Error bounds

∞

X

j=0

ω_jξⁿ, (4.1.15)

Fast Fourier Transform can be used for an efficient computation of {ω_j}_j∈N0 (see [69, Section 7.5]) . Alternatively, a useful recursive expression is given also in [69, formula (7.23)]:

ω₀ = τ^−α, ω_j =



1 − α + 1 j



ω_j−1, ∀j > 0.

For the numerical experiments we exhibit in Section 4.3 we have taken advantage of this identity.

4.2 Error bounds

This section shows error estimates for the numerical scheme discussed in Section 4.1.

The derivation of the error bounds can be carried out following the guidelines from [43]

and [45].

We start by defining the discrete analogue to operators E^αand F^α. Let {φ_h,1, ..., φ_h,N} ⊂ X_h be an orthonormal base of egienfunctions of A_h, the we define

E_h^α(t)v :=

N

X

k=1

E_α,1(−λ_h,kt^α)φ_h,k(v, φ_h,k)_L²_(Ω), (4.2.1) and

F_h^α(t)v :=

N

X

k=1

t^α−1E_α,α(−λ_h,kt^α)φ_h,k(v, φ_h,k)_L²_(Ω). (4.2.2) Discrete analogues to Lemma 2.0.2 and Lemma 2.0.3 (using E_h^α and F_h^α instead of E^α and F^α) can be easily proved.

4.2.1 L

²

and elliptic projection

We start with some estimations on the elliptic or Ritz projection R_h : eH^s(Ω) → X_h. This operator is defined as the one that satisfies

hRhu, ϕis= (Au, ϕ)_L²_(Ω), ∀ϕ ∈ Xh, and we have the following estimation.

Lemma 4.2.1. Let ψ ∈ ˙H^θ(Ω), with θ ∈ [0, 2], then

k(R_h− I)ψk_L²_(Ω) ≤ h^γ(1+θ/2)kψk_θ,s, (4.2.3) k(R_h− I)ψk_1,s≤ h^γθ/2kψk_θ,s, (4.2.4) with γ = min{s, 1/2 − ε}.

Proof. From [17] Proposition 3.3.2 and equation (3.3.3), taking r = 0 and r = −s, we can assert that

k(R_h− I)ψk_L²_(Ω) ≤ Ch^2γkAψk_L²_(Ω) = Ch^2γkψk_2,s, k(Rh− I)ψk_L²_(Ω) ≤ Ch^γkψk_L²_(Ω) = Ch^γkψk0,s,

k(R_h− I)ψk_1,s ≤ Ckψk_L²_(Ω) = Ckψk_0,s, k(R_h− I)ψk_1,s ≤ Ch^γkAψk_L²_(Ω)= Ch^γkψk_2,s,

From this, and using standard interpolation arguments (see [57, Theorem 4.36] for instance) we can obtain (4.2.3) and (4.2.4).

We end this section with classical estimations for Ph.

Lemma 4.2.2 (cf. [44, Lemma 2.1]). For a quasi-uniform mesh, and θ ∈ [0, 1] we

k(P_h− I)ψk_1,θ ≤ h^2−θkψk_H²_(Ω), ∀ψ ∈ H²(Ω) ∩ H₀¹(Ω), (4.2.5) k(P_h− I)ψk_1,θ ≤ h^1−θkψk_H¹_(Ω), ∀ψ ∈ H₀¹(Ω), (4.2.6) k(Ph− I)ψk1,θ ≤ h^−θkψk_L²_(Ω), ∀ψ ∈ L²(Ω). (4.2.7)

4.2.2 Discrete norm and inverse inequality

Let (λ_h,k, φ_h,k), with k = {1, ..., N }, be an eigen-pair of the operator A_h. Another important tool in the error estimation is the following discrete analogue of the norm k · kθ,s. For every u ∈ Xh we define

|||u|||²_θ,s:=

N

X

k=1

λ^θ_h,k(u, φ_h,k)²_L2(Ω).

It can be shown that |||·|||_θ,s and k · k_θ,s are equivalent. Indeed, this assertion can be easily checked for θ = 1, 0, and the case θ ∈ (0, 1) follows by means of interpolation arguments.

