h ]]k^Γh)

≤C_γC_µ

√ν τkuⁿ⁺¹_h −uⁿ_hkkσⁿ⁺¹_h k

≤τkσⁿ⁺¹_h k²+C_γ²C_µ²

4ν τkuⁿ⁺¹_h −uⁿ_hk², (4.8) where the Young’s inequality is used in the last step. As a consequence, if we let ^C

2 γCµ²

4ν τ ≤ ¹₂, i.e,τ ≤ _C^2ν2

γC_µ², then

kuⁿ⁺¹_h k ≤ kuⁿ_hk ≤ · · · ≤ ku⁰_hk. (4.9) Step 2: prove (4.5). Takingv =uⁿ⁺¹_h −uⁿ_h in (4.1a), we get

kuⁿ⁺¹_h −uⁿ_hk² =R₁+R₂+R₃, where

R₁=τH(uⁿ_h,uⁿ⁺¹_h −uⁿ_h)≤C_γτ(k∇uⁿ_hk+p

µh⁻¹k[[uⁿ_h]]kΓh)kuⁿ⁺¹_h −uⁿ_hk

≤C_γC_µ

√ν τkσⁿ_hkkuⁿ⁺¹_h −uⁿ_hk ≤ kuⁿ⁺¹_h −uⁿ_hk²+C_γ²C_µ²

4ν τ²kσⁿ_hk², (4.10a) R₂=τL(σⁿ⁺¹_h ,uⁿ⁺¹_h −uⁿ_h) =−τK(uⁿ⁺¹_h −uⁿ_h, σⁿ⁺¹_h ) =−τ(σⁿ⁺¹_h −σⁿ_h, σⁿ⁺¹_h )

= − τ

2kσⁿ⁺¹_h k²−τ

2kσⁿ⁺¹_h −σⁿ_hk²+τ

2kσⁿ_hk², (4.10b)

R₃=τP(pⁿ⁺¹_h ,uⁿ⁺¹_h −uⁿ_h) =−τQ(uⁿ⁺¹_h −uⁿ_h, pⁿ⁺¹_h ) = 0. (4.10c) In (4.10a) we used Lemmas 2.1 and 2.4, and the Young’s inequality; Lemma 2.3 and (4.1b), (4.1c) are used in (4.10b) and (4.10c). Thus,

τ

2kσⁿ⁺¹_h k²+τ

2kσⁿ⁺¹_h −σⁿ_hk²− τ

2kσⁿ_hk² ≤ C_γ²C_µ²

4ν τ²kσⁿ_hk². (4.11) The simple use of Gronwall’s inequality leads to (4.5).

Step 3: prove (4.6). With the help of Lemma 2.5, we have, forpⁿ⁺¹_h ∈L²₀(Ω), there exists u^⋆ ∈H₀¹(Ω)² such that ku^⋆k¹ ≤α2kpⁿ⁺¹_h k and

kpⁿ⁺¹_h k² ≤C[K(P_hu^⋆, σⁿ⁺¹_h ) +Q(P_hu^⋆, pⁿ⁺¹_h )

| {z }

S

+kσⁿ⁺¹_h k²], (4.12)