Smooth Data Error Estimates

3.2 Standard Galerkin Finite Element Method

3.2.1 Smooth Data Error Estimates

In this section, we focus our attention to derive error estimates for smooth initial data, i.e., u₀ ∈H˙²(Ω).

Error estimates using the Ritz projection. Using the Ritz projection as the comparison function, we split the error u(t)− u_h(t) as usual way like for parabolic problems

(3.16) u(t)−u_h(t) =ρ(t) +θ(t), ρ=u−R_hu, θ=R_hu−u_h.

From the approximation property (3.13) and the stability estimate (2.61), we have the bounds for ρ(t) as

(3.17) kρ(t)k+hk∇ρ(t)k ≤Ch²ku(t)k₂ ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

.

CHAPTER 3. Semidiscretization 43 Now it remains to estimate kθ(t)k and k∇θ(t)k. It is easy to verify that θ satisfies the following error equation

(3.18)

∂_t^αθ(t) +A_hθ(t) = −P_h∂_t^αρ(t) +P_h Z t

0

B(t, s)ρ(s)ds+ Z t

0

B_h(t, s)θ(s)ds

=−Ph∂_t^αρ(t) +PhB˜ρ(t) + ˜Bhθ(t), where (˜Bρ(t), χ) = (Rt

0B(t, s)ρ(s)ds, χ) =Rt

0 B(t, s;ρ(s), χ)ds, for χ∈ S_h. With u_0h = u_h(0) =R_hu₀, we have θ(0) = 0.By Duhamel’s principle (3.5), we write

(3.19)

θ(t) = − Z t

0

Eh(t−τ)(P_h∂_τ^αρ(τ))dτ + Z t

0

Eh(t−τ)P_hB˜ρ(τ)dτ +

Z t 0

Eh(t−τ) ˜B_hθ(τ)dτ =:I₁+I₂+I₃.

We first estimate the L²(Ω)-norm of θ(t) and then its gradient. Applying (3.9) with p = q = 0, the L²-stability of P_h, (3.13), and the estimate (2.62) with p = 2, we have for the term I₁

kI₁k ≤C Z t

0

(t−τ)^α−1k∂_τ^αρ(τ)k dτ

≤Ch² Z t

0

(t−τ)^α−1k∂_τ^αu(τ)k₂ dτ

≤Ch² Z t

0

(t−τ)^α−1 τ^−α[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

dτ

≤Ch² ku₀k₂+kfk_C1(L²(Ω))

. (3.20)

Now we proceed to estimate the term I₂. Following [69], we define the discrete solution operator T_h :L² →S_h of the elliptic problem by

(∇T_hf,∇χ) = (f, χ), ∀χ∈S_h.

Here the operator T_h is self-adjoint, positive definite on S_h and positive semidefinite on L²(Ω). Therefore T_h : S_h → S_h is invertible and it is bounded on S_h. By Bounded Inverse Theorem T_h⁻¹ : S_h → S_h is bounded. Thus, for each χ ∈ S_h, there exists a unique ψ ∈S_h such that T_hψ =χ imply ψ =T_h⁻¹χ, and

kψk ≤ kT_h⁻¹k kχk. With χ=T_hψ, we note that

(3.21) ( Z t

0

B(t, s)ρ(s)ds, χ) = ( Z t

0

B(t, s)ρ(s)ds, Thψ) = Z t

0

B(t, s;ρ(s), Thψ)ds.

CHAPTER 3. Semidiscretization 44 Invoking [69, Lemma 2.1], (3.13) and (2.61), we find that

B(t, s;ρ(s), T_hψ)≤C(hk∇ρ(s)k+kρ(s)k)kψk

≤C(Ch²ku(s)k₂+Ch²ku(s)k₂)kT_h⁻¹k kχk

≤Ch² ku0k₂+kfk_C1(L²(Ω))

kχk. (3.22)