Also we have an inverse inequality [46, Lemma 3.3].

Lemma 4.2.3. There exists a constant C, independent of h, such that for all ψ ∈ X_h and for any p > q

|||ψ|||_p,s≤ Ch^s(q−p)|||ψ|||_q,s. (4.2.8) Proof. It is well known that for a quasi uniform triangulation Th the inverse inequality

|ψ|_H^s_(Rⁿ₎ ≤ Ch^−skψk_L²_(Ω),

holds for any ψ ∈ X_h(see (4.2.7)). From this, observing that |φ_h,k|_H^s_(Rⁿ₎ = (A_hφ_h,k, φ_h,k)^1/2_L2(Ω)= λ^1/2_h,k, we can conclude that max_1≤k≤N λ_h,k ≤ h^−2s. Then we have

|||ψ|||²_p,s ≤ C max

1≤k≤Nλ^p−q_h,k

N

X

k=1

λ^p_h,k(ψ, φ_h,k)²_L2(Ω)≤ Ch^2s(q−p)|||ψ|||²_q,s, and (4.2.8) follows.

4.2.3 Error bounds for the semi-discrete scheme

Here we focus only on the diffusion-wave case (1 < α < 2), where the error bounds becomes more laborious. Of course, ideas used for this case can be straightforwardly generalized for the case 0 < α < 1.

In order to establish the error bounds, we first set an integral representation of u for the homogeneous case f = 0. Define the sector Σ_θ := {z ∈ C, z 6= 0, such that | arg(z)| <

θ}, then u(t) : [0, T ] → L²(Ω) can be analytically extended to Σ_π/2 (see [72, Theorem 2.3]). Applying the Laplace transform in (0.0.3) we obtain

z^αu(z) + Aˆˆ u(z) = z^α−1v + z^α−2b,

where A is the fractional Laplacian with homogeneous Dirichlet conditions. Therefore, via the Laplace inversion formula, we write the integral representation

u(t) = 1 2πi

ˆ

Γ_θ,δ

e^zt(z^αI + A)⁻¹(z^α−1v + z^α−2b) dz, (4.2.9) where Γ_θ,δ = {z ∈ C : |z| = δ, | arg(z)| ≤ θ} ∪ {z ∈ C : z = re^±iθ, r ≥ δ}.

Recalling (2.0.8), if we choose θ such that π/2 < θ < min{π, π/α}, then z^α ∈ Σ_θ⁰ with θ⁰ = αθ for all z ∈ Σ_θ. Considering θ in this mode, there exists a constant C only depending on θ and α such that

k(z^α+ A)⁻¹)k_L²_(Ω) ≤ C|z|^−α. (4.2.10)

As in (4.2.9), we can write an analogous expression for u_h, u_h(t) = 1

2πi ˆ

Γθ,δ

e^zt(z^αI + A_h)⁻¹(z^α−1v_h+ z^α−2b_h) dz. (4.2.11)

The following technical result can be proved analogously to [12, Lemma 3.3].

Lemma 4.2.4. Let ϕ ∈ eH^s(Ω), and z ∈ Σ_θ with π/2 < θ < min{π, π/α}. Then there exists a positive constant c(θ) such that

|z^α| kϕk²_L2(Ω)+ |ϕ|²_Hs(Rⁿ)≤ c 

z^αkϕk²_L2(Ω)+ |ϕ|²_Hs(Rⁿ)

  .

The next lemma sets an error estimate between (z^αI + A)⁻¹f and its discrete ap-proximation (z^αI + A_h)⁻¹P_hf , analogous to [12, Lemma 3.4].

Lemma 4.2.5. Let f ∈ L²(Ω), z ∈ Σθ, ω := (z^αI + A)⁻¹f , ωh := (z^αI + Ah)⁻¹Phf . Then there exists a positive constant C(s, n, θ) such that

kω − ω_hk_L²_(Ω)+ h^γ|ω − ω_h|_H^s_(Rⁿ₎ ≤ Ch^2γkf k_L²_(Ω). As before, γ = min{s, 1/2 − ε}, with ε > 0 arbitrary small.

