Spectral element discretization of the heat equation with variable diffusion coefficient

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Spectral element discretization of the heat equation with variable diffusion coefficient

Y Daikh, W Chikouche

To cite this version:

Y Daikh, W Chikouche. Spectral element discretization of the heat equation with variable diffusion coefficient. 2015. �hal-01143558�

Spectral element discretization of the heat equation with variable diffusion coefficient

Y. Daikh, W. Chikouche Université de Jijel

Laboratoire de Mathématiques Pures et Appliquées B.P. 98, Ouled Aissa

18000 Jijel, Algeria

yasmina-daikh@univ-jijel.dz, wided-chikouche@univ-jijel.dz.

Abstract

We are interested in the discretization of the heat equation with a diffusion coefficient depending on the space and time variables. The discretization relies on a spectral element method with respect to the space variables and Euler’s implicit scheme with respect to the time variable. A detailed numerical analysis leads to optimal a priori error estimates.

Keywords: Heat equation, diffusion coefficient, spectral element methods, a priori esti- mates.

2010 Mathematics Subject Classification: 35K05, 65N35, 35B45.

1 Introduction

An impressive amount of work has been done concerning a priori and a posteriori analysis of parabolic type problems for finite element methods, see [6] and [1] for instance. An exten- sion in spectral element method of some results obtained by Bergam et al. in [1] has been performed recently by N. Chorfi et al. in [4]. They were interested in the a posteriori analysis of the spectral element discretization of the one-dimensional heat equation with constant diffu- sion coefficient. The spectral element method consists on approaching the solution of a partial differential equation by polynomial functions of high degree on each element of a decomposition.

In this paper, we are interested in the discretization of the heat equation with a diffusion coefficient depending on the space and time variables by an implicit Euler’s scheme with respect to the time variable and spectral element method with respect to the space variables in a two- or three-dimensional bounded domain. For the space discretization, we consider a partition of the domain into rectangles in dimension 2 or rectangular parallelepipeds in dimension 3 which is conforming and without overlap. The discrete spaces are constructed from tensorized spaces of polynomials of the same high degree on each subdomain. The full discrete problem is then obtained by Galerkin method with numerical integration.

An outline of the paper is as follows:

• In Section 2, we present the linear heat equation and we study the continuous problem ant its stability.

• In Section 3, we describe its time semi-discretization and the corresponding stability property.

• Section 4 is devoted to the description of the space discretization of the problem by using spectral element method.

The well-posedness of the corresponding problem in each section is proved.

• Optimal error estimates are proved in Section 5.

2 Position of the problem

LetΩ be a connected and bounded open set in R^d (d = 1,2,or 3)with a Lipschitz-continuous boundary. Also let T be a fixed positif integer. We consider the heat equation

(1)





∂_tu−div(λ∇u) = f inΩ×]0, T[

u = 0 on∂Ω×]0, T[

u|_t=0 = u₀ in Ω,

whereλ is a given continuous function on Ω×[0, T] satisfying for some positive constants λ_min and λ_max,

(2) ∀x∈Ω,∀t∈[0, T], λ_min ≤λ(x, t)≤λ_max.

The data are the distribution f and the function u₀; the unknown is the function u.

As usual, we denote by L^p(Ω),1 ≤ p ≤ ∞, the Lebesgue spaces and by H^s(Ω), s > 0, the standard Sobolev spaces. The usual norm and seminorm of H^s(Ω) are denoted by k · k_s,Ω and

| · |_s,Ω respectively. The space H₀¹(Ω) stands for the closure in H¹(Ω) of the space of infinitely differentiable functions with a compact support in Ω, and H⁻¹(Ω) stands for its dual space.

For simplicity, we denote by (·,·) the scalar product on L²(Ω) and by k · k_0,Ω the associated norm. By extension, the duality pairing between H⁻¹(Ω) and H₀¹(Ω), is also denoted by (·,·).

