Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system

Vol. 39, N 6, 2005, pp. 1177–1202 DOI: 10.1051/m2an:2005051

CONVERGENCE OF LOCALLY DIVERGENCE-FREE

DISCONTINUOUS-GALERKIN METHODS FOR THE INDUCTION EQUATIONS OF THE 2D-MHD SYSTEM

Nicolas Besse

and Dietmar Kr¨ oner

Abstract. We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition ∆t∼h^4/3, we obtain error estimates inL² of orderO(∆t²+h^m+1/2) wheremis the degree of the local polynomials.

Mathematics Subject Classification. 76W05, 65J10.

Received: January 17, 2005. Revised: July 4, 2005.

1. Introduction

The magnetohydrodynamic (MHD) equations modelize electrically conducting fluid flow in which the elec- tromagnetic forces can be of the same order or even greater than hydrodynamic ones. The ideal MHD system which combines the equations of gas dynamics with Maxwell equations in which relativistic, viscous and resistive effects are neglected can be written in the following three dimensional conservative form

∂_tρ+∇ ·(ρu) = 0 (mass conservation)

∂_t(ρu) +∇ ·

ρu⊗u+

p+¹₂|B|²

I −B⊗B

= 0 (momentum conservation)

∂_tB+∇ ·(u⊗B−B⊗u) = 0 (induction equation)

∂_t(ρe) +∇ ·

ρe+p+¹₂|B|²

u−B(u·B)

= 0 (energy conservation)

∇ ·B = 0 (divergence constraint)

whereρis the density,uthe velocity field,Bthe magnetic field,pthe pressure,ethe total energy andIidentity matrix. If the initial data are divergence free, that is to say that∇ ·B₀= 0, then an exact solution will satisfy this constraint for all the times. For smooth solution this is obvious because the induction equation can be

Keywords and phrases. Magnetohydrodynamics, discontinuous-Galerkin methods, convergence analysis.

∗The authors acknowledge financial support from the HYKE project No HPRN-CT-2002-00282 on “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications”, funded by European Union.

1 Institut de Recherche Mathematique Avanc´ee, Universit´e Louis Pasteur, CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France. [email protected]

2 Institut f¨ur Angewandte Mathematik Universit¨at Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany.

[email protected]

c EDP Sciences, SMAI 2005

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2005051

rewritten as∂_tB+ curl(B×u) = 0 and we have the identity∇ ·(∇ × ·) = 0. The numerical preservation of the

∇ ·B= 0 condition is an important and much debated problem for numerical MHD codes [3, 4, 10–12, 20, 22].

Because∇ ·Berrors arise in numerical simulations and may increase in time, numerical instabilities can appear and lead to unphysical behaviour of the system. For example numerically incorrect magnetic field topologies lead to unphysical plasma transport orthogonal to the magnetic field. The non enforcement of the∇ ·B constraint leads to the loss of momentum and energy conservation and allows fictitious forces to develop parallel to the magnetic field. These effects are discussed in [3, 4].

In this paper we present the convergence analysis and error estimates of locally divergence-free Runge Kutta discontinuous Galerkin schemes for smooth solutions of the two dimensional induction equation which arises in the MHD system

∂B

∂t + curl(B×u) = 0 (1)

where B = (B_x(t,x), B_y(t,x)) is the magnetic field, andu= (u_x(t,x), u_y(t,x)) is the velocity field, with the notationsx= (x, y). We assume for this paper that the velocity fielduis given.

The construction and the convergence analysis of this discontinuous Galerkin method rest on three ingredients.

First we rewrite the induction equation as a Friedrichs system. Then we write a discontinuous Galerkin formula- tion of this new system in which we choose upwind flux as definition of the flux at the cell interface. In order to have a locally divergence free scheme we use piecewise solenoidal functions that are totally discontinuous across interface cells but which are pointwise divergence free on each element. This basis functions were developed in [2, 16] in the context of nonconforming finite element method for the stationary Navier-Stokes equations and were also used in [17]. We also use the Nedelec finite element in H(curl) [19] in two dimensions which is obtained by rotating the Raviart-Thomas element [21] byπ/2. Then we used the framework developed in [6, 24] to show the convergence and obtained error estimates for our scheme. In [7] the authors develop a locally divergence free discontinuous Galerkin scheme for numerically solving the Maxwell equation. They also show an error estimate of the formO(h^k+1/2) wherekis the degree of the local polynomials for a scheme semi-discretized in space only.