Proof. We consider first the case s ≥ 1/2. By definition of ω and ωh, it holds that z^α(ω, ϕ) + hω, ϕi_H^s_(Rⁿ₎ = (f, ϕ), ∀ϕ ∈ eH^s(Ω),

z^α(ω_h, ϕ) + hω_h, ϕi_H^s_(Rⁿ₎ = (f, ϕ), ∀ϕ ∈ X_h.

If we set e_h := ω − ω_h and subtract these two expressions, we derive

z^α(e_h, ϕ) + he_h, ϕi_H^s_(Rⁿ₎ = 0, ∀ϕ ∈ X_h. (4.2.12) Applying Lemma 4.2.4 and this identity, we arrive to

|z^α| ke_hk²_L2(Ω)+ |e_h|²_Hs(Rⁿ) ≤ c

z^α(e_h, e_h) + he_h, e_hi_H^s_(Rⁿ₎ 

= c

z^α(e_h, ω − ϕ) + he_h, ω − ϕi_H^s_(Rⁿ₎

 ∀ϕ ∈ X_h. Taking ϕ = Π_hω in the former expression, where Π_h is a suitable quasi-interpolation operator (see, for example, [3, Section 4.1]), we deduce

|z^α|ke_hk²_L2(Ω)+ |e_h|²_Hs(Rⁿ)

≤ c kehk_L²_(Ω)h^1/2−ε|ω|_H^s_(Rⁿ₎+ |eh|_H^s_(Rⁿ₎h^1/2−ε|ω|_H^s+1/2−ε_(Rn)
, (4.2.13) where we have used the fact that s ≥ 1/2 implies that h^s≤ h^1/2−ε.

On the other hand, if we choose ϕ = ω in Lemma 4.2.4, we obtain

|z^α| kωk²_L2(Ω)+ |ω|²_Hs(Rⁿ)≤ c

z^α(ω, ω) + hω, ωi_H^s_(Rⁿ₎ 

= c |(f, ω)| ≤ ckf k_L²_(Ω)kωk_L²_(Ω). Consequently,

kωk_L²_(Ω) ≤ c |z|^−αkf k_L²_(Ω) and |ω|_H^s_(Rⁿ₎ ≤ c |z|^−α/2kf k_L²_(Ω). (4.2.14) From Proposition 1.2.1, we know that for the case z = 0 the estimate |ω|_Hs+1/2−ε(Rⁿ) ≤ kf k_L²_(Ω) holds. Utilizing this estimate with −z^αω + f instead of f , we obtain

|ω|_Hs+1/2−ε(Rⁿ) ≤ k − z^αω + f k_L²_(Ω) ≤ ckf k_L²_(Ω),

where in the last inequality we used (4.2.14). Combining this with (4.2.13), we derive

|z^α| ke_hk²_L2(Ω)+ |e_h|²_Hs(Rⁿ)≤ ch^1/2−εkf k_L²_(Ω) z^α/2ke_hk_L²_(Ω)+ |e_h|_H^s_(Rⁿ₎
. This implies that

|z^α| ke_hk²_L2(Ω)+ |e_h|²_Hs(Rⁿ) ≤ ch^1−2εkf k²_L2(Ω), (4.2.15) and gives the bound for |e_h|_H^s_(Rⁿ₎. Next, we aim to estimate ke_hk²_L2(Ω). For this purpose, we proceed via the following duality argument. Given ϕ ∈ L²(Ω), define

ψ := (z^α+ A)⁻¹ϕ and ψ_h := (z^α+ A)⁻¹P_hϕ.

Thus, we write

ke_hk_L²_(Ω) = sup

ϕ∈L²(Ω)

|(e_h, ϕ)|

kϕk_L²_(Ω) = sup

ϕ∈L²(Ω)

z^α(e_h, ψ) + he_h, ψi_H^s_(Rⁿ₎  kϕk_L²_(Ω) .

