Convergence analysis for the boundary value problem

For the second term in (B.4.21) we notice that [[(E−Gh)×n]] = −[[G_h×n]] owing to the regularity of E. The by proceeding as above, we arrive at

({{κ∇×F_h}},[[(E−G_h×n]])_Σ∪Γ≤ch⁻¹²k[[G_h×n]]k_L2(Σ∪Γ)kκ∇×F_hk_L2(T_h)

≤ch⁻¹²k[[(E−G_h)×n]]k_L2(Σ∪Γ)kκ∇×F_hk_L2(T_h), where we used the inverse inequalities

hk∇×κ∇×F_hk_L2(T_h) ≤ckκ∇×F_hk_L2(T_h), h^skκ∇×F_hk_Hs(T_h) ≤ckκ∇×F_hk_L2(T_h). The desired result is obtained by gathering the above estimates.

The following result will be useful in order to prove a convergence result inL²(Ω). Proposition B.4.4 (Adjoint continuity). For any s ∈ 0,¹₂

, there is c > 0, uniform in h such that the following holds for every (E, p),(F, q) ∈ Z^s×H₀¹(Ω), F_h ∈ X_h ∩H_0,curl(Ω), qh∈Mh and(Gh, dh)∈Xh×M_h:

c a_h((E−G_h, p−d_h),(F−F_h, q−q_h))

kE−Gh, p−dhk_h ≤ kF−F_h, q−q_hk_h+h^α−1kF−F_hk_L2(Ω)

+h^skκ∇×(F−Fh)k_Hs(T_h)

+hk∇×κ∇×(F−Fh)k_L2(T_h)

(B.4.22)

+h^−αkq−q_hk_L2(Ω)+h^(12−α)kq−q_hk_L2(Σ). Proof. The proof proceeds similarly as in the proof of Proposition B.4.3. The only dierence here is that we have ({{∇×(E−Gh)}},[[(F−Fh)×n]])_Σ∪Γ = 0.owing to the assumption on F_h. This identity makes the analysis of the consistency term (B.4.21) tractable.

Remark B.4.3. Note that the coercivity and the continuity of ah have been established for anyα∈[0,1].

Theorem B.5.1. Let g ∈ L²(Ω) and denote (E, p) the solution of (B.2.10). Let τ <

min(τ_ε, τ_µ) where τ_ε and τ_µ are dened in Theorem B.2.1. Denote (E_h, p_h) the solution of (B.4.13). Then, for anyα ∈_`(1−τ)

`−τ ,1

, there exists c >0, uniform inh, such that (B.5.1) kE−Eh, p−phk_h 6ch^rkgk_L2(Ω),

with r= min 1−α, α−1 +τ 1−^α_r .

Proof. We rst recall that, owing to Theorem B.2.1, we haveE∈H^τ(Ω)∩H^τ_0,curl(Ω), together with the estimates

kEk_Hτ(Ω)+k∇×Ek_Hτ(Ω)+k∇×(κ∇×E)k_L2(Ω)+k∇pk_L2(Ω) ≤ckgk_L2(Ω). We establish (B.5.1) by using the triangular inequality

kE−E_h, p−p_hk_h 6kE− K_δE,0k_h+kK_δE− C_hK_δE, p− P_hpk_h +kC_hK_δE−E_h,P_hp−p_hk_h,

for some δ >0to be dened later, and by bounding from above the three terms separately.

Using the denition ofk · k_h together with the approximation properties ofK_δ, cf. (B.3.16)-(B.3.17)-(B.3.18), we infer:

kE− K_δE,0k_h ≤cδ^τk∇×Ek_Hτ(Ω)+ch^αδ^τ−1kEk_Hτ(Ω)+h^α−12kK_δEk_L2(Σ).

Note that the estimate (B.3.17) is critical to obtain a bound that depends only onk∇×Ek_Hτ(Ω)

instead of kEk_H1+τ(Ω). To estimate the last term in the above inequality, we apply (B.7.6) withΘ = _2(1−τ)^1−2τ ,

h^α−12kK_δEk_L2(Σ)≤ch^α−12kK_δEk^1−Θ_Hτ(Ω)kK_δEk^Θ_H1(Ω)

≤ch^α−12δ^Θ(τ−1)kEk_Hτ(Ω)≤ch^α−12δ^τ−12kEk_Hτ(Ω). Finally, we arrive at

(B.5.2) kE− K_δE,0k_h≤c

δ^τ+h^αδ^τ−1+h^α−12δ^τ−12

kgk_L2(Ω).

