Error estimation

The error estimation for the numerical scheme presented in the previous section can be obtained in a similar way to the linear case. For the sake of simplicity we are going to consider ε² = 1 throughout this section.

5.2.1 Error estimation for the semidiscrete scheme

First, we give an error estimation for the semi-discrete scheme.

Theorem 5.2.1. Let u and u_h be the the exact and the semi-discrete solution of (2.2.4) and (5.1.1) respectively. And let v ∈ L²(Ω) and v_h = P_hv with kvk_L²_(Ω) ≤ R. Then there exists a positive constant C = C(R, T ) such that

ku(t) − u_h(t)k_L²_(Ω) ≤ Ch^2γ(t^−α+ | log h|²) , t ∈ (0, T ]. (5.2.1) With γ as in Theorem 4.2.6.

Proof. We can write the solution and its semi-discrete approximation as

u = E^α(t)v + ˆ _t

0

F^α(t − s) ˜f (u(s)) ds, and

u_h = E_h^α(t)v_h+ ˆ _t

0

F_h^α(t − s) ˜f (u_h(s)) ds, respectively. Then, defining e = u − u_h, we have

e(t) = E^α− E_h^αP_h
(t)v + ˆ t

0

F_h^α(t − s)P_h f (u(s)) − ˜˜ f (u_h(s))
ds

+ ˆ t

0

F^α− F_h^αP_h
(t − s) ˜f (u(s)) ds.

Using Theorem 4.2.6 in the first term; | ˜f |, | ˜f⁰| ≤ B, and (2.0.6) in the second term;

Theorem 4.2.7 with f = ˜f (u) and | ˜f | < B in the last term, we have

ke(t)k_L²_(Ω) ≤ Ckt^−αh^2γ+ CB ˆ t

0

(t − s)^α−1ke(s)k_L²_(Ω)ds + Ch^2γ`h

Then, applying Lemma 2.2.2 we derive (5.2.1).

5.2.2 Error estimation for the fully discrete scheme

Consider the discrete problem of find V_hⁿ ∈ X_h, n ∈ {1, ..., N }, V_h⁰ = 0 such that

n

X

j=0

ω_jV_h^n−j = −A_hV_hⁿ+ f_hⁿ, (5.2.2)

with f_hⁿ ∈ X_h, for all n ∈ {1, ..., N }. Recalling that ω₀ = τ^−α, and defining E = (I + τ^αA_h)⁻¹, we can rewrite (5.2.2) as

V_hⁿ = EXⁿ

j=1

−τ^αω_jV_h^n−j+ τ^αf_hⁿ

. (5.2.3)

If we define {˜ω_n}_n∈N as the coefficients of the series expansion of (1 − ξ)^α, from the definition of {ω_n}_n∈N (4.1.15) we have ˜ω_n = τ^αω_n for all n ∈ N. And we can write Vhⁿ

as a function of f_hⁿ in a recursive expression

V_hⁿ=

n

X

j=1

E_n−jf_h^j, n > 0, (5.2.4) with E_n recursively defined as

E₀ = τ^αE, E_n= EXⁿ⁻¹

j=0

−˜ω_n−jE_j

. (5.2.5)

As we have observed in the proof of Theorem 5.1.1, we have kEk_L²_(Ω) = k(I + τ^αAh)⁻¹k_L²_(Ω)< 1.

Then, from (5.2.5), and recalling that −˜ω_j > 0 for j ≥ 1, we have

kE₀k_L²_(Ω) ≤ τ^α, kE_nk_L²_(Ω) ≤

n−1

X

j=0

−˜ω_n−jkE_jk_L²_(Ω). (5.2.6)

Defining the sequence

c₀ = 1, c_n=

n−1

X

j=0

−˜ω_n−jc_j, (5.2.7)

it is possible to check that

kE_nk_L²_(Ω)≤ τ^αc_n. (5.2.8) In order to get the error estimation, it will be useful to know something about the asymptotic behaviour of {c_n}_n∈N.

Lemma 5.2.2. Let {˜ω_n}_n∈N0 be the coefficients of the power series expansion of (1−ξ)^α, with α ∈ (0, 1), and {c_n}_n∈N0 the sequence recursively defined in (5.2.7). Then, c_n ∈ O(n^α−1).

Proof. From the definition of {˜ω_n}_n∈N0, we know that

(1 − ξ)^α =

∞

X

j=0

˜ ω_jξ^j.

Then, defining g(ξ) = 1 − (1 − ξ)^α, and recalling that w0 = 1, we have

g(ξ) =

∞

X

j=1

−˜ω_jξ^j.