We aim to bound the supremum in the identity above. Resorting to (4.2.12) and the Cauchy-Schwarz inequality, we bound

 

z^α(e_h, ψ) + he_h, ψi_H^s_(Rⁿ₎   =

z^α(e_h, ψ − ψ_h) + he_h, ψ − ψ_hi_H^s_(Rⁿ₎ 

≤ z^α/2ke_hk_L²_(Ω)z^α/2kψ − ψ_hk_L²_(Ω)+ |e_h|_H^s_(Rⁿ₎|ψ − ψ_h|_H^s_(Rⁿ₎

≤ z^α/2ke_hk_L²_(Ω)+ |e_h|_H^s_(Rⁿ₎

z^α/2kψ − ψ_hk_L²_(Ω)+ |ψ − ψ_h|_H^s_(Rⁿ₎
. Finally, applying (4.2.15) we arrive at

z^α(eh, ψ) + heh, ψiH^s(Rⁿ)



≤ h^1−2εkf k_L²_(Ω)kϕk_L²_(Ω), from where we can derive the desired inequality.

The analysis of the case s ≤ 1/2 can be carried out in analogously. Indeed, using that h^s≥ h^1/2−ε we obtain, instead of (4.2.13), the inequality

|z^α| ke_hk²_L2(Ω)+|e_h|²_Hs(Rⁿ)

≤ c ke_hk_L²_(Ω)h^s|ω|_H^s_(Rⁿ₎+ |e_h|_H^s_(Rⁿ₎h^s|ω|_Hs+1/2−ε(Rⁿ)
, and proceeding as before we arrive at the desired estimate.

Homogeneous case

At this point, we are able to give an error estimation for the case f ≡ 0.

Theorem 4.2.6. Let 1 < α < 2, u be the solution of (0.0.3) with v ∈ eH^q(Ω), b ∈ eH^r(Ω) Proof. First, suppose v and b ∈ L²(Ω). Combining (4.2.9) and (4.2.11), we can obtain an integral representation for eh,

e_h(t) = 1

With the same idea we can obtain the estimate for ke_hk_L²_(Ω), and this shows the asser-tion for v, b ∈ L²(Ω).

where w^v(z) = (z^αI + A)⁻¹Av, w^v_h(z) = (z^αI + A_h)⁻¹A_hR_hv, and w^b(z) and w_h^b(z) are defined analogously. Now Lemma 4.2.5 and the relation A_hR_h = P_hA yield

kw^v(t) − w^v_h(t)k_L²_(Ω)+ h^γ|w^v(t) − w^v_h(t)|_H^s_(Rⁿ₎≤ Ch^2γkAvk_L²_(Ω). Then, we can estimate the first term (i)

k(i)k_L²_(Ω)≤ Ch^2γkAvk_L²_(Ω)

The second term (ii) can be estimated in a similar way:

k(ii)k_L²_(Ω)≤ Ch^2γkAbk_L²_(Ω)

and the L²(Ω)-error estimate follows. The estimate in H^s(Rⁿ) norm can be obtained analogously. Finally, for the choice v_h = P_hv and b_h = P_hb, we have

E^α(t)v − E_h^α(t)P_hv = E(t)v − E_h^α(t)R_hv + E_h^α(t)(R_hv − P_hv), The first term is already bounded. For the second one we have

kE_h^α(t)(P_hv − R_hv)k_p,s ≤ CkP_hv − R_hvk_p,s≤ Ch^2γ−γpkvk_H^2s_(Rⁿ₎, p = 0, 1.

where in the first inequality we have used Lemma 2.0.2, and lemmas 4.2.1 and 4.2.2 (combined with classical interpolation techniques) in the second step. The estimate for b ∈ H^2s(Rⁿ) follows analogously. These estimates along with standard interpolation arguments complete the proof of the theorem.

Non-homogeneous case

To complete the error estimate for the semi-discrete scheme, it still remains to analyze the case v ≡ b ≡ 0 and f 6≡ 0. A proper generalization of [45, Theorem 3.2] can be carried out following the guidelines outlined in that work.

Theorem 4.2.7. Let 1 < α < 2, f ∈ L^∞(0, T ; L²(Ω)), and let u and u_h be the solutions of (0.0.3) and (4.1.1) respectively, with f_h = P_hf , and all the initial data equal to zero.