We define C⁰(0, T;L²(Ω)), as the space of continuous functions, with values in L²(Ω), and also L²(0, T;H₀¹(Ω)),respectivelyL²(0, T;H⁻¹(Ω)),as the space of square-integrable functions with values in H₀¹(Ω), respectively in H⁻¹(Ω).

The problem (1) admits the equivalent variational formulation:

Find u inC⁰(0, T;L²(Ω))∩L²(0, T;H₀¹(Ω)) satisfying

(3) u|_t=0 =u₀ inΩ,

and such that, for a.e. t in]0, T[,

(4) ∀v ∈H₀¹(Ω), (∂_tu(t), v) + (λ(t)∇u(t),∇v) = (f(t), v).

It is well known [5, Ch. 3, § 4] that, for any f in L²(0, T;H⁻¹(Ω)) and u₀ in L²(Ω), problem (3)-(4) admits a unique solution. Moreover, let us introduce the norm

(5) [[v]](t) =

µ

kv(t)k²_0,Ω+ Z _t

0

kλ¹²(s)∇v(s)k²_0,Ω

¶¹₂ .

By taking v equal to u(t) in (4) and integrating on the interval ]0, t[, we easily derive the following estimate [1]: for all t∈[0, T]

(6) [[u]](t)≤

µ

ku0k²_0,Ω+ 1

λ_minkfk²_L²_(0,t;H⁻¹_(Ω))

¶¹

2

.

3 The time semi-discrete problem

In order to describe the time discretization of equation (1), we introduce a partition of the interval [0, T] into subintervals[t_k−1, t_k], 1≤k ≤ K, such that 0 =t₀ < t₁ < ... < t_K =T.We denote by τk := tk−tk−1, by τ the K− tuple (τ1, ..., τK) and by |τ| the maximum of the τk, 1≤k≤K. We also define the regularity parameter

στ = max

2≤k≤K

τ_k τ_k−1.

With each family (v^k)0≤k≤K, we agree to associate the function vτ on [0, T] which is affine on each interval[t_k−1, t_k],1≤k ≤K, and equal tov^k att_k,0≤k ≤K.Equivalently, this function can be written, for 1≤k ≤K, as

(7) ∀t∈[t_k−1, t_k], v_τ(t) = v^k−tk−t

τ_k (v^k−v^k−1).

For simplicity, we introduce the notation λ^k =λ(tk) and f^k =f(tk), which obviously requires the continuity of f with respect to t. The semi-discrete problem issued from Euler’s implicit scheme is now written as

(8)





u^k−u^k−1

τk −div(λ^k∇u^k) = f^k inΩ, 1≤k ≤K,

u^k = 0 on∂Ω, 1≤k≤K,

u⁰ = u0 in Ω.

Equivalently, it admits the variational formulation:

Find (u^k)_0≤k≤K in L²(Ω)×H₀¹(Ω)^K satisfying

(9) u⁰ =u0 in Ω,

and such that, for1≤k ≤K,

(10) ∀v ∈H₀¹(Ω), a^k(u^k, v) =L^k(v), where the bilinear forms a^k,1≤k ≤K, are defined by

a^k(u, v) = (u, v) +τ_k(λ^k∇u,∇v),

and the linear forms L^k,1≤k≤K, are defined as

L^k(v) = (u^k−1, v) +τ_k(f^k, v).

The existence and uniqueness of a solution (u^k)_0≤k≤K for any data f in C⁰(0, T;H⁻¹(Ω)) and u0 inL²(Ω) is now a simple consequence of the Lax-Milgram lemma.

Moreover, by using the notation λ^k_min = inf

x∈Ωλ(x, t_k) and taking v =u^k in (10), we easily derive the following estimate

(11) ku^kk²_0,Ω+τk

°°(λ^k)¹²∇u^k°

°²

0,Ω ≤ ku^k−1k²_0,Ω+ τk

λ^k_minkf^kk²_−1,Ω. We now define the norm on whole sequences v<sup>`</sup>, 0≤`≤k by

(12) [[(v^`)]]_k=

Ã

kv^kk²_0,Ω+ Xk

`=1

τ_`°

°(λ^{`</sup>)<sup>12</sup>∇v<sup>`}°

°²

0,Ω

!¹₂ .