Let us now turn our attention tokK_δE− C_hK_δE, p− P_hpk_h. Owing to the denition ofC_h and the regularity of K_δE, we have C_hK_δE ∈H_0,curl(Ω), so that we only have four terms to bound (the jumps of C_hK_δE across the mesh interfaces and the tangent trace on Γ are zero, cf. Remark B.4.1). Using the properties of K_δ and C_h together with (B.7.5) we deduce that:

kκ¹²∇×(K_δE− C_hK_δE)k_L2(Ω)≤ch<sup>`−1</sup>kK<sub>δ</sub>Ek<sub>H</sub>`(Ω)≤ch^{`−1</sup>δ<sup>τ−`}kEk_Hτ(Ω), h^αk∇·(ε(K_δE− C_hK_δE))k_L2(Ω)≤ch<sup>α+`−1</sup>kK<sub>δ</sub>Ek<sub>H</sub>`(Ω) ≤ch^{α+`−1</sup>δ<sup>τ−`}kEk_Hτ(Ω),

h^1−αkε¹²∇(p− P_hp)k_L2(Ω)≤ch^1−αkpk_H1 0(Ω),

h^α−12k[[ε(K_δE− C_hK_δE)·n]]k_L2(Σ)≤ch^α−12kK_δE− C_hK_δEk_L2(Σ)

≤ch^α−12kK_δE− C_hK_δEk¹⁻

1 2α

L²(Ω)kK_δE− C_hK_δEk

1 2α

H^α(Ω)

≤ch^α−12h^{`</sup>(<sup>1−</sup><sub>2α</sub><sup>1</sup>)h<sup>(`−α)2α1} kK_δEk_H`(Ω)

≤ch^{α+`−1</sup>δ<sup>τ−`}kEk_Hτ(Ω).

These estimates lead to

(B.5.3) kK_δE− C_hK_δE, p− P_hpk_h ≤c

h^{`−1</sup>δ<sup>τ−`}+h^1−α

kgk_L2(Ω).

The last term, kC_hK_δE−Eh,P_hp−phk_h, is a little more subtle to handle. We start from the coercivity ofa_h (B.4.19) and use both the Galerkin orthogonality (B.4.15) and the continuity ofa_h (B.4.20) withs= 1−α to get the following estimate:

kC_hK_δE−E_h,P_hp−p_hk_h

≤cah((C_hK_δE−Eh,P_hp−ph),(C_hK_δE−Eh,P_hp−ph)) kC_hK_δE−E_h,P_hp−p_hk_h

≤ca_h((C_hK_δE−E,P_hp−p),(C_hK_δE−E_h,P_hp−p_h)) kC_hK_δE−E_h,P_hp−p_hk_h

≤c kC_hK_δE−E,P_hp−pk_h+h^α−1kE− C_hK_δEk_L2(Ω)

+h^1−αkκ∇×(E− C_hK_δE)k_H1−α(Ω)+h^−αkp− P_hpk_L2(Ω)

+hk∇×κ∇×(E− C_hK_δE)k_L2(T_h)+h^12−αkp− P_hpk_L2(Σ)

.

We now handle each term in the right hand side separately. Using the triangle inequality kC_hK_δE−E,P_hp−pk_h ≤ kC_hK_δE− K_δE,P_hp−pk_h +kK_δE−E,0k_h and the estimates (B.5.2)-(B.5.3), we obtain

kC_hK_δE−E,P_hp−pk_h ≤c

δ^τ+h^αδ^τ−1+h^α−12δ^τ−12 +h^{`−1</sup>δ<sup>τ−`}+h^1−α

kgk_L2(Ω). Similarly, we obtain

h^α−1kE− C_hK_δEk_L2(Ω)≤c

h^α−1δ^τ+h^{α+`−1</sup>δ<sup>τ−`}

kgk_L2(Ω), h^1−αkκ∇×(E− C_hK_δEk_H1−α(Ω)≤c

h^1−αδ^τ+α−1+h^{`−1</sup>δ<sup>τ−`}

kgk_L2(Ω).