Then, there exists a positive constant C = C(s, n) such that ku − u_hk_L²_(Ω)≤ Ch^2γ| log h|²kf k_L^∞_{([0,T ];L}²_(Ω)).

Proof. Recalling the results of Section 2.3, we know that if v ≡ b ≡ 0 then

u(t) = ˆ _t

0

F^α(t − s)f (t) ds.

So we can estimate

ku(t)k_2−ε,s ≤ ˆ _t

0

kF^α(t − s)f (t)k_2−ε,sds ≤ ˆ _t

0

(t − s)^εα/2−1kf (t)k_L²_(Ω)ds (4.2.17)

≤ Cε⁻¹t^εα/2kf k_L^∞_{([0,T ];L}²_(Ω)), where in the second inequality we have used Lemma 2.0.2.

Now, splitting u − uh as

u − u_h = (u − P_hu) + (P_hu − u_h) = a(t) + c(t),

and using the relation A_hR_h = P_hA we can obtain the following equation for a(t):

C∂_t^αa + Aha = Ah(Rhu − Ahu), (4.2.18) with a(0) ≡ ∂_ta(0) ≡ 0.

Using 4.2.1 and (4.2.17) we can easily estimate

kc(t)k_L²_(Ω) ≤ h^2γ−γε/2ku(t)k_2−ε,s ≤ h^2γ−εku(t)k_2−ε,s (4.2.19)

≤ Ch^2γ−εε⁻¹t^εα/2kf k_L^∞_{([0,T ];L}²_(Ω)), for all t > 0.

On the other hand, using (4.2.18) we can write

a(t) = ˆ t

0

F^α(t − s)A_h(R_hu − A_hu) ds, and from this we can estimate

ka(t)k_L²_(Ω)≤ ˆ _t

0

kF^α(t − s)A_h(R_hu − A_hu)k_L²_(Ω)ds = ˆ _t

0

kF^α(t − s)(R_hu − A_hu)k_2,sds

≤ C ˆ t

0

(t − s)^αε/2−1k(Rhu − Ahu)kε,sds ≤ C ˆ t

0

(t − s)^αε/2−1|||(Rhu − Ahu)|||_ε,sds,

where in the third step we have used Lemma 2.0.2 with p = 2 and q = ε. Further,

where in the third step we have used estimation (4.2.17). Finally, we can obtain the statement of the theorem taking ε = | log h| in the last inequality.

4.2.4 Error bounds for the fully-discrete scheme

Considering all the theory displayed up to this point, error estimates for the fully-discrete scheme can be derived in the same way as in [43].

Theorem 4.2.8. Let u be the solution of problem (0.0.3) with v ∈ eH^q(Ω), b ∈ eH^r(Ω)

For v ∈ H^2s(Rⁿ), we first consider v_h = R_hv. Using the relation G(z) = I − (z^αI + A_h)⁻¹A_h, with G_s(z) = (z^αI + A_h)⁻¹, we have U_hⁿ− u_h(t_n) = (G_s(∂_τ) − G_s(∂_t))A_hv_h. Using (4.2.10) and Lemma 4.1.1 (with µ = α and β = 1) gives

ku_h(t_n) − U_hⁿk_L²_(Ω) ≤ Cτ t^α−1_n kA_hv_hk_L²_(Ω) ≤ cτ t^α−1_n kvk_H^2s_(Rⁿ₎,

where the last step we have used A_hR_h = P_hA. The estimate holds also for the choice v_h = P_hv in view of the L²(Ω) stability of the scheme (4.2.22), and repeating the final argument in the proof of Theorem 4.2.6. The assertion now follows from interpolation.

The case of 1 < α < 2 is analogous, and hence the proof is omitted.

Remark 4.2.9. In the previous theorem –and in Theorem 4.2.6 as well – we wrote the orders of convergence in term of various Sobolev norms of the data. For clarity, hypotheses in theorems 2.1.3 and 2.3.2 just involved either L² or H^snorms of the data.