Indeed, by summing up estimate (11) on k,we derive the semi-discrete analogue of (6):

(13) [[(u^`)]]_k≤

Ã

ku₀k²_0,Ω+ Xk

`=1

τ_`

λ^{`</sup><sub>min</sub>kf<sup>`}k²_−1,Ω

!¹₂ .

The norm [[(u^`)]]_k involved in this estimate is not equal to the norm [[u_τ]](t_k) (see (7) for the definition of the function u_τ). However, when u₀ is supposed to be in H¹(Ω), they are equivalent, as proven in the next lemma [1].

Lemma 3.1 Assume that the function λ is continuously differentiable in time, with maximum value of∂_tλ onΩ×[0, T]denoted byµ_max.There exists a positive real numberα₀,equal to _2µ^λmin_max, such that the following equivalence property holds for |τ| ≤α0 and for any family (v<sup>`</sup>)0≤`≤K in H¹(Ω)^K+1

(14) 1

8[[(v^`)]]²_k ≤[[v_τ]]²(t_k)≤ 3 4

¡1 + 3 2σ_τ¢

[[(v^`)]]²_k+3 4τ₁°

°(λ¹)¹²∇v⁰°

°²

0,Ω. Proof:

Owing to the definitions (5) and (12) of the norms, we have to compare the quantities X_`=

Z _t`

t`−1

°°λ¹²(s)∇v_τ(s)°°²

0,Ωds and Y_{`</sub> =τ<sub>`}°°(λ^{`</sup>)<sup>12</sup>∇v<sup>`}°°²

0,Ω.

Thanks to the definition of µmax, we have the standard estimate

∀s∈[t`−1, t`], kλ(s)−λ<sup>`</sup>k∞,Ω ≤τ` µmax, so that, when |τ| ≤α0,

(15) ∀s∈[t_{`−1</sub>, t<sub>`}], 1

2 ≤

°°

°λ(s) λ^`

°°

°∞,Ω ≤ 3 2.

It can also be noted that, thanks to the definition of v_τ, and for a.e. xin Ω,

(16)

Z _t`

t`−1

|∇v_τ(x, s)|²ds = τ_` 3

¡|∇v^{`</sup>(x)|<sup>2</sup>+|∇v<sup>`−1}(x)|²+∇v^{`</sup>(x)· ∇v<sup>`−1}(x)¢ .

By combining (15) and (16), we obtain X` ≥ τ<sub>`</sub>

6

³°°(λ^{`</sup>)<sup>12</sup>∇v<sup>`}°

°²+°

°(λ^{`</sup>)<sup>12</sup>∇v<sup>`−1}°

°²+ ((λ^{`</sup>)<sup>12</sup>∇v<sup>`},(λ^{`</sup>)<sup>12</sup>∇v<sup>`−1})

´ .

So using the inequality ab≥ −¹₄a²−b² yields X<sub>`</sub> ≥ τ`

8

°°(λ^{`</sup>)<sup>12</sup>∇v<sup>`}°°²

0,Ω = 1 8Y_`, whence the first inequality in (14).

Similarly, by combining (15) and (16) and using the inequality ab≤ ¹₂a²+¹₂b²,we have X_{`</sub> ≤ 3τ<sub>`}

4

¡°°(λ^{`</sup>)<sup>12</sup>∇v<sup>`}°

°²

0,Ω+°

°(λ^{`</sup>)<sup>12</sup>∇v<sup>`−1}°

°²

0,Ω

¢.

When `= 1, we keep this inequality without modification. When ` >1, we use an analogue of (15) to obtain

X_{`</sub> ≤ 3τ<sub>`} 4

°°(λ^{`</sup>)<sup>12</sup>∇v<sup>`}°

°²

0,Ω+3τ_`−1 4

3 2σ_τ°

°(λ^{`−1</sup>)<sup>12</sup>∇v<sup>`−1}°

°²

0,Ω. By summing up the previous lines on `, we derive the second inequality in (14).