Note that the previous computation is valid since 1 −α ≤ τ owing to the assumption α ∈_`(1−τ)

`−τ ,1

. For the last term involving E we use the commuting property ¯δ∇×K_δE = K_δ∇×E, see (B.3.19) as follows:

hk∇×κ∇×(E− C_hK_δE)k_L2(T_h) ≤hk∇×κ∇×Ek_L2(T_h)+hk∇×κ∇×K_δEk_L2(T_h)

+hk∇×κ∇×(K_δE− C_hK_δE)k_L2(T_h)

≤c

hkgk_L2(Ω)+hk∇×K_δEk_H1(Ω)+h<sup>`−1</sup>kK<sub>δ</sub>Ek<sub>H</sub>`(Ω)

≤c

hkgk_L2(Ω)+hkK_δ∇×Ek_H1(Ω)+h^{`−1</sup>δ<sup>τ−`}kEk_Hτ(Ω)

≤c

h+hδ^τ−1+h^{`−1</sup>δ<sup>τ−`}

kgk_L2(Ω).

For the remaining terms involvingp, we use (B.7.5) together with the approximation properties ofC_h:

h^−αkp− P_hpk_L2(Ω) ≤ch^1−αkpk_H1

0(Ω)≤ch^1−αkgk_L2(Ω), h^12−αkp− P_hpk_L2(Σ) ≤h^12−αkp− P_hpk¹⁻

1 2α

L²(Ω)kp− P_hpk

1 2α

H^α(Ω)

≤h^12−αh^1−2α1 h^(1−α)2α1 kpk_H1

0(Ω)≤ch^1−αkgk_L2(Ω).

Gathering all the above estimates together with (B.5.2) and (B.5.3), we nally obtain (B.5.4) kE−Eh, p−phk_h ≤c δ^τ +h^1−α+h+hδ^τ−1+h^{`−1</sup>δ<sup>τ−`}+h^α−1δ^τ

+h^1−αδ^τ+α−1+h^αδ^τ−1+h^α−12δ^τ−12

kgk_L2(Ω).

We want to useδ = h^β for some β ∈ (0,1), i.e., δh⁻¹ → +∞ as h→ 0. Once the negligible terms are removed in (B.5.4), we derive the following estimate:

kE−E_h, p−p_hk_h ≤c h^α−1δ^τ +h^1−α+h^{`−1</sup>δ<sup>τ−`}

kgk_L2(Ω). Usingδ =h^{1−α`</sup> implies thath<sup>α−1</sup>δ<sup>τ</sup> =h<sup>`−1}δ^τ−` and we arrive at

kE−E_h, p−p_hk_h ≤c(h^α−1+τ(^1−α_{`</sub>) +h<sup>1−α</sup>)kgk<sub>L</sub>2(Ω), which leads to (B.5.1) withr:= min 1−α, α−1 +τ 1−<sup>α</sup><sub>`}

. Note that the assumed lower bound on α ensures that we have a convergence result as h→0.

Remark B.5.1. Note that the best choice forα is such that 1−α =α−1 +τ 1−^α_{`</sub> . This choice gives the following convergence rate <sup>τ</sup><sub>2</sub>(1−<sup>1</sup><sub>`})< r=τ<sub>2`−τ</sub><sup>`−1</sup> < ^τ₂.

We now derive a convergence estimate assuming that the solution of (B.2.10) is smooth.

Theorem B.5.2. Let g ∈ L²(Ω) and denote (E, p) the solution of (B.2.10). Assume that E ∈H^k+1(Ω) and p∈ H^k+α(Ω) for some 0 < k≤`−1 and α ∈[0,1]. Denote (Eh, ph) the solution of (B.4.13). Then there exists c >0, uniform in h, such that

(B.5.5) kE−E_h, p−p_hk_h ≤c h^k

kgk_L2(Ω)+kEk_Hk+1(Ω)+kpk_Hk+α(Ω)

.

Proof. The proof is similar to that of Theorem B.5.1. We start from the triangular inequality kE−E_h, p−p_hk_h ≤ kE− C_hE, p− P_hpk_h+kC_hE−E_h,P_hp−p_hk_h.