For instance, assuming that v ∈ eH^s(Ω) is such that (−∆)^sv ∈ L²(Ω) and b ∈ eH^s(Ω), the conclusions of Theorem 4.2.8 read

ku(t_n) − U_hⁿk_L²_(Ω)≤ C t^α−1_n τ + h^s+γ
k(−∆)^svk_L²_(Ω) if 0 < α < 1, ku(tn) − U_hⁿk_L²_(Ω)≤ C t^α−1_n τ + h^s+γ
k(−∆)^svk_L²_(Ω)

+ C t

α

n2τ + t¹⁻

α

n 2h^2γ

|b|_H^s_(Rⁿ₎ if 1 < α < 2.

We emphasize that, as stated in Remark 1.2.3, the identity k(−∆)^svk_L²_(Ω) ≤ C|v|_H^2s_(Rⁿ₎ holds for all v ∈ eH^2s(Ω).

Finally, we state the order of convergence of the fully-discrete scheme for the prob-lems with a non-null source term.

Theorem 4.2.10. Let u be the solution of (0.0.3) with homogeneous initial data and with f ∈ L^∞(0, T ; L²(Ω)); and let U_hⁿ be the solution of (4.1.13) or (4.1.14) with f_h = P_hf . Then, there exists a positive constant C = C(s, n) such that

• For 0 < α < 1, if ´t

0(t − s)^α−1kf⁰(s)k_L²_(Ω)ds < ∞ for t ∈ (0, T ], then ku(tn) − U_hⁿk_L²_(Ω) ≤ C

h^2γ`²_hkf k_L^∞_{([0,T ];L}²_(Ω))+ t^α−1_n τ kf (0)k_L²_(Ω) + τ

ˆ _tn

0

(t_n− s)^α−1kf⁰(s)k_L²_(Ω) .

• If 1 < α < 2, then

ku(t_n) − U_hⁿk_L²_(Ω)≤ C(h^2γ`²_h+ τ )kf k_L^∞_{([0,T ];L}²_(Ω)).

Proof. Defining G(z) = (z^αI + A_h)⁻¹, the semidiscrete solution u_h and fully discrete solution U_hⁿ can be written as u_h = G(∂_t)f_h and U_hⁿ = G(∂_τ)f_h, respectively. Using the splitting fh(t) = fh(0) + (1 ∗ f_h⁰)(t) and the associativity property (4.1.10), we have

u_h(t_n) − U_hⁿ= G(∂_t) − G(∂_τ)
(f_h(0) + (1 ∗ f_h⁰)(t_n))

= G(∂t) − G(∂τ)
fh(0) + (G(∂t) − G(∂τ))1
∗ f_h⁰(tn)) := (i) + (ii).

Then Lemma 4.1.1 (with µ = α and β = 1) gives us a bound for the first term (i) k(i)k_L²_(Ω) ≤ cτ t^α−1_n kf_h(0)k_L²_(Ω) ≤ cτ t^α−1_n kf (0)k_L²_(Ω).

Again, by Lemma 4.1.1 and the L² stability of P_h we have k(ii)k_L²_(Ω) ≤

ˆ tn

0

k (G(∂t) − G(∂τ))1
(tn− s)f_h⁰(s)k_L²_(Ω)ds

≤ cτ ˆ tn

0

(t_n− s)^α−1kf_h⁰(s)k_L²_(Ω)ds ≤ cτ ˆ tn

0

(t_n− s)^α−1kf⁰(s)k_L²_(Ω)ds.

This shows the first assertion. For the scheme (4.1.14) (using the corrected term Gⁿ_h =

∂_τ∂_t⁻¹f_h(t_n) as source) with v = b = 0, we have U_hⁿ = G(z)g_h with g_h = ∂_t⁻¹f_h and G(z) = z(z^αI + A_h)⁻¹. Hence the relation g_h = 1 ∗ f_h, the convolution property (4.1.10) and Lemma 4.1.1 with µ = α − 1 and β = 1 yield

ku_h(t_n) − U_hⁿk_L²_(Ω) ≤ cτ ˆ _tn

0

(t_n− s)^α−2kf (s)k_L²_(Ω)ds ≤ c_Tτ kf k_L^∞_{(0,T ;L}²_(Ω)), from which follows the second assertion.

In document Numerical methods for non-local evolution problems (Page 69-79)