It follows from (3.21) and (3.22) that

k Z t

0

B(t, s)ρ(s)dsk_0,h:= sup

χ∈S_h χ6=0

(Rt

0 B(t, s)ρ(s)ds, χ)

kχk ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

, (3.23)

where k · k_0,h denotes the discrete L²(Ω)-norm. With p = 0, q = 0 in the stability estimate (3.9) of the operator Eh(t) and (3.23), we get

kI2k ≤C Z τ

0

(t−τ)^α−1k Z t

0

B(τ, s)ρ(s)dsk0,hdτ

≤C Z t

0

(t−τ)^α−1Ch² ku₀k₂+kfk_C1(L²(Ω))

dτ

≤Ch² ku₀k₂ +kfk_C1(L²(Ω))

. (3.24)

To estimate I₃, we proceed as in the case of I₂ and apply the inverse estimate and the change of order of integration to obtain

kI3k ≤C Z t

0

(t−τ)^α−1k Z τ

0

Bh(τ, s)θ(s)dsk0,hdτ

≤C Z t

0

(t−τ)^α−1 Z τ

0

kθ(s)k+hk∇θ(s)k

ds

dτ

≤C Z t

0

(t−τ)^α−1 Z τ

0

kθ(s)k ds

dτ

≤C Z t

0

Z t s

(t−τ)^α−1kθ(s)k dτ

ds

=C Z t

0

(t−s)^αkθ(s)k ds.

(3.25)

Using (3.20), (3.24), (3.25) and applying Gronwall’s lemma, we obtain (3.26) kθ(t)k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

.

CHAPTER 3. Semidiscretization 45 Our next task is to bound

∇θ(t)

. Puttingp= 1, q= 0 in (3.9), theL²-stability ofPh, (3.13), and (2.62), we have

k∇I₁k ≤C Z t

0

(t−τ)^α/2−1k∂_τ^αρ(τ)k dτ

≤Ch Z t

0

(t−τ)^α/2−1k∂_τ^αu(τ)k₁ dτ (3.27)

≤Ch[ku₀k₂+kf(0)k]

Z t 0

(t−τ)^α/2−1τ^−α/2dτ +Chkfk_C1(L²(Ω))

Z t 0

(t−τ)^α/2−1dτ

≤Chku₀k₂B(α/2,−α/2 + 1) +Chkfk_C1(L²(Ω))

≤Ch ku₀k₂+kfk_C1(L²(Ω))

, (3.28)

where B(·,·) is the Beta function. Again, with p= 1, q= 0 in (3.9) and (3.23) yield k∇I₂k ≤C

Z t 0

(t−τ)^−1+α/2k Z τ

0

B(τ, s)ρ(s)dsk_0,hdτ

≤Ch² ku₀k₂+kfk_C1(L²(Ω))

Z t

0

(t−τ)^−1+α/2dτ

≤Ch² ku₀k₂+kfk_C1(L²(Ω))

. (3.29)

Arguing as I₃, we estimatek∇I₃k as k∇I₃k ≤C

Z t 0

(t−τ)^−1+α/2k Z τ

0

B_h(τ, s)θ(s)dsk_0,hdτ

≤C Z t

0

(t−s)^α/2kθ(s)k ds ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

, (3.30)

where we have used (3.26). Altogether (3.28)-(3.30) now lead to k∇θ(t)k ≤Ch ku₀k₂+kfk_C1(L²(Ω))

.

Combining the estimates forρand θ we have the following theorem for smooth data error estimate.

Theorem 3.2.1. Let u and u_h be the solutions of (3.1) and (3.2), respectively. With u₀ ∈H˙²(Ω), f ∈C¹(L²(Ω)) and u_0h =R_hu₀, f_h =P_hf, we have for 0≤t ≤T,

ku(t)−uh(t)k+hk∇(u(t)−uh(t))k ≤Ch² ku0k₂+kfk_C1(L²(Ω))

.

CHAPTER 3. Semidiscretization 46 Remark 3.2.1. (i) One can improve the estimate of k∇θ(t)k to O(h²) with an addi- tional factor t^−α/2 by using the estimate

k∂_τ^αρ(τ)k ≤Ch²k∂_τ^αu(τ)k₂ ≤Ch² τ^−α[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

in (3.27) together with the estimates (3.29)-(3.30).

(ii) When the initial data u₀ ∈ H˙¹(Ω), the Ritz projection R_hu₀ is well-defined.