We bound the two terms in the right hand side separately. For the rst one, we use the approximation properties of C_h to get:

kE− C_hE, p− P_hpk_h ≤c

h^kkEk_Hk+1(Ω)+h^k+αkEk_Hk+1(Ω)+h⁻¹²kE− C_hEk_L2(Σ∪Γ)

+h^1−αh^k+α−1kpk_Hk+α(Ω)+h^α−12kE− C_hEk_L2(Σ)

. Using (B.7.5) for anyσ∈ ¹₂,1

, we have kE− C_hEk_L2(Σ)≤ckE− C_hEk¹⁻

1 2σ

L²(Ω)kE− C_hEk

1 2σ

H^σ(Ω)≤c h^k+12kEk_Hk+1(Ω). As a result, we obtain

(B.5.6) kE− C_hE, p− P_hpk_h ≤ch^k

kEk_Hk+1(Ω)+kpk_Hk+α(Ω)

.

Now we turn our attention tokC_hE−E_h,P_hp−p_hk_h. We use the coercivity ofah, the Galerkin orthogonality and the continuity of a_h (for anyσ∈ 0,¹₂

) to get

kC_hE−E_h,P_hp−p_hk_h≤c kE− C_hE, p− P_hpk_h+h^α−1kE− C_hEk_L2(Ω)

+h^σkκ∇×(E− C_hE)k_Hσ(T_h)

+hk∇×κ∇×(E− C_hE)k_L2(Th)

+h^−αkp− P_hpk_L2(Ω)+h^12−αkp− P_hpk_L2(Σ)

. Using the approximation properties ofC_h together with (B.5.6), we infer

kE− C_hE, p− P_hpk_h ≤ch^k

kEk_Hk+1(Ω)+kpk_Hk+α(Ω)

, h^α−1kE− C_hEk_L2(Ω) ≤ch^k+αkEk_Hk+1(Ω),

h^σkκ∇×(E− C_hE)k_Hσ(T_h) ≤ch^kkEk_Hk+1(Ω), h^−αkp− P_hpk_L2(Ω) ≤ch^kkpk_Hk+α(Ω). For the last term involving p, we use (B.7.5) for some σ∈ ¹₂,1

: h^12−αkp− P_hpk_L2(Σ)≤ch^12−αkp− P_hpk¹⁻

1 2σ

L²(Ω)kp− P_hpk

1 2σ

H^σ(Ω)

≤ch^12−αh^k+α−12kpk_Hk+α(Ω) =ch^kkpk_Hk+α(Ω).

For the last term involving E, we distinguish two cases depending whether k < 1 or k ≥1.

If k < 1, we use an inverse inequality together with the approximation properties of C_h to deduce that

hk∇×κ∇×(E− C_hE)k_L2(Th)≤hk∇×κ∇×Ek_L2(Ω)+chkC_hEk_H2(Th)

≤hkgk_L2(Ω)+h^kkEk_Hk+1(Ω). If k≥1, we use the approximation properties ofC_h to get

hk∇×κ∇×(E− C_hE)k_L2(T_h)≤chkE− C_hEk_H2(T_h)≤ch^kkEk_Hk+1(Ω). In both cases, we have:

hk∇×κ∇×(E− C_hE)k_L2(Th)≤ch^k

kEk_Hk+1(Ω)+kgk_L2(Ω)

. Gathering all the above estimates and using (B.5.6) gives the desired result (B.5.5).

Remark B.5.2. Note that the error estimate (B.5.5) is optimal since it implies thatk∇×(E− E_h)k_L2(Ω) ≤ c h^k, which is the best that can be expected from the piece-wise polynomial approximation of degreek. Note also that there is no lower bound on α to get convergence when the solution of (B.2.10) is smooth, i.e., anyα in the range[0,1] is acceptable.

B.5.2 Convergence in the L²-norm.

Before proving that the discrete solution converges to the exact solution in the L²-norm, we prove a global version of Lemma B.7.4 that will be useful in the proof of Theorem B.5.3.