The a priori error estimate has been established in [1]: we observe that the family(e^k)_0≤k≤K, withe^k =u(t_k)−u^k,satisfiese⁰ = 0 and also, by integrating∂_tubetweent_k−1 andt_kand using equation (10) and equation (4) at time t =tk,

(17) ∀v ∈H₀¹(Ω), (e^k, v) +τk(λ^k∇e^k,∇v) = (e^k−1, v) +τk(²^k, v), where the consistency error ²^k is given by

(²^k, v) = µ1

τ_k Z _tk

tk−1

(∂tu)(s)ds−(∂tu)(tk), v

¶ .

So, applying (13) to this new problem, we derive the estimate: If the solution u is such that

∂_t²u belongs to L²(0, T;H⁻¹(Ω)), and for 1≤k ≤K, [[(u(t<sub>`</sub>)−u<sup>`</sup>)]]_k ≤ 2

3λ_min¹²

³

1≤`≤kmaxτ<sub>`</sub>´°

°∂_t²u°°²

L²(0,tk;H⁻¹(Ω)).

Thanks to Lemma 3.1, this also induces a similar bound for the norm [[u−u_τ]](t_k).

4 The time and space discrete problem

From now on, we assume that Ω admits a partition without overlap into a finite number of subdomains

(18) Ω =∪^R_r=1Ω_r and Ω_r∩Ω_r⁰ =∅, 1≤r < r⁰ ≤R,

which satisfy the further conditions:

(i) Each Ω_r, 1 ≤ r ≤ R, is a rectangle in dimension d = 2 or a rectangular parallelepiped in dimension d= 3.

(ii) The intersection of two subdomains Ω_r and Ω_r⁰,1 ≤ r < r⁰ ≤ R, if not empty, is either a vertex or a whole edge or a whole face of both Ω_r and Ω_r⁰.

We introduce the space P_N(Ω_r) of restrictions to Ω_r of polynomials with d variables and degree≤N with respect to each variable. Relying on this definition, we introduce the discrete spaces, for an integer N ≥2,

Y_N ={w∈L²(Ω) | w_r=w|_Ωr ∈P_N(Ω_r), r = 1, ..., R}, X⁰_N =Y_N ∩H₀¹(Ω).

Setting ξ₀ =−1 and ξ_N = 1, we introduce the N −1 nodes ξ_j,1≤j ≤N −1, and the N + 1 weights ρ_j,0 ≤ j ≤ N, of the Gauss-Lobatto quadrature formula on Λ := [−1,1]. We recall that the following equality holds

(19) ∀ϕ_N ∈P_2N−1(Λ),

Z ₁

−1

ϕ(ζ)dζ = XN

j=0

ϕ(ξ_j)ρ_j.

We also recall [2, form. (13.20)] the following property, which is useful in what follows

(20) ∀ϕ_N ∈P_N(Λ),kϕ_Nk²_L2(Λ) ≤ XN

j=0

ϕ²_N(ξ_j)ρ_j ≤3kϕ_Nk²_L2(Λ).

Denoting by F_r the affine mapping that sends Λ^d onto Ω_r, we introduce the local discrete products, defined on continuous functions u and v onΩ_r by

(21)

(u, v)^r_N =







meas(Ωr) 4

P_N

i=0

P_N

j=0u◦Fr(ξi, ξj)v◦Fr(ξi, ξj)ρiρj if d= 2,

meas(Ωr) 8

P_N

i=0

P_N

j=0

P_N

p=0u◦F_r(ξ_i, ξ_j, ξ_p)v ◦F_r(ξ_i, ξ_j, ξ_p)ρ_iρ_jρ_p if d= 3.

The global product is then defined on continuous functions uand v onΩ by

(22) ((u, v))_N =

XR

r=1

(u|_Ωr, v|_Ωr)^r_N.