2. The numerical scheme 2.1. The Friedrichs formulation

Thanks to the divergence constraint,∇ ·B= 0, the induction equation (1) can be rewritten as the following Friedrichs system:

∂B

∂t +∂(A_xB)

∂x +∂(A_yB)

∂y +CB= 0 in Ω×[0, T], (2)

where

Ax(t,x) =

u_x(t,x) 0 0 u_x(t,x)

, Ay(t,x) =

u_y(t,x) 0 0 u_y(t,x)

, and

C(t,x) =−

∂_xu_x(t,x) ∂_yu_x(t,x)

∂_xu_y(t,x) ∂_yu_y(t,x)

.

Sometimes we will use the notation (v1,v2) instead of (vx,vy) wherevis a vector or matrix fields. For suitable boundary conditions on ∂Ω Friedrichs [14] has proved the existence and uniqueness in L²(Ω) of a weak and strong solution to (2), under appropriate smoothness conditions and the additional assumption

C+C^T +∂Ax

∂x +∂Ay

∂y ≥αI, in Ω, (3)

with α a strictly positive constant. We suppose that u ∈ L^∞(0, T;C^∞(Ω)), ∇ ·u ∈ L^∞(0, T;L^∞(Ω)) and C ∈L^∞(0, T;C^∞(Ω)). We recall that ifB₀ ∈H^s(^Ê2) with s∈^Ê, then the system (2) admits a unique weak solutionB∈C⁰

[0,+∞[;H^s(^Ê2)

∩C¹

[0,+∞[;H^s−1(^Ê2)

(see [1]). In particular, ifs >2 the weak solution lies inC¹

[0,+∞[×^Ê2

and is in fact a classical solution.

Remark 1. The condition (3) can be easily satisfied by usingB(t, x) =e^−λtB(t,x), withλ >0 instead of B.

ThenB satisfies

∂B

∂t +∂(AxB)

∂x +∂(AyB)

∂y +CB = 0 in Ω×[0, T], (4)

whereC=C+λI. Since we assume∇u∈L^∞(Ω) we can chooseλsufficiently large such that condition (3) is satisfied for (4).

2.2. The approximation spaces

First we introduce the space^∞(0, T;X) defined by ^∞(0, T;X) :=

f :{t⁰, ..., t^NT} →X| ||f||^∞_(0,T;X)= max

1≤n≤NT||f(tⁿ)||_X <∞

where X denotes a functional space. For a subdomain D⊂Ω,H^m(D) denotes the usual Sobolev space which is a Hilbert space with the inner product (u, v)_m,D =

|α|≤m D∂^αu∂^αvdx, u, v∈ H^m(D) and the square of seminorm|v|²_Hm(D)=

|α|=m D|∂^αv|².Then we define the spaceH^m(D) = (H^m(D))² with inner product (v,w)_m,D = ₂

i=1(v_i, w_i)_m,D, and the solenoidal vector fields S^m(D) = {v∈H^m(D)| ∇ ·v= 0 inD}, m ≥ 1. Let T_h a family of partitions of Ω that possesses properties described in [2, 16]. For integer k ≥ 0,

È

k(D) will denote the linear space of polynomials in two variables, of degree less than or equal tokonD. Then we define ^k(D) =

È

k(D)2

andV^k(D) =

v∈^k(D)| ∇ ·v= 0 inD

.We haveV^k(D)⊂S^m(D) for all m≥1. Fork≥0 we defineV_h^k(Ω) =

K∈ThV^k(K) where K is a finite element (or cell) andN the number of finite elements. The way of constructing local bases for V^k and hence forV_h^k are described in [2, 16]. For example, fork= 1 the bases functions can be constructed using the set of functions

Ξ¹= 1

0

, 0

1

, y

0

, 0

x

, x

−y

.

To obtain a basis function fork= 2 it suffices to augment the above set by Ξ²=

y² 0

,

0 x²

,

x²

−2xy

,

−2xy y²

.

Then we see that the basis functions for the approximation space V_h^k can be constructed by adding suitable terms toV_h^k−1.