Lemma B.5.1. Lets∈ 0,¹₂

. Then there existsc >0, uniform inh, such that the following holds, for any ψ ∈Hcurl(Ω)∩H^s(Ω) and any Fh ∈Xh:

(B.5.7) |(ψ,[[Fh×n]])_Σ∪Γ|

≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) h^skψk_Hs(Ω)+hk∇×ψk_L2(Ω)

. Proof. Let us considerψ ∈Hcurl(Ω)∩H^s(Ω)and Fh∈Xh. Notice that the left hand side is well dened owing to Lemma B.7.4. We start from

ψ,[[F_h×n]]

Σ∪Γ

≤ |(ψ− K_δψ,[[F_h×n]])_Σ∪Γ|

| {z }

:=I1

+|(K_δψ,[[F_h×n]])_Σ∪Γ|

| {z }

:=I2

,

for someδ to be dened later. We handle the two termsI₁,I₂ separately. For the rst one, we apply Lemma B.7.4 withv= [[Fh×n]],φ=ψ− K_δψ and σ=s, and we sum over all the facesF ∈Σ∪Γ. This leads to

I1 ≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) h^skψ− K_δψk_Hs(T_h)

+hk∇×(ψ− K_δψ)k_L2(T_h)+kψ− K_δψk_L2(Ω)

≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) h^skψ− K_δψk_Hs(T_h)

+hk∇×ψk_L2(T_h)+hk∇×K_δψk_L2(T_h)+kψ− K_δψk_L2(Ω)

. Using the approximation properties ofK_δ (B.3.16) and (B.3.18), we arrive at

I1 ≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) h^skψk_Hs(Ω)

+hk∇×ψk_L2(Ω)+δ^skψk_Hs(Ω)+hkK_δψk_H1(Ω)

≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) (h^s+δ^s+hδ^s−1)kψk_Hs(Ω)+hk∇×ψk_L2(Ω)

. We handleI2 by using the Cauchy-Schwarz inequality on every∂Ωi,i= 1,· · · , N.

I2 ≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) N

X

i=1

h¹²kK_δψk_L2(∂Ωi). We use (B.7.6) on everyΩi withΘ := _2(1−s)^1−2s , this leads to

I2 ≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ) N

X

i=1

h¹²kK_δψk^1−Θ_Hs(Ωi)kK_δψk^Θ_H1(Ωi)

≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ)h¹²kK_δψk^1−Θ_Hs(Ω)kK_δψk^Θ_H1(Ω),

where the constant c depends on N, which we recall is a xed number. Using again the approximation properties ofK_δ we infer that

I₂ ≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ)h¹²δ^(s−1)Θkψk_Hs(Ω)

≤c h⁻¹²k[[F_h×n]]k_L2(Σ∪Γ)h¹²δ^s−12kψk_Hs(Ω).

Then (B.5.7) is obtained by gathering the above estimates and settingδ=h.

Remark B.5.3. The proof of Lemma B.5.7 can done by using the decomposition ψ = ψ− C_hψ+C_hψ instead of ψ =ψ− K_δψ+K_δψ.

Theorem B.5.3. Let g ∈ L²(Ω) and denote (E, p) the solution of (B.2.10). Let τ <

min(τ_ε, τ_µ) where τ_ε and τ_µ are dened in Theorem B.2.1. Denote (E_h, p_h) the solution of (B.4.13). Then, for any α ∈_`(1−τ)

`−τ ,1

, there exists c >0, uniform inh, such that (B.5.8) kE−Ehk_L2(Ω)≤c h^2rkgk_L2(Ω),

withr := min 1−α, α−1 +τ 1−^α_`

. If in additionEandpare smooth, sayE∈H^k+1(Ω) andp∈H^k+α(Ω)for some 0< k < `−1, then the following holds:

(B.5.9) kE−E_hk_L2(Ω) ≤c h^k+r

kgk_L2(Ω)+kEk_Hk+1(Ω)+kpk_Hk+α(Ω)

.

Proof. We are going to use a duality argument à la Nitsche-Aubin. In the following we denotea¹_h the extension to

(Z^τ(Ω) +X_h)×H₀¹(Ω)2

of the bilinear form dened on[X_h×M_h]² in (B.4.12) withθ= 1. Then the following symmetry property holds:

a¹_h((F, q),(G, r)) =a¹_h((G,−r),(F,−q)). for all ((F, q),(G, r)) ∈

(Z^τ(Ω) +X_h)×H₀¹(Ω)2

. Let (w, q) ∈ H_0,curl(Ω)×H₀¹(Ω) be the solution of the following (adjoint) problem:

∇×κ∇×w−ε∇q=ε(E−E_h), ∇·(εw) = 0.

