H(div) conforming and DG methods for incompressible Euler’s equations

doi:10.1093/imanum/drnxxx

H(div) conforming and DG methods for incompressible Euler’s equations

JOHNNYGUZMAN´ †

Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

AND

FILANDER´ A. SEQUEIRA‡

Escuela de Matem´atica, Universidad Nacional de Costa Rica, Heredia, Costa Rica.

Present address: CI²MA and Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.

AND

CHI-WANGSHU§

Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 [Received on 26 June 2015]

H(div) conforming and discontinuous Galerkin (DG) methods are designed for incompressible Euler’s equation in two and three dimension. Error estimates are proved for both the semi-discrete method and fully-discrete method using backward Euler time stepping. Numerical examples exhibiting the perfor- mance of the methods are given.

Keywords: Euler equation, discontinuous Galerkin method

1. Introduction

In this paper we study H(div) conforming and DG finite element methods for the incompressible Euler equations in both two and three dimensions. Our methods are based on the velocity-pressure formula- tion. LetΩ be a bounded and simply connected polygonal domain in R^d, d∈ {2,3}, with boundaryΓ^. The velocity u∈H¹₀(Ω):= [H₀¹(Ω)]^d, and the pressure p∈L²₀(Ω)satisfy

u_t+u·∇u+∇p=0 in (0,T)×Ω, (1.1a)

div(u) =0 in (0,T)×Ω, (1.1b)

u·n=0 on (0,T)×Γ, (1.1c)

u(0,x) =u₀(x) in Ω, (1.1d)

where u_t=∂tu is the time derivative,∇u is the tensor gradient of u, and T >0.

The goal of this paper is to define methods that are L²stable, and, for DG methods, are also locally conservative. The methods are inspired by the work Cockburn et al. (2005) where they developed locally conservative DG methods for the steady state Navier-Stokes equations. There they take Newton iterations to solve numerically the equations and in each step they postprocess the DG approximation to

†Email: johnny [email protected]

‡Email: [email protected] / [email protected]

§Email: [email protected]

c The author 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

get a new approximation that belongs to H(div) and is divergence-free. Here we apply this idea to DG methods in each time step for Euler’s equations. However, we first consider H(div) conforming elements as they seem natural for incompressible Euler’s equations and are easier to analyze. In order to make the H(div) elements L²stable, one has to add numerical fluxes of the nonlinear term on the interfaces of the triangulation. We start with the semi-discrete method, using both central and upwind fluxes, and then analyze a backward Euler time stepping method. Once we have developed H(div) conforming methods, we develop DG methods using the post-processing idea used in Cockburn et al. (2005). In Cockburn et al. (2005) upwind fluxes are used, but it is important to note that central numerical fluxes can also guarantee L²stability for Euler’s equations.

The development and study of finite element methods for incompressible flows have a long history;

see for example the books of Temam (see Temam (1984)) and Girault and Raviart (see Girault & Raviart (1986)). More recently there has been an interest in using H(div) conforming methods for these prob- lems (see, e.g., Cockburn et al. (2007)) since they produce divergence-free approximations. However, to the best of our knowledge, an analysis of these methods for the inviscid problem (i.e. Euler’s equation) has not been considered. On the other hand, there has been recent work on proving convergence rates for other finite element methods for problems with arbitrarily low viscosity (see Burman & Fern´andez (2007)).

We give an error analysis for both the semi-discrete methods and the backward Euler time stepping methods. The error estimate for the velocity in the L²norm converges with rateO(h^k)if the velocity space contains the polynomials of degree k. Notice that this is sub-optimal by one order. However, numerical experiments suggest that these results are not sharp for some polynomial orders and using a central numerical flux. More specifically, when using even degree polynomial order for the velocity it seems the methods with central flux converge optimally. In particular, the error estimate will not give an error estimate for the lowest-order Raviart-Thomas element. However, on structured grids our numerical experiments show that the lowest-order Raviart-Thomas elements seem to be converging.