Remark 2. In the case whereT_his a triangulation in^Ê2the spacesV^k(K) are equivalent to the spacesRT_k⁰(K) defined byRT_k⁰(K) ={v∈RT_k(K)|divv= 0}, whereRT_k(K) is the usual Raviart-Thomas space onKused to approximateH(div). In fact each element ofV^k(K) orRT_k⁰(K) is constructed as the curl of a stream function which belongs to^Èk(K) and the dimension of the both space is equal to dim (^Èk(K)−1) = (k+ 1)(k+ 4)/2.

The spaces V_h^m possess optimal approximation properties in relation to the spacesS^m(Ω) which has been proved in [2]. We recall some of those.

Theorem 1. Let m≥0. Ifv∈H^m+1(K), then there exists χ∈^m(K)such that v−χ _Hj(K)≤Ch^m+1−j_K |v|_H^m+1_(K), 0≤j≤m+ 1 whereh_K is the diameter of the largest ball contained in the cellK.

Let m≥0. Ifv∈S^m+1(K), then there existsχ∈V^m(K)such that

v−χ _Hj(K)≤Ch^m+1−j_K |v|_H^m+1_(K), 0≤j≤m+ 1. (5)

Proof. See [2].

Now we define the local L² projection operator π_h from S^m(Ω) ontoV_h^k(Ω), with m≥1 by the equations π_h=

K∈Thπ_K^½_K(x) and

K(π_Kv−v)·φ= 0, ∀φ∈V^k(K), ∀K∈ Th. (6) where ^½_K(x) denotes the characteristic functions of K which is equal to 1 on K and 0 elsewhere. Thanks to the properties of the partitionTh(see [2]) there exists a constantσsuch thath_K≤σh,∀K∈ Th. For example we can takeh= min_K∈Thh_K. Then we have the following approximation result.

Theorem 2. Let Th be a partition ofΩsatisfying the properties assumed in [2], i.e.

i) ∪K∈ThK= Ω,K∩Q=∅, K, Q∈ Th, K=Q.

ii) ∃ µ₁ a constant independent ofN andK ∈ Th such thatρ_K ≤µ₁h_K whereρ_K is the diameter of the smallest ball containingK.

iii) ∃µ₂a constant independent ofN andK∈ T_hsuch thath_Q≤µ₂ρ_K wheneverQandKhave a common 1-dimensional interface.

iv) K is a regular domain in ^Ê2, i.e. the divergence theorem holds onK.

If v∈S^m+1(Ω) then there exists a constantC such that

π_hv−v _L2(Ω)≤Ch^m+1|v|_H^m+1_(Ω). (7) Moreover we have the following inverse inequalities

|v_h|_H¹_(K)≤ C

h v_{h L}2(K), |v_h|_L²_(∂K)≤ C

h^1/2 v_{h L}2(K), ∀v_h∈V^m(K) (8) and the following approximation result

πhv−v _Hs(K)≤Ch^m+1−s|v|_H^m+1_(K), ∀v∈S^m+1(Ω) s∈^Ê+, s≤m+ 1. (9)

Proof. See Appendix 5.

Now we introduce another discontinuous interpolation operator Π_h based on the two-dimensional N´ed´elec element in H(curl) (see [19]). Let_k denote the space of homogeneous polynomials of degreekin^Ê2and con- sider the following subspace of^k,R_k =^k−1⊕S_kwhereS_kis defined byS_k={p∈k; p(x)·x= 0, for all x= (x₁, x₂)}.Then we defineW^k(K) ={v∈ R_k(K)}andW_h^k(Ω) =

K∈ThW^k(K).Following [15], if we define the two sets of moments ofv∈H^swiths≥1/2 onK: M_e(v) ={ _e(v×ν_e)qdΓ ∀q∈^Èk−1(e) for all edgese ofK}whereν_eis the outward unit normal to the edgeseofK, andM_K(v) = _Kv·qdx, ∀q∈^k−2(K)

, then a vector fieldv ofRk is entirely determined in a triangle Kby its two sets of moments: Me(v), MK(v).