Now, defining f (ξ) = P∞

j=0cjξ^j, from the definition of {cn}_n∈N0, and using the Cauchy product for power series, the following equality can be easily checked,

f (ξ)g(ξ)

ξ = f (ξ) − c0

ξ . (5.2.9)

Recalling that c₀ = 1, and −˜ω₁ = α, from (5.2.9) we can obtain an explicit expression for f ,

f (ξ) = (1 − ξ)^−α. (5.2.10)

It is well known that series expansion of f is

f (ξ) =

∞

X

j=0

(−1)^j−α j

 ξ^j. Then

c_n = (−1)ⁿ−α n



, (5.2.11)

where

−α n



= Γ(1 − α)

Γ(1 + n)Γ(1 − n − α). (5.2.12)

Finally, by means of basic Gamma function properties, we can verify that ^−α_n
∈ O(n^α−1), and hence {c_n}_n∈N0 ∈ O(n^α−1).

At this point, we are able to obtain a bound for the error.

Theorem 5.2.3. Let u and U_hⁿ = U_h(t_n) be the solution of (5.0.2) and (5.1.3) respec-tively. Consider v ∈ L²(Ω) and v_h = P_hv with kvk_L²_(Ω) ≤ R. Then, under the condition 0 < τ^α< δ < 1, there exists a positive constant C = C(R, T, α, δ) such that

ku(t_n) − U_h(t_n)k_L²_(Ω) ≤ Ch^2γ(t^−α_n + | log h|²) + Cτ t⁻¹_n , (5.2.13) t_n∈ [0, T ].

Proof. In view of Theorem 5.2.1, we only need to get an estimation for ku_h(t_n) − Uh(tn)k_L²_(Ω), with uh the semi-discrete solution. Defining the function G(z) = (z^αI + A_h)⁻¹, analytic in a sector Σ_θ with θ ∈ (π/2, π) (see Section 4.1.2), from the semi-discrete and fully semi-discrete scheme we have

u_h = G(∂_t)∂_t^αv_h+ G(∂_t)P_hf (u˜ _h), and

U_h = G(∂_τ)∂_τ^αv_h+ G(∂_τ)P_hf (U˜ _h).

Subtracting both expressions we obtain a formula for e_h := u_h− U_h,

e_h = (G(∂_t)∂_t^α− G(∂_τ)∂_τ^αP_h)v + G(∂_t)P_hf (u˜ _h) − G(∂_τ)P_hf (U˜ _h) = (5.2.14) (G(∂_t)∂_t^α− G(∂_τ)∂_τ^αP_h)v_h+ (G(∂_t) − G(∂_τ))P_hf (u˜ _h) + G(∂_τ)P_h( ˜f (u_h) − ˜f (U_h))

= (i) + (ii) + (iii).

The norm of the first term (i) can be estimated by means of Lemma 4.1.1. That is, taking µ = 0, β = 1 in that lemma, we obtain

k(i)k_L²_(Ω) ≤ Ct⁻¹_n τ kv_hk_L²_(Ω) ≤ Ct⁻¹_n τ,

with C = C(R).

For the estimation of the second term, using property (4.1.10), we can split (ii) as follow

(ii) = G(∂_t) − G(∂_τ)

P_hf (u˜ _h(0)) + (1 ∗ P_h∂_tf (u˜ _h(t_n))

= G(∂_t) − G(∂_τ)
P_hf (u˜ _h(0)) + ( G(∂_t) − G(∂_τ)
1) ∗ P_h∂_tf (u˜ _h(t_n))

= I + II.

Using Lemma 4.1.1 with µ = α, β = 1, along with the fact that | ˜f | < B, we can estimate

kIk_L²_(Ω)≤ Ct^α−1_n τ.

On the other hand, noticing that estimation (2.2.16) can be easily extended to ∂_tu_h in such a way that k∂_tu_hk_L²_(Ω) ≤ Ct^α/2−1 (since v_h ∈ eH^s(Ω)); using again Lemma 4.1.1, estimation | ˜f⁰| < B, and writing ∂_tf (u˜ _h(t)) = ˜f⁰(u_h(t))∂_tu_h, we have

kIIk_L²_(Ω)≤ ˆ tn

0

k( G(∂t) − G(∂τ)
1)(tn− s) ˜f⁰(uh(s))∂tuh(s)k_L²_(Ω)ds

≤ Cτ ˆ _tn

0

(t_n− s)^α−1k∂_tu_h(s)k_L²_(Ω)ds ≤ Cτ ˆ _tn

0

(t_n− s)^α−1s^α/2−1ds

≤ Cτ t

3 2α−1

n ,

where in the last inequality we have estimated the integral in terms of the beta function B(α/2, α), as in Theorem 2.2.1.