Moreover, when using the upwind numerical flux numerical experiments suggest that the method is optimal. However, at the present time we are not able to prove this result. Our estimates assume that the velocity belongs to W^1,∞. Of course, these a-priori estimates are not known (and might not hold) in three dimensions for general smooth initial data. However, in two dimensions the a-priori estimates were proved by Kato (see Kato (1967)) for smooth initial data.

In addition to providing numerical experiments to check the order of convergence of our methods, we give numerical experiments to show how the methods behave in high gradient flows. We see that using upwind flux the method seems to do very well and comparable to DG methods that use the vorticity- potential formulation (see Liu & Shu (2000)).

One of the advantage of the H(div) conforming methods is that it gives approximations that are pointwise divergence-free. An advantage of the DG methods is that they give locally conservative methods (see below). The other advantage for the upwind versions of both H(div) conforming and DG methods is that numerically they converge optimally on structured meshes. Indeed, it is well known that on general quasi-uniform mesh all the standard methods (Galerkin, streamline diffusion, DG) lose at least a half-order accuracy for scalar problems. On the other hand, upwind DG methods can be shown to convergence optimally (also assuming the correct regularity) for scalar problems on classes of meshes (see Cockburn et al. (2010a), Cheng & Shu (2010)). In fact, as far as we know these are the only methods that have been proving to have this property. To extend these results to the current setting seems non-trivial, but we will pursue this in the future. Moreover, DG methods can be stabilized by adding consistent terms (i.e. jump terms) whereas methods like the streamline diffusion method add in- consistent terms with parameter that need to be tuned to stabilize the method. The paper is organized as

follows. In the next section we present the semi-discrete methods and prove error estimates. In section 3 we present the backward Euler methods. Finally, in section 4 we provide some numerical examples.

2. Semi-discrete methods

We begin by introducing some preliminary notations. LetT_h be a shape-regular and quasi-uniform triangulation ofΩ without the presence of hanging nodes, and letE_hbe the set of edges/faces F ofT_h. In addition, we denote byEⁱ

handE^∂

h the set of interior and boundary faces, respectively, ofE_h, and we set∂^Th:=∪{∂^{K : K}∈T_h}.

Next, let(·,·)U denote the usual L²and L²:= [L²]^d inner product over the domain U ⊂R^d, and similarly leth·,·iGbe the L²and L²inner product over the surface G⊂R^d−1. Then, we introduce the inner products:

(·,·)^T_h :=

∑

K∈T_h

(·,·)K and h·,·i∂T_h :=

∑

K∈T_h

h·,·i∂K.

On the other hand, let n⁺ and n⁻ be the outward unit normal vectors on the boundaries of two neighboring elements K⁺and K⁻, respectively. We use(τ^±,v^±,q^±)to denote the traces of(τ,v,q)on F :=K⁺∩K⁻from the interior of K^±, whereτ, v and q are second-order tensorial, vectorial and scalar functions, respectively. Then, we define the means{{·}}and jumps[[·]]for F∈Eⁱ

h, as follows {{τ}} := 1

2 τ⁺+τ⁻, {{v}} := 1

2 v⁺+v⁻

, {{q}} := 1

2 q⁺+q⁻ , [[τn]] := τ⁺ⁿ⁺+τ⁻ⁿ⁻, [[v·n]] := v⁺·n⁺+v⁻·n⁻, [[qn]] := q⁺n⁺+q⁻n⁻. The method is derived using the conservative or divergence form of the equation. To this end, denoting⊗as the usual dyadic or tensor product, that is, (u⊗v)i j= (u^tv)i j=u_iv_j, we consider the formula

div(u⊗v) = v·∇u +div(v)u, (2.1) together with the divergence-free condition, to write the problem (1.1) in the form

ut +div(u⊗u) +∇p = 0 in (0,T)×Ω, div(u) = 0 in (0,T)×Ω, u·n = 0 on (0,T)×Γ, u(0,x) = u₀(x) in Ω,

(2.2) where div denotes the usual divergence operator div acting along each row of the corresponding tensor.