Moreover the tangential components of von a given edgeeofK depend only the momentsMe(v) defined on that edges. Besides we have the following approximation properties (see [15, 19]):

Lemma 1. Let v∈H^m(curl; Ω), then there exists a constant C independent of hsuch that

v−Π_hv _L2(K)≤Ch^m |v|_Hm(K) (10) v−Π_hv _H(curl;K) ≤Ch^m

|v|_Hm(K)+|curlv|_Hm(K)

(11) and

(v−Π_hv)×νe H^s(e)≤Ch^m−s|v|_Hm(e), ∀e∈∂K s≤m (12) with Π_hv_|K = Π_Kv whereΠ_Kv∈W^m(K)is determined byMK(Π_Kv−v) ={0},andMe(Π_Kv−v) ={0}.

If m = 1 then R1 = ⁰⊕S₁ where S₁ = span(x^⊥) and then it is obvious that div R1 ≡ 0. Then the vector fields of R1 is divergence free. Unfortunately the spaces Rk with k >1 are not divergence free spaces.

Now we present a trace result and a continuous embedding which will be useful later. LetKa convex polygon with Lipschitz boundary in ^Ê2. First we define the space V(K) and W(K) by W(K) = {v ∈ H(curl;K)∩

H(div;K), v×ν∈L²(∂K)}equipped with the norm v _W(K)= v _L2(K)+ curlv _L2(K)+ divv _L2(K)+ v×ν _L2(∂K) and V(K) = {v∈H(curl;K)∩H(div;K), v·ν∈L²(∂K)

equipped with the norm v _V(K)= v _L2(K)+ curlv _L2(K)+ divv _L2(K)+ v·ν _L2(∂K).It is proved in [8,9] thatV(K) =W(K) and then that the mappingv−→γ₀vdefined onD(K) has a unique linear continuous extension as an operator from W(K) ontoL²(∂K), where γ₀v is the boundary values of v on ∂K. Now let us define Q(K) = {v∈H(curl;K)∩H(div;K), v×νe∈H^1/2(e) ∀e∈∂K

equipped with the norm v _Q(K) = v _L2(K)+ curlv _L2(K)+ divv _L2(K)+

e∈∂K v×νe H^1/2(e). Then we have the following theorem where a proof can be found in [13]

Theorem 3. LetKbe a convex polygon in^Ê2with Lipschitz boundary. Then we have the continuous embedding:

Q(K) →H¹(K). (13)

2.3. The discontinuous Galerkin method

In this section we describe our discontinuous Galerkin method. If we take the scalar product of the equa- tion (1) with a test functionϕ, integrate the scalar product on a cellK and use the following Green formula

K

v∂_iwdx=−

K

w∂_ivdx+

e∈∂K

e

wvν_e,Kⁱ dΓ (14)

whereνe,K = (ν_e,K¹ , ν_e,K² )^T = (ν_e,K^x , ν^y_e,K)^T denotes the outward unit normal to the faceeofK, we obtain

∂_t

K

B·ϕdx

− 2 i=1

KAiB·∂_iϕdx+ 2 i=1

e∈∂K

eAiB·ϕν_e,Kⁱ dΓ +

KCB·ϕdx= 0. (15) Now we replace respectively B and ϕ by B_h and ϕ_h in (15) where B_h, ϕ_h ∈ V_h^k(Ω). Nevertheless the terms arising from the boundary of the cell K in (15) are not well defined or have no sense sinceB_h and ϕ_h are discontinuous on the boundary ∂K of the element K. Then we replace these terms by a numerical upwind flux that we are going to define in the following lines. Let us defineA_e,K(t,x) =₂

i=1(A_i)_|eν_e,Kⁱ and C_e,K(t,x) = −(Ae,K(t,x))⁻, D_e,K(t,x) = (Ae,K(t,x))⁺ where A⁻ and A⁺ denote respectively the negative and the positive part of A. Then we define the upwind numerical flux g(ν_e,K,v,w) = −Ce,Kw+D_e,Kv.

