5. Spectral Methods: Notation and Preliminary results 39
6.7. Existence and Uniqueness of the Numerical Solutions
The above inequality is indeed verified at every time t ∈ (0, T). Thus there holds the uniform bound
sup
t∈(0,T)
kR(t)kH−1(Ω)≤Clog(N)1/2
N + (∆t)1/2log(N)
. (6.28)
Note that this estimate only holds under the assumption that the initial discrete energy is a bounded quantity.
6.7. Existence and Uniqueness of the Numerical Solutions
We present in this section an existence theorem for the solutions(unN)Jn=0of the numerical scheme. More precisely, we shall show, using a fixed point argument, that, for each n ∈ {0,1, . . . , J −1} and unN ∈ XN given, there exists a unique solution un+1N ∈ XN to (6.1). On writingu=un+1N andv=unN, Equation (6.1) can be rewritten in the following equivalent form: given v∈XN, findu∈XN such that
hu, φiN +∆t
2 hu,LεφiN =hv, φiN−∆t
2 hv,LεφiN + ∆thNε(u, v), φiN (6.29) for all φ∈XN. On defining the adapted inner product
hu, vi∆t:=hu, viN +∆t
2 hLεu, viN with associated time-increment dependent norm
kvk2∆t:=hv, vi∆t=kvk2N +∆t 2 kvk2L
ε, Equation (6.29) can conveniently be written as follows:
hu, φi∆t=hv, φi∆t+hNε(u, v)− Lεv, φiN for allφ∈XN. (6.30) Note that the inner-product h·,·i∆t = h·,·i∆t,ε depends on the parameterε too, but we shall drop this dependence in the notation for more simplicity. Motivated by the form of the previous equation, we define, for a given v ∈XN, the mappingTv : XN 7→ XN such that foru∈XN,U =Tvu satisfies
hU, φi∆t=hv, φi∆t+hNε(u, v)− Lεv, φiN for allφ∈XN. (6.31) By virtue of the Riesz representation theorem, the operator Tv is well defined. The task of showing the existence of a unique un+1N ∈ XN, given unN ∈ XN, is then equivalent to showing the existence of a unique fixed pointuofTv forv:=unN given, and then defining un+1N :=u.
In order to prove existence and uniqueness of the numerical solution we shall proceed
6.7. Existence and Uniqueness of the Numerical Solutions
in two steps: considering v ∈ XN to be fixed, we first show that the mapping Tv maps a specific ball centered in v into itself. Then we show that under some appropriate rate conditions between the time step ∆t and the gridsize N, this mapping is contractive.
Based on these two statements and invoking finally a fixed point argument, we infer the existence and uniqueness result.
Bound on Tvv−v
Lemma 6.9. Let v∈XN. There exists a constantC >0 such that kTvv−vk2∆t≤C (∆t)2N2(EN(v))2+ ∆tEN(v)
.
Corollary 6.10. Suppose EN(v) <∞. Then there exists a constantα > 0 such that if
∆tN < α, there holds
kTvv−vk∆t≤ 1 4.
Proof. Letv ∈XN. On setting u=v (and hence U =Tvv) and φ=Tvv−v in equation (6.31), we obtain
hTvv, Tvv−vi∆t=hv, Tvv−vi∆t+ ∆thNε(v)− Lεv, Tvv−viN, and hence
kTvv−vk2∆t= ∆thNε(v)− Lεv, Tvv−viN.
On separating the terms appearing on the right-hand side of the above equation, there holds in particular
kTvv−vk2∆t≤∆t|hNε(v), Tvv−viN|+ ∆t|hLεv, Tvv−viN|.
By virtue of Young’s inequality, there further holds
|hNε(v), Tvv−viN| ≤ 1
2∆tkTvv−vk2N +∆t
2 kNε(v)k2N, as well as
|hLεv, Tvv−viN| ≤ 1
4kTvv−vk2L
ε +kvk2L
ε, so that, on merging both estimates together, we obtain
kTvv−vk2∆t≤(∆t)2kNε(v)k2N + 2∆tkvkLε.
The second term of the right-hand side of the above inequality can be estimated by the associated discrete energy. Indeed, there holds trivially
kvkLε ≤CEN(v).
Concerning the first term, we have kNε(v)k2N = c20
ε2 1 N2
X
j∈N2N
v2(xj)(1−v2(xj))2≤C(ε, c0)kvk2L∞(Ω)EN(v).
On using the estimate of Lemma 5.1, we obtain further kNε(v)k2N ≤CN2kvk2L2(Ω)EN(v).
The term kvk2L2(Ω) can also easily be bounded by the discrete energy. Indeed, by virtue of Parseval’s theorem, we have
kvk2L2(Ω)= X
k∈Z2N
|ˆv(k)|2 ≤CEN(v),
so that finally we obtain the bound on Tvv−v
kTvv−vk2∆t≤C (∆t)2N2(EN(v))2+ ∆tEN(v) .
On taking ∆t.1/N withN sufficiently large, we obtain the result of the corollary.
Bound on Tvu−Tvv
Lemma 6.11. Let v ∈XN. For all u∈XN, we have
kTvu−Tvvk∆t≤C∆t 1 +N2ku−vk2∆t+N2EN(v)
ku−vk∆t.
Corollary 6.12. Suppose EN(v) <∞. Then there exists a constant β1 >0 such that if
∆tN2< β1, then there holds kTvu−Tvvk∆t≤ 1
4+ 1
4ku−vk3∆t+ 1
4ku−vk∆t for all u∈XN.