Finally, given an integerℓ>0 and a subset U of R^d, we denote by P_ℓ(U)the space of polynomials defined in U of total degree at mostℓ, with Pℓ(U):= [Pℓ(U)]^d. Furthermore, for each K∈T_h, we define the local Raviart-Thomas space of orderℓ(see, e.g. Brezzi & Fortin (1991); Roberts & Thomas (1991))

RT_ℓ(K) := P_ℓ(K) + P_ℓ(K)x where x=

x₁

...

x_d

is a generic vector of R^d. In addition, we set ND_ℓ(K) := P_ℓ(K) + P_ℓ(K)×x be the local N´ed´elec space of orderℓon K∈T_h.

2.1 H(div) conforming methods

In this section, we define H(div) conforming finite element schemes associated with the model prob- lem (2.2). We start by introducing the method using the central flux, but in a later section we present the method using the upwind flux. For simplicity we only consider the Raviart-Thomas finite element spaces, but we note that one can use instead the BDM finite elements (see, e.g. Brezzi & Fortin (1991);

Roberts & Thomas (1991)). The globally defined Raviart-Thomas spaces are given by V_hfor the veloc- ity and Q_hfor the pressure, given by

V_h :=

v∈H(div;Ω) : v|K∈RT_k(K) ∀K∈T_h and v·n=0 on Γ , Q_h :=

q∈L²₀(Ω) : q|K∈P_k(K) ∀K∈T_h .

Now, the finite element method is defined by: Find(uh,p_h)∈V_h×Q_h, such that

(∂tu_h,v_h)^T_h −(uh⊗u_h,∇hv_h)^T_h −(ph,div(vh))^T_h +hbσ(uh,p_h)n,v_hi∂T_h = 0,

(qh,div(uh))T_h = 0, (2.3) u_h(0,x) = u_h,0(x) inΩ,

for all(vh,qh)∈V_h×Q_h, where∇his the broken gradient, u_h,0is some projection of u₀on V_h, and σb(uh,p_h)represents the numerical flux of u⊗u+pIonE_h. In particular, we takeσb(uh,p_h):=u_h⊗ u_h+p_hIonE^∂

h and forEⁱ

hwe define

σb(uh,p_h) := {{u_h}} ⊗ {{u_h}}+{{p_h}}I. (2.4) This is the method using the central flux. In a later section we introduce the method using the upwind flux which seems to do better numerically.

Next, using the above definition forσb, together with the formula (2.1), the fact that u_his divergence- free (from the second equation in (2.3)), and integration by parts, we can rewrite (2.3) as: Find(uh,p_h)∈ V_h×Q_h, such that

(∂tu_h,v_h)T_h+ (uh·∇hu_h,v_h)T_h−

∑

F∈Eⁱ

h

h[[(uh⊗u_h)n]],{{v_h}}iF−(ph,div(vh))T_h = 0,

(qh,div(uh))^T_h = 0, (2.5)

u_h(0,x) = u_h,0(x) inΩ, for all(vh,qh)∈V_h×Q_h.

It will be useful to rewrite the[[(u_h⊗u_h)n]]|F. Let F=K⁺∩K⁻. Then, [[(u_h⊗u_h)n]] = [[(u_h·n)u_h]] = (u⁺_h ·n⁺)u⁺_h + (u⁻_h·n⁻)u⁻_h. In addition, from the fact that u⁺_h·n⁺=u⁻_h·n⁺, since u_h∈H(div;Ω), it follows that

[[(uh⊗u_h)n]] = (u⁺_h·n⁺)u⁺_h −(u⁻_h·n⁺)u⁻_h = (u⁺_h ·n⁺)(u⁺_h −u⁻_h).

From now on we will use the notation (without loss of generality)[[[v]]]:=v⁺−v⁻. Also, we use the notation(uh·n)|F= (u⁺_h·n⁺)|F. Hence, we write

[[(uh⊗u_h)n]] = (uh·n)[[[u_h]]].

Now from this we see that the third term in the right-hand side of first equation in (2.5) is consistent, since[[[u]]] =0 onEⁱ

hwhen u is smooth.

LEMMA2.1 (Conservation of energy) Given u_h∈V_hthe solution of (2.5), we have d

dtku_hk²_L2(Ω) = 0.