Recall that Theorem B.2.1 implies thatw∈Z^τ(Ω)∩H^τ(Ω)for anys < τ and that kwk_Hτ(Ω)+kκ∇×wk_Hτ(Ω)+k∇×κ∇×wk_L2(Ω)≤ckE−Ehk_L2(Ω). (B.5.10)

The denition of the pair(w, q)implies that(ε∇q,∇ϕ)_Ω=−(ε(E−E_h,∇ϕ)for allϕ∈H₀¹(Ω), and the following identities hold:

kε¹²(E−E_h)k²_L2(Ω) =a¹_h((w,−q),(E−E_h, p_h−p)) +c_αh^2(1−α)(ε∇q,∇(p_h−p))_Ω

=a¹_h((E−E_h, p−p_h),(w, q)) +cαh^2(1−α)(ε(E−E_h),∇(p−p_h))_Ω

=ah((E−Eh, p−ph),(w, q)) +cαh^2(1−α)(ε(E−Eh),∇(p−ph))_Ω + (1−θ) ({{κ∇×w}},[[−E_h×n]])_Σ∪Γ

We now use the Galerkin orthogonality and introduceC_hK_δw,P_hq, withδ=h^1−α`: (B.5.11) kε¹²(E−E_h)k²_L2(Ω) =a_h((E−E_h, p−p_h),(w− C_hK_δw, q− P_hq))

+cαh^2(1−α)(ε(E−Eh),∇(p−ph))_Ω−(1−θ) (κ∇×w,[[Eh×n]])_Σ∪Γ. Note that we replaced{{κ∇×w}}byκ∇×wsince the tangent component ofκ∇×wis contin-uous across the interfaces owing to∇×(κ∇×w)∈L²(Ω).

We now handle the three terms in the right hand side separately. For the rst one, we use Proposition B.4.4 withs= 1−α, F=wand Fh=C_hK_δw(note that Fh ∈Xh∩H0,curl(Ω)

sinceK_δw∈ C₀^∞(Ω)); we then infer that

a_h (E−E_h, p−p_h),(w− C_hK_δw, q− P_hq) ≤

ckE−Eh, p−phk_h kw− C_hK_δw, q− P_hqk_h

+h^α−1kw− C_hK_δwk_L2(Ω)+h^−αkq− P_hqk_L2(Ω)+h^12−αkq− P_hqk_L2(Σ)

+hk∇×κ∇×(w− C_hK_δw)k_L2(T_h)+h^1−αk∇×(w− C_hK_δw)k_H1−α(Ω)

. The right hand side has already been estimated in the proof of Theorem B.5.1. We then have (B.5.12)

a_h (E−E_h, p−p_h),(w− C_hK_δw,P_hq−q)

≤ckE−E_h, p−p_hk_hh^rkE−E_hk_L2(Ω).

The second term in (B.5.11) is estimated by using the Cauchy-Schwarz inequality, the deni-tion of the norm k · k_h and inequalityr≤1−α,

h^2(1−α)(ε(E−E_h),∇(p−p_h))_Ω

≤c h^1−αk∇(p−p_h)k_L2(Ω)h^1−αkE−E_hk_L2(Ω)

≤ckE−E_h, p−p_hk_hh^rkE−E_hk_L2(Ω). (B.5.13)

The last term in (B.5.11) is estimated by using Lemma B.5.1 with ψ:=κ∇×wand s:=τ:

(1−θ) κ∇×w,[[Eh×n]]

Σ∪Γ

≤ckE−E_hk_h h^τkκ∇×wk_Hτ(Ω)+hk∇×(κ∇×w)k_L2(Ω)

≤ckE−E_hk_hh^rkE−E_hk_L2(Ω), (B.5.14)

where we have used (B.5.10) and r ≤ ^τ₂ < τ. Upon inserting (B.5.12)-(B.5.13)-(B.5.14) in (B.5.11) we obtain

kε¹²(E−E_h)k²_L2(Ω) ≤ch^rkE−E_hk_L2(Ω)kE−E_h, p−p_hk_h. Owing to the uniform positivity of ε, this leads to:

kE−Ehk_L2(Ω)≤ch^rkE−Eh, p−phk_h.

Now we consider two cases. Assuming only minimal regularity, Theorem B.5.1 gives a bound on kE−E_h, p−p_hk_h that leads to (B.5.8). If E and p are smooth, then we can apply Theorem B.5.2 and we obtain (B.5.9).