We denote by i_N the interpolation operator at the nodes ξ_j,0 ≤ j ≤ N. We need the lo- cal Lagrange interpolation operators I_N^r : For each function ϕ continuous on Ω_r,I_N^rϕ be- longs to PN(Ωr) and is equal to ϕ at all nodes Fr(ξi, ξj),0 ≤ i, j ≤ N in dimension 2 and at F_r(ξ_i, ξ_j, ξ_p),0≤i, j, p≤ N in dimension 3. Finally, for each function ϕcontinuous on Ω,I_Nϕ denotes the function equal to I_N^rϕon each Ω_r,1≤r ≤R.

The fully discrete problem is now constructed from (9)-(10) by using the Galerkin method combined with numerical integration. It reads:

Find ¡ u^k_N¢

0≤k≤K inY_N ×(X⁰_N)^K, satisfying

(23) u⁰_N =I_Nu₀ inΩ,

and such that, for1≤k ≤K,

(24) ∀v_N ∈X⁰_N(Ω), a^k_N(u^k_N, v_N) =L^k_N(v_N), where the bilinear forms a^k_N(·,·),1≤k ≤K, are defined by

a^k_N(uN, vN) = ((uN, vN))N +τk((λ^k∇uN,∇vN))N, and the bilinear forms L^k_N are defined by

L^k_N(v_N) = ((u^k−1_N , v_N))_N +τ_k((f^k, v_N))_N.

It follows from (20) combined with Cauchy-Schwarz inequalities, that the forms a^k_N and L^k_N are continuous on X⁰_N ×X⁰_N and X⁰_N respectively, and a^k_N are coercive with norms bounded independently of N.

In all that follows, cstands for a generic constant which can vary from one line to the next one but is always independent ofN. The proof of the next proposition is now standard.

Proposition 4.1 For any data f continuous on Ω×[0, T]and a continuous u₀ on Ω,problem (23)-(24) has a unique solution (u^k_N)_0≤k≤K in Y_N×(X⁰_N)^K. Moreover this solution satisfies for a constant c independent of N

(25) [[(u^`_N)]]k ≤c

³

1 + λ_max λ_min

´¹

2

Ã

kINu0k²_0,Ω+

³

1 + λ_max λ_min

´X^k

`=1

τ<sub>`</sub> λ<sup>`</sup>_min

°°INf^`°

°²

0,Ω

!¹

2

.

Proof: Taking v_N equal to u^k_N in (24), we have thanks to Cauchy-Schwarz inequality ((u^k_N, u^k_N))N +τk((λ^k∇u^k_N,∇u^k_N))N ≤((u^k−1_N , u^k−1_N ))_N¹² ·((u^k_N, u^k_N))_N¹²

+τ_k((I_Nf^k,I_Nf^k))_N²¹ ·((u^k_N, u^k_N))_N¹², (26)

using (20), Poincaré-Friedrichs inequality and the inequality ab≤ _2ε¹a²+^ε₂b²,∀ε >0,we obtain 1

2

°°u^k_N°

°²

0,Ω+τ_k((λ^k∇u_N^k,∇u^k_N))_N ≤ 1 2

°°u^k−1_N °

°²

0,Ω+ 1 2ε

cτ_k λ^k_min

°°I_Nf^k°

°²

0,Ω+ ε 2τ_k°

°(λ^k)¹²∇u^k_N°

°²

0,Ω.

Summing up on k,and using (23), to obtain 1

2

°°u^k_N°

°²

0,Ω+ Xk

`=1

τ<sub>`</sub>((λ<sup>`</sup>∇u^{`</sup><sub>N</sub>,∇u<sup>`}_N))_N ≤ 1 2

°°I_Nu₀°

°²

0,Ω+ c 2ε

Xk

`=1

τ<sub>`</sub> λ<sup>`</sup>_min

°°I_Nf^`°

°²

0,Ω

+ε 2

Xk

`=1

τ`

°°(λ^{`</sup>)<sup>12</sup>∇u<sup>`}_N°

°²

0,Ω. (27)