Proof. Taking v_h:=u_hin the first equation of (2.5) and using that u_his divergence-free, it follows that 1

2 d

dtku_hk²_L²_(Ω)+ (uh·∇hu_h,uh)T_h −

∑

F∈Eⁱ

h

h(uh·n)[[[uh]]],{{u_h}}iF = 0. (2.6) Thus, note that

(u_h·∇hu_h,u_h)T_h = 1

2

∑

K∈T

h

Z K

u_h·∇(|u_h|²) = 1

2

∑

K∈T

h

− Z

K

div(u_h)|u_h|² + Z

∂K(u_h·n)|u_h|²

= 1

2

∑

K∈T_h Z

∂K

(uh·n)|u_h|² = 1

2

∑

F∈Eⁱ

h

Z

F{{u_h}} ·[[|u_h|²n]]

=

∑

F∈Eⁱ

h

Z F

(u_h·n)[[[u_h]]]· {{u_h}}, (2.7)

which, together with (2.6) complete the proof.

We remark here that, from the previous lemma, integrating in time over(0,t), we can deduce that ku_h(t,·)kL²(Ω)=ku_h,0kL²(Ω)for each t∈(0,T). That is, we proved that the scheme (2.5) is stable.

2.1.1 Error estimates. Our next goal is to obtain error estimates for the scheme (2.5). In order to do that, we now introduce the Raviart-Thomas interpolation operator (see Brezzi & Fortin (1991); Roberts

& Thomas (1991))Π^k_h^{: H1}(Ω)→V_h, which satisfies the following approximation properties: for each v∈H^m(Ω), with 16m6k+1, there holds

kv−Π^k_h(v)kL²(K)+h_Kk∇(v−Π^k_h(v))kL²(K) 6 C h^m_K|v|m,K ∀K∈T_h. (2.8) Moreover, we also have the following bounds

kv−Π^k_h(v)kL^∞(K)+hKk∇(v−Π^k_h(v))kL^∞(K) 6 C hKk∇vkL^∞(K) ∀K∈T_h. (2.9) In addition, letP^k

h: L²(Ω)→Q_hbe the L²-orthogonal projector. Hence, for each q∈H^m(Ω), with 06m6k+1, there holds (see, e.g. Ciarlet (1978))

kq−P^k

h(q)kL²(K) 6 C h^m_K|q|H^m(K) ∀K∈T_h. (2.10) We now aim to derive the a priori error estimates for the scheme (2.5). To this end, thanks to the triangle inequality, we only need to provide estimates for the approximation errors, namely, E^u:=

Π^k_h(u)−u_h and E^p:=P^k

h(p)−p_h. To do this, we use the fact that the exact solution satisfies the approximation method (2.5), in order to obtain the error equations:

(∂t(u−u_h),v_h)^T_h+ (u·∇hu−u_h·∇hu_h,v_h)^T_h

−

∑

F∈Eⁱ

h

h(uh·n)[[[u−u_h]]],{{v_h}}iF−(p−p_h,div(vh))T_h = 0, (qh,div(u−u_h))^T_h = 0,

for all(vh,q_h)∈V_h×Q_h. In addition, from the property div(Π^k_h(u)) =P^k

h(div(u)) =0, we can rewrite the error equations in the form

(∂tE^u,v_h)^T_h + (u·∇hu−u_h·∇hu_h,v_h)^T_h −

∑

F∈Eⁱ

h

h(uh·n)[[[E^u]]],{{v_h}}i^F −(E^p,div(vh))^T_h

= (∂t(Π^kh(u)−u),v_h)^T_h −

∑

F∈Eⁱ

h

h(uh·n)[[[Π^kh(u)−u]]],{{v_h}}iF−(P^k

h(p)−p,div(vh))^T_h, (2.11) (qh,div(E^u))^T_h = 0,

for all(v_h,q_h)∈V_h×Q_h, where it is important to remark here that E^uis divergence-free.