Remark B.5.4. Let τ ∈ (0,¹₂) and denote (E, p) the solution of (B.2.10). Assume that E ∈ H^τ(Ω)and E∈/ H^τ+(Ω)for all τ⁺ > τ. Then the best choice ofα is α= ^{`(2−τ)</sup><sub>2`−τ</sub> , which gives the convergence rate 2r =τ<sub>`−</sub><sup>`−1}τ

2; this convergence rate approaches the optimal rate, τ, when the approximation degree`is large.

Remark B.5.5. It is interesting to notice that the degree of the polynomials used for M_h is not involved in the convergence rate when minimal regularity is assumed. This means that we can use dierent degrees of polynomials for X_h and M_h, and that it is sucient to take polynomials of degree1 for M_h.

B.5.3 Numerical illustrations

We present in this section some numerical illustrations of the performance of the method on a boundary value problem. We consider the L-shaped domain

Ω = (−1,1)²\([0,+1]×[−1,0]). We assume thatΩis composed of three subdomains:

Ω1 = (0,1)², Ω2 = (−1,0)×(0,1), Ω3= (−1,0)².

We useκ≡1inΩ,ε|Ω₂ = 1 and ε|Ω₁ =ε|Ω₃ =:εr. Denotingλ >0 a real value such that tan

λπ 4

tan

λπ 2

=εr,

we dene the scalar potentialSλ(r, θ) = r^λφλ(θ), where (r, θ) are the polar coordinates, and φ_λ is dened by

φ_λ(θ) = sin(λθ) if 0≤θ < π

2, φ_λ(θ) = sin ^λπ₂

cos ^λπ₄ cos

λ

θ−3π 4

if π

2 ≤θ < π, φ_λ(θ) = sin

λ

3π 2 −θ

if π ≤θ≤ 3π 2 . Then we solve the problem

∇×∇×E= 0, ∇·(εE) = 0, E×n_|∂Ω =∇S_λ×n.

The exact solution is given byE =∇S_λ. We present two series of simulations. In table B.1, we useλ= 0.535, which leads toεr= 0.499±10⁻³. In table B.2, we useλ= 0.24, which leads toε_r'7.55 10⁻². In both case, we have computed the relative error in theL²-norm, and the column COC stands for the computed order of convergence. We have used several values of α, to show the eect of λ and α on the convergence rates. It seems that the convergence

h α= 0.4 α= 0.6 α= 0.9

rel.tol. coc rel.tol. coc rel.tol. coc

0.2 2.332E-1 - 1.444E-1 - 1.249E-1

-0.1 2.473E-1 -0.08 1.168E-1 0.31 8.846E-2 0.50 0.05 2.631E-1 -0.09 9.452E-2 0.31 6.186E-2 0.52 0.025 2.797E-1 -0.09 7.700E-2 0.30 4.289E-2 0.53 0.0125 2.968E-1 -0.09 6.312E-2 0.29 2.962E-2 0.53

Table B.1: L²-errors and computed order of convergence forλ= 0.535. We expect a conver-gence rate that is at most0.535: it is almost optimal withα= 0.9

rate improves when α is close to 1, which seems to be in contradiction with Remark B.5.4.

Actually, if we write the example used here in the form of (B.2.10), we can use g which is divergence free. Then one can notice in the convergence proofs that in this case, we have only

r=α−1 +τ 1−α

`

, which increases withα.

h α= 0.4 α= 0.6 α= 0.9 rel.tol. coc rel.tol. coc rel.tol. coc

0.2 5.773E-1 - 4.739E-1 - 4.426E-1

-0.1 6.209E-1 -0.11 4.507E-1 0.07 3.801E-1 0.22 0.05 6.711E-1 -0.11 4.413E-1 0.03 3.259E-1 0.22 0.025 7.180E-1 -0.10 4.452E-1 -0.01 2.788E-1 0.23 0.0125 7.564E-1 -0.08 4.602E-1 -0.05 2.380E-1 0.23

Table B.2: L²-errors and computed order of convergence for λ= 0.24. We expect a conver-gence rate that is at most0.24: it is almost optimal with α= 0.9

Dans le document Analyse théorique et numérique des équations de la magnétohydrodynamique : (Page 112-121)