By noting that |A| = A⁺−A⁻ and A = A⁺+A⁻ we can rewrite g(ν_e,K,v,w) as the Lax-Friedrichs flux g(ν_e,K,v,w) =Ae,K(v+w)

2 − |Ae,K|^(w−v)₂ ,where v(resp. w) denotes the interior (resp. exterior) value of the solution from the cellK. Now we obtain the following semi-discretized scheme in space

∂_t

K

B_h·ϕ_hdx

−²

i=1

KA_iB_h·∂_iϕ_hdx+ 2 i=1

e∈∂K

eg(ν_e,K,B_K,B_Ke)·ϕ_KdΓ +

KCB_h·ϕ_hdx= 0 for allϕ_h∈V_h^k whereK_eis the neighbourhood ofK along the facee. LetT be the final time and ∆t=T /N_T the time step. Now we use a second order Runge-Kutta scheme for the discretization in time. Then the fully

discretized scheme reads

K

Yⁿ_h·ϕ_hdx =

K

Bⁿ_h·ϕ_hdx+ ∆tF_Kⁿ(Bⁿ_h,ϕ_h) (16)

K

Bⁿ⁺¹_h ·ϕ_hdx =

K

1 2Bⁿ_h+1

2Yⁿ_h

·ϕ_hdx+∆t

2 F_Kⁿ⁺¹(Yⁿ_h,ϕ_h) (17) where

F_Kⁿ(W,ϕ_h) = 2

i=1

KAⁿ_iWⁿ·∂_iϕ_hdx+ 2 i=1

e∈∂K

egⁿ(W)_e·ϕ_KdΓ−

KCⁿWⁿ·ϕ_hdx with

gⁿ(W)_e=gⁿ(ν_e,K,W_Kⁿ,W_Kⁿ_e) =Aⁿ_e,K(Wⁿ_K+W_Kⁿ

e)

2 − |Aⁿ_e,K|(Wⁿ_Ke−Wⁿ_K)

2 (18)

andAⁿ_e,K =Ae,K(tⁿ,x), Cⁿ=C(tⁿ,x).LetX^k be V^k orW^k. If we expandB_h andϕ_h, that is to say

Bⁿ_h(x) =

K∈Th

Bⁿ_K(x)^½_K(x), Bⁿ_K(x) =

dim(X^k) i=1

σ_K^i,nΨ_i(x), Y_Kⁿ(x) =

dim(X^k) i=1

θ^i,n_K Ψ_i(x)

with Ψ_i∈ X^k, then we obtain the scheme







[Θ_K]ⁿ = [Σ_K]ⁿ+ ∆tLⁿ_K([Σ_h]ⁿ) [Σ_K]ⁿ⁺¹ = 1

2([Σ_K]ⁿ+ [Θ_K]ⁿ) + ∆tLⁿ⁺¹_K ([Θ_h]ⁿ) where [Σ_K]ⁿ=

σ_K^1,n, . . . , σ_K^{dim (Xk),n} _T

, [Σ_h]ⁿ=

σ^1,n, . . . , σ^{card(Th)dim(Xk),n} _T

and for alli∈ {1, ...,dim(X^k)}

[Lⁿ_K([Σ_h]ⁿ)]_i= 2 l=1

dim(X^k) m,j=1

σ_K^j,n

KAⁿ_l(x)Ψ_j(x)· M^−T_K

m,i∂_lΨ_m(x)dx

+ 2 i=1

e∈∂K

e dim(X^k)

m,j=1

[σ_K^j,nDⁿ_e,K(x(Γ))−σ_K^j,n_eC_e,Kⁿ (x(Γ))]Ψ_j(x(Γ))· M^−T_K

m,iΨ_m(x(Γ))dΓ

−

dim(X^k) m,j=1

σ_K^j,n

KCⁿ(x)Ψ_j(x)· M^−T_K

m,iΨ_m(x)dx with (MK)_ij = _KΨ_i(x)·Ψ_j(x)dx.

3. Analysis of the semi-discretized scheme in space

In this section we show the L²-stability, the convergence and present some error estimates for the semi- discretized scheme in space, continuous in time.

Theorem 4. Let u andB₀ sufficiently, regular, typically we consider that u∈ L^∞

[0,+∞[;W^1,∞(^Ê2) and B₀∈H^m+1(^Ê2) then there exists a constantC=C( ∇u _L∞([0,T]×Ω), T)independent of hsuch that

B_{h L}∞(0,T;L²(Ω))≤C B_h(0) _L2(Ω),

whereB_h(0)∈V_h^morB_h(0)∈W_h^mwithm= 1. Let[B_h]_e be the jump ofB_hacross an edgeeof Eh, the set of all the edges of the partition Th then, there exists a constantC=C( ∇u _L∞([0,T]×Ω), T, B _L∞(0,T;H^m+1(^Ê2))) independent of hsuch that

B−B_{h L}∞(0,T;L²(Ω))+ _T

0

e∈Eh

e|A_e|[B_h]_e·[B_h]_edΓdt≤Ch^m+η whereη =¹₂ if B_h∈V_h^m andη=−¹₂ if B_h∈W_h^m withm= 1.