THEOREM2.1 Assume that u∈W^1,∞([0,T]×Ω)^dis uniformly bounded. In addition, given an integer k>1, suppose that u₀∈H^k+1(Ω), u∈L²(0,T ; H^k+1(Ω)), and ut∈L²(0,T ; H^k+1(Ω)). Then, there exists C>0, independent of h, such that

k(u−u_h)(T,·)kL²(Ω) 6 C(u)h^kB(u), where

C(u) := (1+C(1+Cu))exp(C(1+C_u)T), with C_u:=kukW^1,∞([0,T]×Ω). Also,

B(u) := hku₀kH^k+1(Ω)+kukL²(0,T ;H^k+1(Ω))+hku_tkL²(0,T ;H^k+1(Ω)). Proof. We begin by choosing v_h:=E^uin (2.11). Thus, we have

1 2

d

dtkE^uk²_L2(Ω) = −(u·∇hu−u_h·∇hu_h,E^u)T_h

| {z }

I1

+

∑

F∈Eⁱ

h

h(u_h·n)[[[E^u]]],{{E^u_h}}iF

| {z }

I2

+ (Π^kh(ut)−u_t,E^u)^T_h −

∑

F∈Eⁱ

h

h(uh·n)[[[Π^kh(u)−u]]],{{E^u}}iF

| {z }

I₃

, (2.12)

where we have used the fact that∂tΠ^k_h(u) =Π^k_h(ut). Next, note that

I₁ = −(u·∇h{u−Π^k_h(u)},E^u)T_h −((u−u_h)·∇hΠ^k_h(u),E^u)T_h −(uh·∇hE^u,E^u)T_h,

= −(u·∇h{u−Π^kh(u)},E^u)^T_h −((u−u_h)·∇hΠ^kh(u),E^u)^T_h −I₂,

where in the last term, we apply the same arguments of (2.7) by using E^uinstead of u_hin the last two functions. Furthermore, using (2.9) we deduce that

I₁+I₂ 6 C_uk∇h{Π^kh(u)−u}kL²(Ω)kE^ukL²(Ω)+CC_uku−u_hkL²(Ω)kE^ukL²(Ω)

6 Cuk∇h{Π^k_h(u)−u}kL²(Ω)kE^ukL²(Ω)+CCu

n

kΠ^k_h(u)−ukL²(Ω)+kE^ukL²(Ω)

o

kE^ukL²(Ω)

6 CCukE^uk²_L²_(Ω)+CCu

n

kΠ^k_h(u)−uk²_L²_(Ω)+k∇h{Π^k_h(u)−u}k²_L²_(Ω)o

. (2.13)

On the other hand, for I₃it follows I₃ = −

∑

F∈Eⁱ

h

h(E^u·n)[[[Π^k_h(u)−u]]],{{E^u}}iF+

∑

F∈Eⁱ

h

h(Π^k_h(u)·n)[[[Π^k_h(u)−u]]],{{E^u}}iF

6 Ch⁻¹kΠ^k_h(u)−ukL^∞(Ω)

∑

F∈Eⁱ

h

h_Fk{{E^u}}k²_L²_(F)

+CkΠ^k_h(u)kL^∞(Ω)





∑

F∈Eⁱ

h

h⁻_F¹k[[[Π^k_h(u)−u]]]k²_L2(F)





1/2



∑

F∈Eⁱ

h

h_Fk{{E^u}}k²_L2(F)





1/2

. (2.14)

In addition, given v∈H¹(T_h)and applying a discrete trace inequality, we note that there existsCb>0, independent of h, such that

∑

F∈Eⁱ

h

h⁻_F¹k[[[v]]]k²_L²_(F) 6 Cbn

h⁻²kvk²_L²_(Ω) +k∇hvk²_L²_(Ω)o

, (2.15)

and, in the same way together with an inverse inequality we obtain

∑

F∈Eⁱ

h

h_Fk{{E^u}}k²_L²_(F) 6 CbkE^uk²_L²_(Ω). (2.16) Hence, replacing (2.15) and (2.16) in (2.14) and using (2.9) we deduce that