On the other hand and thanks to (20), we have

°°(λ^{`</sup>)<sup>12</sup>∇u<sup>`}_N°°²

0,Ω ≤λ_max((∇u^{`</sup><sub>N</sub>,∇u<sup>`}_N))_N ≤ λmax

λ_min((λ^{`</sup>∇u<sup>`}_N,∇u<sup>`</sup><sub>N</sub>))<sub>N</sub>, 1≤` ≤k, so, (27) implies that

1 2

°°u^k_N°°²

0,Ω+ Xk

`=1

τ<sub>`</sub>°°(λ<sup>`</sup>)¹²∇u^`_N°°²

0,Ω ≤ 1

2(1 + λmax

λ_min)°°I_Nu₀°°²

0,Ω+ c

2ε(1 + λmax

λ_min) Xk

`=1

τ`

λ^`_min

°°I_Nf^`°°²

0,Ω

+ ε

2(1 + λ_max λ_min)

Xk

`=1

τ_`°

°(λ^{`</sup>)<sup>12</sup>∇u<sup>`}_N°

°²

0,Ω

Finally, we choose ε= _λ ^λmin

max+λmin to obtain (25).

5 Error estimate

We now wish to establish the error estimate between the solution(u^k)0≤k≤K of problem (9)-(10) and the solution (u^k_N)_0≤k≤K of problem (23)-(24).

LetΠ^1,0_N denote the orthogonal projection operator fromH₀¹(Ω) ontoX⁰_N for the scalar product associated with the norm | · |_1,Ω. For 0≤` ≤k,Π<sup>1,0</sup><sub>N−1</sub>u<sup>`</sup> will be denoted by v^`_N.

Proposition 5.1 Assume thatf andu₀ are continuous onΩ×[0, T]and Ωrespectively. Then the following estimate holds for the error between the solution (u^k)_0≤k≤K of problem (9)-(10) and the solution (u^k_N)0≤k≤K of problem (23)-(24)

[[(u^{`</sup>−u<sup>`}_N)]]_k≤c µ

[[(u^{`</sup>−v<sub>N</sub><sup>`} )]]_k+³

1 + λ_max λmin

´¹

2³

ku₀−v_N⁰k_0,Ω+ku₀ − I_Nu₀k_0,Ω

+

³

1 + λ_max λ_min

´¹

2 Xk

`=1

³ τ<sub>`</sub> λ<sup>`</sup>_min

´¹

2¡

E_{N,`</sub><sup>a,1</sup> +E<sub>N,`}^a,2 +E_{N,`</sub><sup>f</sup> ¢´¶ , where the quantities E<sub>N,`}^a,1, E_{N,`</sub><sup>a,2</sup> and E<sub>N,`}^f are defined by

E_N,`^a,1 = sup

vN∈X⁰_N

(u^k−u^k−1

τ_k , vN)−((v^k_N −v_N^k−1

τ_k , vN))N

|v_N|_1,Ω ,

E_N,`^a,2 = sup

vN∈X⁰_N

(λ^k∇u^k,∇v_N)−((λ^k∇v_N^k,∇v_N))_N

|vN|1,Ω

, E_N,`^f = sup

vN∈X⁰_N

(f, v_N)−((f, v_N))_N

|vN|1,Ω

.

Proof: We have

(28) [[(u^{`</sup>−u<sup>`}_N)]]_k≤[[(u^{`</sup>−v<sub>N</sub><sup>`} )]]_k+ [[(u^{`</sup><sub>N</sub> −v<sub>N</sub><sup>`} )]]_k,

so we have to estimate the term [[(u^{`</sup><sub>N</sub> −v<sub>N</sub><sup>`} )]]_k. It follows from (10) and (24) that

(29) ((u^k_N −v^k_N, vN))N +τk((λ^k∇(u^k_N −v_N^k),∇vN))N = ((u^k−1_N −v^k−1_N , vN))N +τkM_N^k(vN),

where M_N^k is the linear form onX⁰_N defined by M_N^k(vN) = (u^k−u^k−1

τ_k , vN)−((v_N^k −v^k−1_N

τ_k , vN))N + (λ^k∇u^k,∇vN)−((λ^k∇v_N^k,∇vN))N

+ ((f^k, v_N))_N −(f^k, v_N).