Now, we observe that the last term (iii) is a solution for (5.2.2), with f_hⁿ = P_h( ˜f (u_h) − ˜f (U_h)). Then, in view of (5.2.4) and (5.2.8), and using again that | ˜f⁰| < B, we have

k(iii)k_L²_(Ω) ≤ τ^α

n

X

j=1

c_n−jke_h(t_j)k_L²_(Ω), where {c_n}_n∈N is the sequence defined in (5.2.7).

Using

Cτ (t⁻¹_n + t

3 2α−1

n + t^α−1_n ) ≤ Cτ t⁻¹_n , with C = C(T ), we can derive the estimation

ke_h(t_n)k_L²_(Ω) ≤ Cτ t⁻¹_n + τ^α

n

X

j=1

c_n−jke_h(t_j)k_L²_(Ω).

Now, recalling hypothesis 0 < τ^α < δ < 1, we have

ke_h(t_n)k_L²_(Ω)≤ Cτ t⁻¹_n + Cτ^α

n−1

X

j=1

c_n−jke_h(t_j)k_L²_(Ω),

with C = C(δ). We can apply a discrete Gronwall type inequality to the former expression, and get

keh(tn)k_L²_(Ω) ≤ Cτ t⁻¹_n e^{(CταPn−1j=1cn−j)}. (5.2.15) Now, from Lemma 5.2.2, we know that cn ∼ n^α−1, and hence τ^αPn

j=1cn−j . τ^αn^α ≤ τ^αN^α = T^α. From this, (5.2.15), and (5.2.1), we can derive (5.2.13).

5.2.3 L

^∞

bounds

In order to prove that the solution remains bounded between 1 and −1, we are going to define first the semi-discrete in time problem. That is, find Uⁿ ∈ eH^s(Ω), with n ∈ {1, ..., N }, such that

 ∂_τ^αUⁿ+ AUⁿ = ∂_τ^αv + ˜f (Uⁿ)

U⁰ = v. (5.2.16)

Before giving an existence result, we set the following auxiliary lemma.

Lemma 5.2.4. Let {a_j}_j∈N and {b_j}_j∈N be two sequences of real numbers, with {a_j}_j∈N non-increasing and positive valued. Suppose

n−1

X

j=0

a_j(b_n−j − b_n−j−1) < 0, for some n ≥ 2. Then there exists j₀ < n such that b_j0 > b_n. Proof. Suppose b_j ≤ b_n for all j < n. Since a_j−1 ≥ a_j > 0 we have

0 >

n−1

X

j=0

a_j(b_n−j− b_n−j−1) ≥ a₁(b_n− b_n−2) +

n−1

X

j=2

a_j(b_n−j− b_n−j−1)

≥ a₂(b_n− b_n−3) +

n−1

X

j=3

a_j(b_n−j − b_n−j−1) ≥ ... ≥ a_n−1(b_n− b₀) ≥ 0,

and the contradiction came from the assumption b_j ≤ b_n for all j < n. Then there exists j₀ < n such that b_j0 > b_n and the lemma is proved.

The proof of the existence and uniqueness of solutions for problem (5.2.16) is similar to the one given for the fully discrete case. For the solutions of this problem, we have the following result.

Theorem 5.2.5. Consider the semi-discrete in time scheme (5.2.16) with U⁰ ∈ L^∞(Ω), then there exists τ > 0 small enough such that (5.2.16) has a unique solution Uⁿ, n ∈ {0, ..., N }, with Uⁿ∈ C^s(Rⁿ) for all n > 0. Moreover if |U⁰(x)| ≤ 1 for all x ∈ Ω, then |Uⁿ(x)| ≤ 1 for all x ∈ Ω and n ∈ {1, ...N }.

Proof. Suppose we have a solution with the desired properties for all m < n. From (5.2.16) we have the relation

Uⁿ= (I + τ^αA)⁻¹ Xⁿ

First we want to show that there exists Uⁿ ∈ L²(Ω) that satisfies equation 5.2.17.

In order to do that, we define the map T : L²(Ω) → L²(Ω)

Then, for a small τ we have that T is a contraction. Hence, there exists a unique solution Uⁿ∈ L²(Ω) for (5.2.17), and the relation is satisfied. Since the right hand side belongs to L^∞(Ω), applying Theorem 1.2.4, we can conclude that Uⁿ∈ C^s(Rⁿ) ∩ C^2s(Ω_ρ) for all 0 < ρ < ρ₀.

Now, we want to see that if the initial data is regular enough, then the solution remains bounded between 1 and −1. Suppose |U^m(x)| ≤ 1 for all x ∈ Ω, for all m < n.