I₃ 6 CC_ukE^uk²_L²_(Ω)+CC_u n

h⁻²kΠ^kh(u)−uk²_L²_(Ω)+k∇h{Π^kh(u)−u}k²_L²_(Ω)o

. (2.17) Now, we return to (2.12), which satisfies that

1 2

d

dtkE^uk²_L2(Ω) 6 1

2kE^uk²_L2(Ω)+1

2kΠ^k_h(ut)−u_tk²_L2(Ω)+ (I1+I₂) +I₃, where, replacing (2.13) and (2.17), we obtain that

d

dtkE^uk²_L2(Ω) 6 C(1+Cu)kE^uk²_L2(Ω)+CkΠ^k_h(ut)−u_tk²_L2(Ω)

+CCu

n

h⁻²kΠ^kh(u)−uk²_L²_(Ω)+k∇h{Π^kh(u)−u}k²_L²_(Ω)o

. (2.18) Hence, applying (2.8) we get

d

dtkE^uk²_L²_(Ω) 6 C(1+Cu)kE^uk²_L²_(Ω) +C(1+Cu)h^2k

h²ku_tk²_Hk+1(Ω)+kuk²_Hk+1(Ω)

.

which, applying the Gronwall’s inequality (see, e.g. Evans (2010)), yields kE^u(T,·)k²_L²_(Ω) 6 exp(C(1+C_u)T)n

kE^u(0,·)k²_L²_(Ω) +C(1+Cu)

h²kutk²_L²_{(0,T ;H}k+1(Ω))+kuk²_L²_{(0,T ;H}k+1(Ω))

o .

Finally, we use thatkE^u(0,·)k²_L2(Ω)6C h^2(k+1)ku₀k²_Hk+1(Ω)to complete the proof.

The next goal is to establish error estimates for the pressure variable. To do this, we first obtain an estimate for∂t(u−u_h), which is the subject of the next result.

LEMMA2.2 Assume the same hypotheses of Theorem 2.1. Then, there exists C>0, independent of h, such that

k∂tE^u(T,·)kL²(Ω) 6 (C(u)h^k−d/2B(u) +Cu)h^k−1n

C(u)B(u) +ku(T,·)kH^k+1(Ω)

o

+Ch^k+1kut(T,·)kH^k+1(Ω).

Proof. First, we take v_h:=∂tE^uin (2.11) and using that div(∂tE^u) =∂tdiv(E^u) =0, we obtain k∂tE^uk²_L2(Ω) = −(u·∇hu−u_h·∇hu_h,∂tE^u)T_h +

∑

F∈Eⁱ

h

h(u_h·n)[[[E^u]]],{{∂tE^u}}iF

+ (Π^kh(ut)−u_t,∂tE^u)^T_h −

∑

F∈Eⁱ

h

h(uh·n)[[[Π^kh(u)−u]]],{{∂tE^u}}iF

6 ku·∇hu−u_h·∇hu_hkL²(Ω)k∂tE^ukL²(Ω)+kΠ^k_h(ut)−u_tkL²(Ω)k∂tE^ukL²(Ω)

+Cku_hkL^∞(Ω)





∑

F∈Eⁱ

h

h⁻_F¹k[[[E^u]]]k²_L2(F)





1/2



∑

F∈Eⁱ

h

h_Fk{{∂tE^u}}k²_L2(F)





1/2

+Cku_hkL^∞(Ω)





∑

F∈Eⁱ

h

h⁻_F¹k[[[Π^k_h(u)−u]]]k²_L2(F)





1/2



∑

F∈Eⁱ

h

h_Fk{{∂tE^u}}k²_L2(F)





1/2

.