Due to the Riesz’s theorem, there exists a unique polynomial F_N^k inX⁰_N such that (30) ∀v_N ∈X⁰_N, M_N^k(v_N) = ((F_N^k, v_N))_N.

Thus the family (u^k_N −v_N^k)_0≤k≤K is a solution of the discrete problem (23)-(24) with I_Nu₀ − v_N⁰ instead of INu0 and F_N^k instead of f^k. Consequently, following the same arguments as in Proposition 4.1, we get

[[(u^{`</sup><sub>N</sub> −v<sup>`}_N)]]_k ≤c

³

1 + λ_max λ_min

´¹

2³

kI_Nu₀−v_N⁰ k²_0,Ω+

³

1 + λ_max λ_min

´X^k

`=1

τ<sub>`</sub> λ<sup>`</sup>_min(ϕ^`)²

´¹

2,

where ϕ^` = sup

vN∈X⁰_N

((F_N^`, v_N))_N

|vN|1,Ω

.We conclude the proof thanks to (30).

In order to estimate the term E<sub>N,`</sub><sup>a,1</sup>, we denote by w<sup>`</sup> the quantity ^{u`−u</sup><sub>τ</sub><sup>`−1}

` and we observe that v<sub>N</sub><sup>`</sup> −v_N^`−1

τ<sub>`</sub> = Π<sup>1,0</sup><sub>N−1</sub>w<sup>`</sup>, so as a consequence of the exactness property (19), the terms (Π^1,0_N−1w^`, v_N¢

0,Ω and ((Π^1,0_N−1w^`, v_N¢

)_N coincide and thus,

(31) E_N,`^a,1 ≤°

°w^{`</sup>−Π<sup>1,0</sup><sub>N−1</sub>w<sup>`}°

°0,Ω.

Now, in order to evaluate the term E_N,`^a,2, we introduce the orthogonal projection operator Π^r,1_N fromH¹(Ω_r)ontoP_N(Ω_r)for the scalar product associated with the normk · k_1,Ωr. IfN˜ stands for the integer part of ^N−1₂ ,so as a consequence of the exactness property (19),we have for any v_N ∈X⁰_N

(λ^{`</sup>∇u<sup>`},∇v_N)−((λ^{`</sup>∇v<sub>N</sub><sup>`} ,∇v_N))_N = (λ^{`</sup>(∇u<sup>`}− ∇v_N^` ),∇v_N) +

XR

r=1

³ Z

Ωr

(λ^{`</sup>∇v<sub>N</sub><sup>`} − I_N^r_˜λ^{`</sup>∇(Π<sup>r,1</sup><sub>N˜</sub> u<sup>`}))(x)· ∇v_N(x)dx +(I_N^r_˜λ^{`</sup>∇(Π<sup>r,1</sup><sub>N˜</sub> u<sup>`})−λ^{`</sup>∇v<sub>N</sub><sup>`} ,∇v_N)^r_N

´ . (32)

Due to Cauchy-Schwarz inequality and by using the notationλ^`_max = sup

x∈Ω

λ(x, t_`), the first term in the right hand side of (32) can be estimated as

(λ^{`</sup>(∇u<sup>`}− ∇v_N^{`</sup> ),∇v<sub>N</sub>)≤λ<sub>max</sub><sup>`} |u^{`</sup>−v<sub>N</sub><sup>`} |_1,Ω|v_N|_1,Ω. Similar arguments also lead to

XR

r=1

Z

Ωr

(λ^{`</sup>∇v<sub>N</sub><sup>`} − I_N^r_˜λ^{`</sup>∇(Π<sup>r,1</sup><sub>N˜</sub> u<sup>`}))(x)· ∇v_N(x)dx ≤

³ λ^`_max¡

|u^{`</sup>−v<sup>`}_N|_1,Ω+ XR

r=1

¯¯u^{`</sup>−Π<sup>r,1</sup><sub>N˜</sub> u<sup>`}¯

¯1,Ωr

¢

+ XR

r=1

°°λ^{`</sup>− I<sub>N</sub><sup>r</sup><sub>˜</sub>λ<sup>`}°°

∞,Ωr

°°Π^r,1_N˜ u^`°°

1,Ωr

´

|v_N|_1,Ω.