Arguing as in Theorem 3.2.1, it is easy to show that for the homogeneous problem (Eq.

(2.1) with f = 0), by Theorem 2.4.1,

k∇(u(t)−u_h(t))k ≤Cht^−α/2ku₀k₁, t >0.

Error estimates using the Ritz-Volterra projection. In this part we provide an alternate approach to derive error estimates for smooth initial data, where the Ritz- Volterra projection is used as an intermediate solution to split the error as

(3.31) u(t)−u_h(t) = (u(t)−V_hu(t)) + (V_hu(t)−u_h(t)) = ˆρ(t) + ˆθ(t).

Using the approximation property of the Ritz-Volterra projection (3.15) and the stability estimate (2.61), the bounds of ˆρ(t) are obtained as follows:

(3.32) kˆρ(t)k+hk∇ˆρ(t)k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

.

It remains to bound ˆθ(t) in the L²(Ω) and gradient norms which relies on a bound for the termk∂_t^αρ(t)k. Unlike the Ritz projection, in this caseˆ ∂_t^αV_h 6=V_h∂_t^α, in general. As a result, the bounds forθin bothL²(Ω)- andH¹(Ω)-norms are not straightforward from the approximation properties of the Ritz-Volterra projection V_h. The following lemma provides a bound for k∂_t^αρ(t)k.ˆ

Lemma 3.2.2. Let ρ(t)ˆ be defined as in (3.31). Then

k∂_t^αρ(t)kˆ +hk∇∂_t^αρ(t)k ≤ˆ Ch² t^−α[ku0k₂+kf(0)k] +kfk_C1(L²(Ω))

, t >0.

Proof. We first prove the gradient norm estimate, and thenL²(Ω)-norm estimate follows by duality argument. Taking α-th order Caputo fractional time derivative to (3.14), we

CHAPTER 3. Semidiscretization 47 have

(3.33)

(∇∂_t^αρ(t),ˆ ∇χ) =

d

X

i,j=1

( 1

Γ(1−α) Z t

0

(t−τ)^−αb_ij(x;τ, τ)∂ρ(τ)ˆ

∂x_i dτ, ∂χ

∂x_j) +

d

X

i,j=1

( 1

Γ(1−α) Z t

0

(t−τ)^−αZ τ 0

∂b_ij

∂τ (x;τ, s)∂ρ(s)ˆ

∂xi

ds

dτ, ∂χ

∂xj

)

+

d

X

i=1

( 1

Γ(1−α) Z t

0

(t−τ)^−αb_i(x;τ, τ)∂ρ(τ)ˆ

∂x_i dτ, χ) +

d

X

i=1

( 1

Γ(1−α) Z t

0

(t−τ)^−αZ τ 0

∂bi

∂τ(x;τ, s)∂ρ(s)ˆ

∂x_i ds dτ, χ)

+ ( 1

Γ(1−α) Z t

0

(t−τ)^−αb₀(x;τ, τ) ˆρ(τ)dτ, χ)

+ ( 1

Γ(1−α) Z t

0

(t−τ)^−αZ τ 0

∂b₀

∂τ (x;τ, s) ˆρ(s)ds dτ, χ)

=:J₁+J₂+J₃+J₄+J₅ +J₆.

It is easy to check by the Cauchy-Schwarz inequality that kJ₁k ≤Ch ku₀k₂+kfk_C1(L²(Ω))

k∇χk, kJ₂k ≤Ch ku₀k₂+kfk_C1(L²(Ω))

k∇χk.

Since χ∈S_h, by the Cauchy-Schwarz inequality and the Poincar´e inequality we have kJ₃k+kJ₄k ≤Ch ku₀k₂+kfk_C1(L²(Ω))

k∇χk, and

kJ₅k+kJ₆k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

k∇χk. Hence, by (3.33), for any χ∈S_h we have

(3.34) (∇∂_t^αρ(t),ˆ ∇χ)≤Ch ku0k₂+kfk_C1(L²(Ω))

k∇χk. We now split the term ∂_t^αρ(t) asˆ

∂_t^αρ(t) = (∂ˆ _t^αu(t)−∂_t^αR_hu(t)) + (∂_t^αR_hu(t)−∂_t^αV_hu(t)) =: ˆρ₁+ ˆθ₁. By (3.34) and the Cauchy-Schwarz inequality, we obtain

k∇∂_t^αρ(t)kˆ ² = (∇∂_t^αρ(t),ˆ ∇∂_t^αρ(t))ˆ

= (∇∂_t^αρ(t),ˆ ∇θˆ₁(t)) + (∇∂_t^αρ(t),ˆ ∇ˆρ₁(t))