Next, using (2.15) and (2.16), we deduce after some algebraic manipulation that k∂tE^ukL²(Ω) 6 C

n

h⁻¹ku_hkL^∞(Ω)kE^ukL²(Ω)+ku·∇hu−u_h·∇hu_hkL²(Ω)

+kΠ^k_h(ut)−u_tkL²(Ω)+ku_hkL^∞(Ω)(h⁻¹kΠ^k_h(u)−ukL²(Ω)+k∇_h(Π^k_h(u)−u)kL²(Ω))o

. (2.19) To bound the nonlinear term we add and subtract terms to get

ku·∇hu−u_h·∇hu_hkL²(Ω) = k(u−u_h)·∇hu+u_h·∇h(u−u_h)kL²(Ω)

6 C_uku−u_hkL²(Ω)+ku_hkL^∞(Ω)k∇h(u−u_h)kL²(Ω)

6 Cuku−u_hkL²(Ω)+ku_hkL^∞(Ω)(k∇h(u−Π^k_hu)kL²(Ω)+Ch⁻¹kE^ukL²(Ω)) 6 C_ukE^ukL²(Ω)+CukΠ^k_h(u)−ukL²(Ω)

+ku_hkL^∞(Ω)(k∇h(u−Π^khu)kL²(Ω)+Ch⁻¹kE^ukL²(Ω)), where we have used an inverse estimate. Therefore,

k∂tE^ukL²(Ω) 6 Cn

(h⁻¹ku_hkL^∞(Ω)+C_u)kE^ukL²(Ω)+kΠ^k_h(ut)−u_tkL²(Ω)

+ku_hkL^∞(Ω)k∇h(Π^k_h(u)−u)kL²(Ω)+ (h⁻¹ku_hkL^∞(Ω)+Cu)kΠ^k_h(u)−ukL²(Ω)

o. We can boundku_hkL^∞(Ω)using an inverse estimate

ku_hkL^∞(Ω) 6 kE^ukL^∞(Ω)+kΠ^k_h(u)kL^∞(Ω) 6 C h^−d/2kE^ukL²(Ω)+CC_u. (2.20)

Hence,

k∂tE^u(t)kL²(Ω) 6 Cn

h⁻¹(h^−d/2kE^ukL²(Ω)+Cu)kE^ukL²(Ω)+kΠ^k_h(ut)−u_tkL²(Ω)

+ (h^−d/2kE^ukL²(Ω)+Cu)

k∇h(Π^k_h(u)−u)kL²(Ω)+h⁻¹kΠ^k_h(u)−ukL²(Ω)

o.

Finally, using Theorem 2.1 and (2.8) establishes the result.

Note that in the above proof we have also proved

k(u·∇hu−u_h·∇hu_h)(T,·)kL²(Ω) 6 (C(u)h^k−d/2B(u) +Cu)h^k−1n

C(u)B(u) +ku(T,·)kH^k+1(Ω)

o

+Ch^k+1k∂tu(T,·)kH^k+1(Ω). (2.21) We end this section with the a-priori error estimate for the pressure, which is established next.

THEOREM 2.2 Assume the hypothesis of Theorem 2.1. Also, suppose that p∈L²(0,T ; H^k+1(Ω)).

Then, there exists C>0, independent of h, such that

k(p−p_h)(T,·)kL²(Ω) 6 (C(u)h^k−d/2B(u) +Cu+C h)h^k−1n

C(u)B(u) +ku(T,·)kH^k+1(Ω)

o +Ch^k+1

n

kut(T,·)kH^k+1(Ω)+kp(T,·)kH^k+1(Ω)

o . Proof. We begin by recalling here the discrete inf-sup given by

βkq_hkL²(Ω) 6 sup

v_h∈V_h v_h6=0

(qh,div(vh))^T_h

kv_hkH(div;Ω) ∀q_h∈Q_h, (2.22) which, in particular for q_h:=E^p, it follows

kE^pkL²(Ω) 6 1 βv^suph∈Vh

v_h6=0

(E^p,div(vh))^T_h

kv_hkH(div;Ω) . (2.23)

Now, from the error equation (2.11) and proceeding as in the proof of Lemma 2.2, we have (E^p,div(v_h))T_h = (∂tE^u,v_h)T_h + (u·∇hu−u_h·∇hu_h,v_h)T_h−