Proof. Let v ∈ XN. By Young’s inequality, setting φ = U −U0 and substracting the identities (6.31) obtained respectively for U =Tvuand U0 =Tvv yields
kTvu−Tvvk2∆t≤(∆t)2kNε(u, v)− Nε(v)k2N. Using the estimate of Corollary 6.7, we obtain
kNε(u, v)− Nε(v)k2N ≤C 1 +kuk2L∞(Ω)+kvk2L∞(Ω)
2
ku−vk2N
≤C 1 +ku−vk2L∞(Ω)+kvk2L∞(Ω)
2
ku−vk2N.
We shall now estimate the quantities of the above inequality. Taking the estimate of Lemma 5.1, we first have
ku−vk2L∞(Ω)≤N2ku−vk2L2(Ω)≤N2ku−vk2∆t,
while, as mentioned previously, there holdskvk2L∞(Ω)≤CN2EN(v). Merging the different estimates together, we finally obtain
kTvu−Tvvk∆t≤C∆t 1 +N2ku−vk2∆t+N2EN(v)
ku−vk∆t.
6.7. Existence and Uniqueness of the Numerical Solutions
On taking∆t.1/N2 withN sufficiently large, we have kTvu−Tvvk∆t≤ 1
4+1
4ku−vk3∆t+1
4ku−vk∆t.
Bound on Tvu−v
Proposition 6.13. Let v ∈XN with EN(v) <∞. There exists a constant β2 >0 such that for ∆tN2< β2, there holds
kTvu−vk∆t≤ 1 2 +1
4ku−vk3∆t+1
4ku−vk∆t for all u∈XN.
Proof. Using the triangle inequality and the results of Corollary 6.10 and Corollary 6.12 yields the result.
On defining
B∆tr (v) :={u∈XN :ku−vk∆t< r}
the ball of radiusr >0centered atv in(XN,k · k∆t), the previous proposition shows that Tv maps the ballB∆t1 into itself. More precisely we infer:
Corollary 6.14. Consider the sequence(uk)∞k=0⊂XN whereuk+1 :=Tvuk with u0:=v.
Then for ∆tN2 < β2, we have
uk∈B∆t1 (v) for all k∈N.
Proof. Trivially, we have u0 := v ∈ B∆t1 (v). Let nowuk ∈ B∆t1 (v) for given k ∈ N. On using the estimate of Proposition 6.13, we obtain
kuk+1−vk∆t=kTvuk−vk∆t≤ 1 2 +1
4kuk−vk3∆t+ 1
4kuk−vk∆t<1, and hence uk+1 ∈B∆t1 (v).
Contraction Property of Tv
We shall now prove that the mapping Tv is a contraction; we infer the following proposi-tion:
Proposition 6.15. Let v ∈XN with EN(v) <∞. There exists a constant β3 >0 such that for ∆tN2< β3, we have
kU−U0kN ≤
kU −U0k∆t≤ 1
2ku−u0kN
≤ 1
2ku−u0k∆t for all u, u0 ∈B∆t1 (v), where U =Tvu andU0 =Tvu0.
Proof. Letv∈XN. We consideru andu0 ∈B∆t1 (v). As previously, on settingφ=U−U0 and substracting the inequalities (6.31) obtained respectively for U =Tvu and forU0 =
Tvu0 and then applying Young’s inequality, we obtain
kU−U0k2∆t≤(∆t)2kNε(u, v)− Nε(u0, v)k2N. On using again the estimate of Corollary 6.7, there holds
kNε(u, v)− Nε(u0, v)k2N ≤C 1 +N2ku−vk2∆t+N2ku0−vk2∆t+N2EN(v)2
ku−u0k2N. Since u andu0∈B∆t1 , we obtain
kU−U0k2∆t≤C ∆t+ ∆tN2+ ∆tN2EN(v)2
ku−u0k2N. Consequently, taking ∆t.1/N2 andN sufficiently large, we have
kU −U0k2∆t≤ 1
2ku−u0k2N.
Existence Theorem
On settingβ = min(β1, β2, β3), using the results of Proposition 6.13 and Proposition 6.15 as well as the Banach fixed point theorem, we infer the following existence theorem.
Theorem 6.16. Let v∈XN with EN(v)<∞. There exists a constantβ depending only on v and the parameter ε such that for ∆tN2 < β, the mapping Tv has a unique fixed point u∈XN.
From a more constructive point of view, let us consider the sequence (uk)∞k=0 ⊂ XN, whereuk+1:=Tvuk withu0:=v. Then for` > k+ 1we have
ku`−uk+1k∆t≤
`−1
X
i=k+1
kui+1−uik∆t.
On using the bound of Proposition 6.15, there holds for ∆tN2< β ku`−uk+1k∆t≤
1 2
k
− 1
2 `−1!
ku1−u0k∆t
for all ` > k+ 1. Thus (uk)∞k=1 is a Cauchy sequence in the finite-dimensional normed linear space XN, with respect to the norm k · k∆t. Hence, by completeness of XN with respect to the norm k · k∆t, the sequence(uk)∞k=0 ⊂XN, whereuk+1 :=Tvuk with initial valueu0 :=v, converges inL2(Ω)to a fixed pointu∈XN for anyv∈XN. On passing to the limit `→ ∞in the above inequality, we obtain that
ku−uk+1k∆t≤2−kku1−vk∆t
On using the bound on the initial stepu1−u0given in Lemma 6.9 and provided∆tN2 < β, we deduce that there exists a constantC >0 such that
ku−uk+1k2∆t≤4−kC∆t (EN(v))2+EN(v)
for anyk≥0.