≤Ch ku₀k₂+kfk_C1(L²(Ω))

k∇θˆ₁k+k∇∂_t^αρ(t)k k∇ˆˆ ρ₁(t)k. (3.35)

CHAPTER 3. Semidiscretization 48 Note that,

k∇ρˆ₁(t)k=k∇(∂_t^αu(t)−R_h∂_t^αu(t))k ≤Ch t^−α[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

. Since k∇θˆ₁(t)k ≤ k∇ˆρ₁(t)k+k∇∂_t^αρ(t)k, an application of Young’s inequality to (3.35)ˆ yields

k∇∂_t^αρ(t)kˆ ² ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

2

+Ch² t^−α[ku₀k₂+kf(0)k]

+kfk_C1(L²(Ω))

2

+Ch² t^−α[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

2

≤Ch² t^−α[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

2

,

from which the desired estimate for k∇∂_t^αρ(t)kˆ follows immediately. Next, to estimate

∂_t^αρ(t) in theˆ L²(Ω)-norm, we proceed by duality argument. For ϕ ∈ L²(Ω), let g ∈ H˙²(Ω) be the solution of the auxiliary problem

Ag=ϕ in Ω, g = 0 on∂Ω,

satisfying the regularity estimate kgk₂ ≤Ckϕk. For g_h ∈S_h, we have

(3.36)

(∂_t^αρ(t), ϕ) = (∇∂ˆ _t^αρ(t),ˆ ∇g)

= (∇∂_t^αρ(t),ˆ ∇(g−g_h) + (∇∂_t^αρ(t),ˆ ∇g_h)

=:L₁+L₂.

With g_h = R_hg, the Ritz projector of g, we apply the Cauchy-Schwarz inequality and the gradient norm estimate of ∂^α_tρ(t) to obtainˆ

(3.37) kL₁k ≤Ch² t^−α[ku₀k₂ +kf(0)k] +kfk_C1(L²(Ω))

kϕk. In view of (3.33), the term L₂ can be expressed as

L₂ =J₁+J₂+J₃+J₄+J₅+J₆,

CHAPTER 3. Semidiscretization 49 where

(3.38)

J₁ =

d

X

i,j=1

( 1

Γ(1−α) Z t

0

(t−τ)^−αb_ij(x;τ, τ)∂ρ(τ)

∂xi

dτ,∂g_h

∂xj

)

=

d

X

i,j=1

1 Γ(1−α)

Z t 0

(t−τ)^−α(bij(x;τ, τ)∂ρ(τ)

∂x_i ,∂(g_h−g)

∂x_j )dτ

−

d

X

i,j=1

1 Γ(1−α)

Z t 0

(t−τ)^−α(ρ(τ), ∂

∂x_i(b_ij(x;τ, τ) ∂g

∂x_j))dτ

=:J₁¹+J₁².

Since g_h =R_hg, an easy calculation shows that

kJ₁¹k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

kϕk, kJ₁²k ≤Ch² ku0k₂+kfk_C1(L²(Ω))

kϕk. Using (3.38), it now follows that

kJ₁k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

kϕk. Analogous to J1, we can show that

kJ₂k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

kϕk. Similarly, we split J_i as J_i =J_i¹+J_i² for i= 3,4,5,6. Using the facts

kg_h−gk ≤Ch²kgk₂ ≤Ch²kϕk, k∇gk ≤Ckgk₂, and kgk ≤Ckgk₂, we have for i= 3,4,5,6,

kJik ≤ kJ_i¹k+kJ_i²k ≤Ch² ku0k₂+kfk_C1(L²(Ω))

kϕk. Thus,

(3.39) kL₂k ≤

6

X

i=1

kJ_ik ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

kϕk. Utilizing (3.37), (3.39), and (3.36), it now leads to

(3.40) k∂_t^αρ(t)k ≤ˆ Ch² t^−α[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

, and this completes the proof.