Now, thanks to Cauchy-Schwarz inequality and (20), the last term in the right hand side of (32) can be estimated as follows

XR

r=1

(I_N^r_˜λ^{`</sup>∇(Π<sup>r,1</sup><sub>N˜</sub> u<sup>`})−λ^{`</sup>∇v<sup>`}_N,∇v_N)^r_N ≤ c³ λ^`_max¡

|u^{`</sup>−v<sub>N</sub><sup>`} |_1,Ω+ XR

r=1

¯¯u^{`</sup>−Π<sup>r,1</sup><sub>N˜</sub> u<sup>`}¯

¯1,Ωr

¢

+ XR

r=1

°°λ^{`</sup>− I<sub>N</sub><sup>r</sup><sub>˜</sub>λ<sup>`}°

°∞,Ωr

°°Π^r,1_N˜ u^`°

°1,Ωr

´

|v_N|_1,Ω.

so that

E_N,`^a,2 ≤ c µ

λ^`_max

³

|u^{`</sup>−v<sup>`}_N|_1,Ω+ XR

r=1

¯¯u^{`</sup>−Π<sup>r,1</sup><sub>N˜</sub> u<sup>`}¯

¯1,Ωr

´

+ XR

r=1

°°λ^{`</sup>− I<sub>N</sub><sup>r</sup><sub>˜</sub>λ<sup>`}°

°∞,Ωrku^`k_1,Ωr

¶ . (33)

Finally, in order to estimate the term E_N,`^f , we introduce the orthogonal projection operator Π^r_N from L²(Ω_r) intoP_N(Ω_r). Indeed, using (19) leads to, for any v_N inX⁰_N,

Z

Ωr

f^{`</sup>(x)·v<sub>N</sub>(x)dx−(f<sup>`}, v_N)^r_N = Z

Ωr

(f^{`</sup>(x)−Π<sup>r</sup><sub>N−1</sub>f<sup>`})(x)·v_N(x)dx−(I_N^rf^{`</sup>−Π<sup>r</sup><sub>N−1</sub>f<sup>`}, v_N)^r_N, so that, owing to (20),

(34) E_N,`^f ≤c

XR

r=1

³°°f^{`</sup>−Π<sup>r</sup><sub>N−1</sub>f<sup>`}°

°0,Ωr +°

°f^{`</sup>− I<sub>N</sub><sup>r</sup>f<sup>`}°

°0,Ωr

´ .

Now, to make complete the evaluation of E_{N,`</sub><sup>a,1</sup>, E<sub>N,`}^a,2 and E_N,`^f , we need the following results.

First, we recall from [2, Thm. 7.1, Thm. 7.3 and Thm. 14.2] the approximation properties of the operators Π^r_N,Π^r,1_N and I_N^r,1≤r≤R: for any function ϕin H^s(Ω_r), s ≥0

°°ϕ−Π^r_Nϕ°

°0,Ωr ≤cN^−skϕk_s,Ωr,

for any function ϕin H^s(Ω_r), s≥1

°°ϕ−Π^r,1_N ϕ°

°1,Ωr ≤cN^1−skϕk_s,Ωr,

and for any function ϕin H^s(Ω_r), s > ^d₂

°°ϕ− I_N^rϕ°

°0,Ωr ≤cN^−skϕks,Ωr.

The following result is derived from [3, Lem. VI.2.5] thanks to an interpolation argument, for any real number s ≥ 1, and any function ϕ in H₀¹(Ω) such that each ϕ|_Ωr,1 ≤ r ≤ R, belongs toH^s(Ω_r)

(35) °°ϕ−Π^1,0_N ϕ°°

1,Ω ≤cN^1−s XR

r=1

kϕk_s,Ωr.