Proof. The stability result is proved in the Appendix 5. We will show simultaneously the convergence of the scheme and derivate some error estimates. First we begin to derivate estimates which are valid as well for B_h∈V_h^m thanB_h∈W_h¹. Then we continue the proof by distinguishing the case whereB_h∈V_h^mand the case whereB_h∈W_h¹. First by noting that

L_h(B_h,ϕ_h) = 0 and L_h(B,ϕ_h) = 0 ∀ϕ_h∈V_h^m(Ω) or ∀ϕ_h∈W_h¹(Ω) then

L_h(e_h,ϕ_h) = 0, ∀ϕ_h∈V_h^m(Ω) or ∀ϕ_h∈W_h¹(Ω) (19) wheree_h=B−B_h. By noting thatπ_hB_h=B_handπ_he_h−e_h=π_h(B−B_h)−(B−B_h),then using (19) we obtain

Lh(π_he_h, π_he_h) =Lh(π_he_h−e_h, π_he_h) =Lh(π_hB−B, π_he_h). (20) From the stability analysis done in Appendix 5 the left hand side of (20) is the expression

Lh(π_he_h, π_he_h) =1

2 πhe_h(T) ²_L2(Ω)−1

2 πhe_h(0) ²_L2(Ω)+1 2

_T

0

e∈Eh

e|Ae|[πhe_h]_e·[π_he_h]_edΓdt +

_T

0

Ω

Cπ_he_h·π_he_hdxdt+1 2

_T

0

Ω

∇ ·u π_he_h ²₂dxdt. (21) Now it remains to estimate the right hand side of (20) that is to say find an estimate for

Lh(π_hB−B,ϕ_h), ∀ϕ_h∈V_h^m(Ω) or ∀ϕ_h∈W_h¹(Ω).

Let us decomposeL_h(π_hB−B,ϕ_h) in several terms which will be estimated in the sequel.

L_h(π_hB−B,ϕ_h) = _T

0

Ω

∂_t(π_hB−B)·ϕ_hdxdt (22)

− 2 i=1

_T

0

Ω

Ai(π_hB−B)·∂_iϕ_hdxdt (23)

− _T

0

e∈Eh

eg(π_hB−B)_e·[ϕ_h]_edΓdt (24) +

_T

0

Ω

C(π_hB−B)·ϕ_hdxdt. (25)

Case where B_h∈V_h^m

From the definition ofπ_h for the term (22) we have _T

0

Ω

(π_h(∂_tB)−∂_tB)·ϕ_hdxdt= 0. (26)

Now we define a new projection operatorP_h by (P_hg)_|K =P_Kg= 1

|K|

Kgdx, ∀g∈L¹(Ω)∩W^1,∞(Ω), and P_h=

K∈Th

P_K^½_K.

Then by Taylor expansion it can be easily shown that there exists a constantC independent ofhsuch that

P_hg−g _L∞(Ω)≤Ch g _W1,∞(Ω). (27)

Then thei-th term of (23) can be recasted in

− _T

0

Ω

Ai(π_hB−B)·∂_iϕ_hdxdt= _T

0

Ω

(P_hAi− Ai)(π_hB−B)·∂_iϕ_hdxdt

− _T

0

Ω

(π_hB−B)·P_hAi∂_iϕ_hdxdt= _T

0

Ω

(P_hAi− Ai)(π_hB−B)·∂_iϕ_hdxdt (28) becauseAi is symmetric,P_hAi∂_iϕ_h=P_hu_i∂_iϕ_h∈V_h^m(Ω) and because of the definition ofπ_h. Now we give an estimate for the term (28). Thanks the approximation properties (7) and (27), the inverse properties (8), and the Cauchy-Schwarz inequality we obtain

_T

0

Ω

(P_hA_i− A_i)(π_hB−B)·∂_iϕ_hdxdt

≤ _T

0

K∈Th

K πhB−B ²₂dx

_1/2

K (PhAi− Ai)∂_iϕ_h ²₂dx _1/2

dt

≤C

T, Ai L^∞(0,T;W^1,∞(Ω)),|B|_L^∞_(0,T;H^m+1(Ω))

h^m+1 _T

0

ϕ_{h L}²(Ω)dt. (29)