The semi-discrete in time scheme gives us the relation

n

which can be rewritten as

n−1

X

j=0

aj(U^n−j− U^n−j−1) = −AUⁿ+ ˜f (Uⁿ), with a_n =Pn

j=0ω_j. Suppose there exists some x₀ such that Uⁿ achieves its maximum on that point, and Uⁿ(x0) > 1. Recall that kU^mk_L^∞_(Ω) ≤ 1 for all m < n. Assuming

Observing the fact that {an} is a positive valued and decreasing sequence, we can apply Lemma 5.2.4 to show that there exists m₀ < n, such that U^m0(x₀) > Uⁿ(x₀), and then 1 ≥ U^m0(x₀) > Uⁿ(x₀) > 1. The contradiction came from the assumption that Uⁿ(x0) > 1.

Now we want to see that the former property remains valid for less regular initial data. To this end, applying a density argument, suppose Uⁿ is a solution for (5.2.16) with U⁰ ∈ L^∞(Ω), kU⁰kL^∞(Ω) ≤ 1. Consider {U_k⁰}_k∈N ⊂ C_c^∞(Ω), with kU_k⁰kL^∞(Ω) ≤ 1 and taking norms we obtain

keⁿ_kk_L²_(Ω) ≤ ke⁰_kk_L²_(Ω)+

n

X

j=1

−˜ω_jke^n−j_k k_L²_(Ω)+ τ^αCkeⁿ_kk_L²_(Ω). (5.2.19)

Choosing τ such that τ^αC < 1, recalling that 0 < Pn

j=1−˜ω_j < 1, and applying a Gronwall type inequality we have

keⁿ_kk_L²_(Ω) ≤ ke⁰_kk_L²_(Ω)e^1/1−ταC

1 − τ^αC = Cke⁰_kk_L²_(Ω), with C = C(τ, α, C), and then, keⁿ_kk_L²_(Ω) → 0.

Since kU_k⁰k_L^∞_(Ω) ≤ 1, then kU_kⁿk_L^∞_(Ω) ≤ 1 for all k, and for all n ∈ {1, ..., N }.

Hence, for a fixed n, we can construct a sub-sequence {U_kⁿ_j}_j∈N, such that U_kⁿ_j → Uⁿ a.e. , and conclude that kUⁿk_L^∞_(Ω) ≤ 1.

Remark 5.2.6. Suppose we have U^m ∈ C²(Ω) ∩ C^s(Rⁿ) for all m < n. If we take a fixed ρ⁰ > 0 and ρ = 2ρ⁰ in Theorem 1.2.5, a repeated application of this result, along with the fact that ˜f ∈ C²(R), implies that Uⁿ ∈ C^2+2s(Ω_kρ0) for some k ∈ N, only depending on s. Since ρ0 can be arbitrary small, we can assert that Uⁿ ∈ C²(Ω), and then, Uⁿ ∈ C²(Ω) ∩ C^s(Rⁿ).

Finally, proceeding analogously as in Theorem 5.2.3, we can derive the following error estimation.

Theorem 5.2.7. Let u and Uⁿ = U (t_n) be the solution of (5.0.2) and (5.2.16) respec-tively, with v ∈ L²(Ω), kvk_L²_(Ω) ≤ R. Then, under the condition 0 < τ^α < δ < 1, there exists a positive constant C = C(R, T, α, δ) such that

ku(t_n) − U (t_n)k_L²_(Ω)≤ Ct⁻¹_n τ, t_n ∈ [0, T ]. (5.2.20) Now, consider kvk_L^∞_(Ω)≤ 1, a family of nested subdivisions of [0,T] with τ = T /N_k with k ∈ N ,and a fixed tn ∈ (0, T ]. Let U_k(t_n) be the solution of (5.2.16), and u the solution of (2.2.4), using 5.2.7 we have that U_k(t_n) → u(t_n) in L²(Ω). So, we are able to extract a sub-sequence {U_kj(t_n)}_j∈N such that U_k(t_n) → u(t_n) a.e. . Using 5.2.5, we know that kU_k(t_n)k_L^∞_(Ω) ≤ 1, and then, ku(t_n)k_L^∞_(Ω) ≤ 1. We can summarize this observation in the following result.

Theorem 5.2.8. Let u a solution of (2.2.5) with kvk_L^∞_(Ω) ≤ 1. Then ku(t)k_L^∞_(Ω) ≤ 1 for all t ∈ (0, T ].

This theorem implies that all the analysis displayed up to here remains valid replac-ing ˜f by f .

In document Numerical methods for non-local evolution problems (Page 92-100)