CHAPTER 3. Semidiscretization 50 We are now ready to bound ˆθ(t) and its gradient in the L²(Ω) norms. It is easy to verify that ˆθ satisfies the relation

∂_t^αθ(t) +ˆ A_hθ(t) =ˆ −P_h∂_t^αρ(t) + ˜ˆ B_hθ(t).ˆ (3.41)

With u_0h = V_hu₀, we have ˆθ(0) = 0. Then the solution ˆθ(t) for the above equation is represented by

θ(t) =ˆ Z t

0

E^h(t−τ)(−Ph∂_τ^αρ(τˆ ))dτ + Z t

0

E^h(t−τ) ˜Bhθ(τ)ˆ dτ

= :I₄+I₅.

Using (3.9) for p= 0 =q, the L²-stability of P_h, and Lemma 3.2.2 we obtain kI₄k ≤C

Z t 0

(t−τ)^−1+αk∂_τ^αρ(τˆ )kdτ

≤Ch²[ku₀k₂+kf(0)k]

Z t 0

(t−τ)^−1+ατ^−αdτ +Ch²kfk_C1(L²(Ω))

Z t 0

(t−τ)^−1+αdτ

≤Ch² ku₀k₂+kfk_C1(L²(Ω))

. (3.42)

Similarly, taking p= 1, q = 0 in (3.9), using theL²-stability of P_h and Lemma 3.2.2, it follows that

k∇I₄k ≤C Z t

0

(t−τ)^α/2−1k∂_τ^αρ(τˆ )k dτ

≤Ch²[ku0k₂+kf(0)k]

Z t 0

(t−τ)^α/2−1τ^−αdτ +Ch²kfk_C1(L²(Ω))

Z t 0

(t−τ)^−1+α/2dτ

≤Ch² t^−α/2[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

. (3.43)

The bounds for I₅ and its gradient in the L²(Ω)-norms are analogous to those of the Ritz projection and hence, they are obtained as

(3.44) kI₅k ≤C

Z t 0

(t−s)^αkθ(s)kˆ ds and

(3.45) k∇I5k ≤C

Z t 0

(t−s)^α/2kθ(s)kˆ ds.

CHAPTER 3. Semidiscretization 51 To bound ˆθ(t) in the L²(Ω)-norm, we take the help of (3.42), (3.44) and Gronwall’s lemma to obtain

(3.46) kθ(t)k ≤ˆ Ch² ku₀k₂ +kfk_C1(L²(Ω))

,

and for the gradient norm estimate, we use inverse estimate (3.10) and (3.46) to obtain k∇θ(t)k ≤ˆ Ch⁻¹kθ(t)k ≤ˆ Ch ku₀k₂+kfk_C1(L²(Ω))

. (3.47)

Altogether these estimates lead to the following theorem.

Theorem 3.2.2. Let u be the solution of (3.1) with f ∈ C¹(L²(Ω)) and u₀ ∈ H˙²(Ω).

Further, let u_h be the solution of the discrete problem (3.2) with u_0h =V_hu₀ and f_h = P_hf. Then there exists a positive constant C such that

ku(t)−u_h(t)k+hk∇(u(t)−u_h(t))k ≤Ch² ku₀k₂+kfk_C1(L²(Ω))

, 0≤t≤T.

Remark 3.2.2. (i)The main advantage of using the Ritz-Volterra projection as a com- parison function is that the term Rt

0 B(t, s) ˆρ(s)ds does not appear in the error equation of θ(t)ˆ (cf. (3.41)) in contrast to that of the Ritz projection (cf. (3.18)), which greatly simplifies the analysis.

(ii) The gradient norm error estimate for θˆin (3.47) can be improved toO(h²) with an additional factor t^−α/2 by utilizing (3.43), (3.45), (3.46), and we obtain

k∇θ(t)k ≤ˆ Ch² t^−α/2[ku₀k₂+kf(0)k] +kfk_C1(L²(Ω))

.