For the term (25) we get

_T

0

Ω

C(π_hB−B)·ϕ_hdxdt ≤

_T

0

K∈Th

KC(π_hB−B)·ϕ_hdxdt

≤ C _L^∞_(0,T;L^∞(Ω))

_T

0

K∈Th

K πhB−B ²₂dx _1/2

ϕ_{h L}²_(K)dt

≤C

T, C _L^∞_(0,T;L^∞(Ω)),|B|_L^∞_(0,T;H^m+1(Ω))

h^m+1 _T

0

ϕ_{h L}²(Ω)dt. (30)

Now it remains to estimate the term (24). First have g(π_hB−B)_e = Ae,K(π_hB−B)−1

2|Ae,K|[(π_hB−B)]_e

= Ae,K− |Ae,K|

2 (π_hB−B)_Ke+Ae,K+|Ae,K|

2 (π_hB−B)_K

≤ |Ae,K|(|π_hB−B|Ke+|π_hB−B|K).

Since|Ae,K|is symmetric, using a Young inequality we obtain _T

0

e∈Eh

eg(π_hB−B)_e·[ϕ_h]_edΓdt≤ _T

0

e∈Eh

e|Ae,K|^−1/2g(π_hB−B)_e· |Ae,K|^1/2[ϕ_h]_edΓdt

≤ _T

0

e∈Eh

e

|Ae,K|^−1/2g(π_hB−B)_e²

2dxdt+1 4

_T

0

e∈Eh

e|Ae|[ϕ_h]_e·[ϕ_h]_edΓdt. (31) Now we want to estimate the first term of (31). Using (9) and the trace result γ₀v _L2(∂K)≤C v _H1/2(K)we get

_T

0

e∈Eh

e

|Ae,K|^−1/2g(π_hB−B)_e²

2dxdt≤2 u _L∞(0,T;L^∞(Ω))

_T

0

K∈Th

πhB−B ²_L2(∂K)dt

≤2 u _L∞(0,T;L^∞(Ω))

_T

0

K∈Th

π_hB−B ²_H1/2(K)dt≤C

T, u _L∞(0,T;L^∞(Ω)),|B|L^∞(0,T;H^m+1(Ω))

h^2m+1. (32) Finally, from (20)–(26), (29)–(32) we get

π_he_h(T) ²_L2(Ω)− π_he_h(0) ²_L2(Ω)+1 2

_T

0

e∈Eh

e|Ae|[π_he_h]_e·[π_he_h]_edΓdt≤C₁h^2m+1 +C₂h^m+1

_T

0

π_he_h(t) _L²_(Ω)+ _T

0

2 C _L^∞_(Ω)+ ∇ ·u _L∞(Ω)

π_he_h(t) ²_L2(Ω)dt. (33)

Using a Young inequality, followed by a Gronwall inequality, from (33) we get, ∀t≤T, π_he_h(t) _L²_(Ω)≤C

π_he_h(0) _L²_(Ω)+h^m+1/2

e(CL∞([0,T]×Ω)+¹₂∇·uL∞([0,T]×Ω)+C2/2)t≤Ch^m+1/2. Finally we get,∀t≤T,

e_h(t) _L²_(Ω) ≤ πhe_h(t) _L²_(Ω)+ πhe_h(t)−e_h(t) _L²_(Ω)

≤ π_he_h(t) _L²_(Ω)+ π_hB(t)−B(t) _L²_(Ω)≤Ch^m+1/2 which finishes the proof whenB_h∈V_h^m.

Case where B_h∈W_h¹

We have to give new estimates for the terms (22)–(25). For the term (22), using Cauchy-Schwarz inequality and the approximation property (10) we get

_T

0

Ω

(Π_h(∂_tB)−∂_tB)·ϕ_hdxdt≤C

T,|∂_tB|_L^∞_(0,T;H¹(Ω))

h _T

0

ϕ_{h L}²(Ω)dt. (34)

Now let us estimate the term (24). The inequality (31) is still valid. It remains to give a new estimate for the first term of the right hand side of the inequality (31). Using the approximation properties (10)–(12), and the