∑

F∈Eⁱ

h

h(u_h·n)[[[E^u]]],{{v_h}}iF

−(Π^kh(ut)−ut,v_h)^T_h +

∑

F∈Eⁱ

h

h(uh·n)[[[Π^kh(u)−u]]],{{v_h}}i^F + (P^k

h(p)−p,div(vh))^T_h 6 k∂tE^ukL²(Ω)kv_hkL²(Ω)+ku·∇hu−u_h·∇hu_hkL²(Ω)kv_hkL²(Ω)+Ch⁻¹kE^ukL²(Ω)kv_hkL²(Ω)

+C n

h⁻¹kΠ^kh(u)−ukL²(Ω)+k∇h(Π^kh(u)−u)|L²(Ω)

o

kv_hkL²(Ω)

+kΠ^k_h(ut)−u_tkL²(Ω)kv_hkL²(Ω)+kP^k

h(p)−pkL²(Ω)kdiv(vh)kL²(Ω). The above result together with (2.23) establishes

kE^pkL²(Ω) 6 C n

k∂tE^ukL²(Ω)+ku·∇hu−u_h·∇hu_hkL²(Ω)+h⁻¹kE^ukL²(Ω)

+h⁻¹kΠ^k_h(u)−ukL²(Ω)+k∇h(Π^k_h(u)−u)kL²(Ω)+kΠ^k_h(ut)−utkL²(Ω)+kP^k

h(p)−pkL²(Ω)

o .

Therefore, thanks tokp−p_hkL²(Ω)6kE^pkL²(Ω)+kP^k

h(p)−pkL²(Ω), (2.21), Lemma 2.2 and the ap- proximation properties (2.8) and (2.10), we can easily complete the proof.

Notice the the error estimate for the pressure predictsO(h^k−1)(for k>2) in two and three dimen- sions.

2.1.2 Using an upwind flux. Here, we introduce an alternative version of the conforming method (2.5), analyzed in previous sections. In order to do that, we begin by redefining the numerical fluxσb (cf. (2.4)) in a new general form, given by:

σb(uh,p_h) := bu^w_h⊗ {{u_h}} +{{p_h}}I,

wherebu^w_h is a new numerical trace for u_hrelated with the convective term. In particular, takingbu^w_h :=

{{u_h}}=¹₂ u^int_h +u^ext_h

we arrive exactly to the scheme (2.5). That is, the method (2.5) correspond to a central scheme.

On the other hand, for some problems with high gradients, it is more natural to use an upwind scheme, in order to get better accuracy and order of convergence. In Section 4 we will present some examples of this. In fact, we see numerically that using upwind flux gives optimal convergence rates for both the velocity and pressure variables.

According to above, we consider the following upwind flux b

u^w_h :=

( u^int_h if u_h·n>0, u^ext_h if u_h·n<0.

This definition is given in the same way of that presented in Liu & Shu (2000) for the vorticity, and it is not difficult to check that we can obtain again the method (2.5), with an extra term given by a weighted full jumps ontoEⁱ

h. That is, we seek u_h∈V_hand p_h∈Q_h, such that (∂tu_h,v_h)T_h+ (u_h·∇hu_h,v_h)T_h−

∑

F∈Eⁱ

h

h(u_h·n) [[[u_h]]],{{v_h}}iF

+

∑

F∈Eⁱ

h

h|u_h·n|[[[u_h]]],[[[v_h]]]iF−(ph,div(vh))^T_h = 0 ∀v_h∈V_h, (2.24) (q_h,div(u_h))T_h = 0 ∀q_h∈Q_h,

u_h(0,x) = u_h,0(x) inΩ.

It is important to remark here, that the introduction of this new term does not pose any difficulty in order to prove stability and convergence. In fact, both follow the same arguments, using that when v_h=u_hthis term is positive. In particular, the error estimates are basically the same and the stability, see remark after the proof of Lemma 2.1, now is given byku_h(t,·)kL²(Ω)6ku_h,0kL²(Ω)for each t∈(0,T).

2.2 DG schemes

In this section, we introduce a discontinuous Galerkin method for the model problem (2.2). The velocity space will consist of polynomials of degree k+1 for the fully discontinuous subspace

V^dg_h :=

v∈L²(Ω) : v|K∈P_k+1(K) ∀K∈T_h and v·n=0 on Γ ,