This document summarizes a presentation on rigorously verifying the accuracy of numerical solutions to semi-linear parabolic partial differential equations using analytic semigroups. It introduces the considered problem of finding the solution to a semi-linear parabolic PDE. It then discusses using a piecewise linear finite element discretization in space and time to obtain an initial numerical solution. The goal is to rigorously enclose the true solution within a radius ρ of this numerical solution in the function space L∞(J;H10(Ω)). Key steps involve using properties of the analytic semigroup generated by the operator A and estimating discretization errors to compute the enclosure radius ρ.
5. 半線形放物型方程式
Let Ω be a bounded polygonal domain in R2
.
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(t0, x) = u0(x) in Ω,
▶ J := (t0, t1], 0 ≤ t0 < t1 < ∞ or J := (0, ∞),
▶ f : twice differentiable nonlinear mapping,
▶ u = 0 on ∂Ω is the trace sense,
▶ u0 ∈ H1
0 (Ω).
Lp
(Ω): the set of Lp
-functions,
H1
(Ω): the first order Sobolev space of L2
(Ω),
H1
0 (Ω) := {v ∈ H1
(Ω) : v = 0 on ∂Ω}.
5/55
6. 記号
A : D(A) ⊂ H1
0 (Ω) → L2
(Ω) is defined by
A := −
∑
1≤i,j≤2
∂
∂xj
(
aij(x)
∂
∂xi
)
,
where aij(x) = aji(x) is in W1,∞
(Ω) and satisfies
∑
1≤i,j≤2
aij(x)ξiξj ≥ µ|ξ|2
, ∀x ∈ Ω, ∀ξ ∈ R2
with µ > 0.
6/55
7. 記号
We endow L2
(Ω) with the inner product:
(u, v)L2 :=
∫
Ω
u(x)v(x)dx.
Use the usual norms:
∥u∥L2 :=
√
(u, u)L2 , ∥u∥H1
0
:= ∥∇u∥L2 ,
and
∥u∥H−1 := sup
0̸=v∈H1
0 (Ω)
∥v∥H1
0
=1
|⟨u, v⟩| ,
where ⟨·, ·⟩ is a dual product between H1
0 (Ω) and H−1
(Ω)1
.
1
The topological dual space of H1
0 (Ω).
7/55
8. 精度保証付き数値計算
(PJ ) の弱解: For t ∈ J, u(t) := u(t, ·) ∈ H1
0 (Ω) with the
initial function u0 such that
(∂tu(t), v)L2 + a(u(t), v) = (f(u(t)), v)L2 , ∀v ∈ H1
0 (Ω),
where a : H1
0 (Ω) × H1
0 (Ω) → R is a bilinear form:
a(u, v) :=
∑
1≤i,j≤2
(
aij(x)
∂u
∂xi
,
∂v
∂xj
)
L2
satisfying
a(u, u) ≥ µ∥u∥2
H1
0
, ∀u ∈ H1
0 (Ω),
|a(u, v)| ≤ M∥u∥H1
0
∥v∥H1
0
, ∀u, v ∈ H1
0 (Ω).
8/55
9. 精度保証付き数値計算
(PJ ) の弱解: For t ∈ J, u(t) := u(t, ·) ∈ H1
0 (Ω) with the
initial function u0 such that
(∂tu(t), v)L2 + a(u(t), v) = (f(u(t)), v)L2 , ∀v ∈ H1
0 (Ω),
where a : H1
0 (Ω) × H1
0 (Ω) → R is a bilinear form:
a(u, v) :=
∑
1≤i,j≤2
(
aij(x)
∂u
∂xi
,
∂v
∂xj
)
L2
satisfying
a(u, u) ≥ µ∥u∥2
H1
0
, ∀u ∈ H1
0 (Ω),
|a(u, v)| ≤ M∥u∥H1
0
∥v∥H1
0
, ∀u, v ∈ H1
0 (Ω).
8/55
10. 精度保証付き数値計算
(PJ ) の弱解: For t ∈ J, u(t) := u(t, ·) ∈ H1
0 (Ω) with the
initial function u0 such that
(∂tu(t), v)L2 + a(u(t), v) = (f(u(t)), v)L2 , ∀v ∈ H1
0 (Ω),
where a : H1
0 (Ω) × H1
0 (Ω) → R is a bilinear form:
a(u, v) :=
∑
1≤i,j≤2
(
aij(x)
∂u
∂xi
,
∂v
∂xj
)
L2
satisfying
a(u, u) ≥ µ∥u∥2
H1
0
, ∀u ∈ H1
0 (Ω),
|a(u, v)| ≤ M∥u∥H1
0
∥v∥H1
0
, ∀u, v ∈ H1
0 (Ω).
8/55
12. 精度保証付き数値計算に関する先行研究
M.T. Nakao, T. Kinoshita and T. Kimura,
“On a posteriori estimates of inverse operators for linear
parabolic initial-boundary value problems”, Computing
94(2-4), 151–162, 2012.
M.T. Nakao, T. Kimura and T. Kinoshita,
“Constructive A Priori Error Estimates for a Full Discrete
Approximation of the Heat Equation”, Siam J. Numer. Anal.,
51(3), 1525–1541, 2013.
T. Kinoshita, T. Kimura and M.T. Nakao,
“On the a posteriori estimates for inverse operators of linear
parabolic equations with applications to the numerical
enclosure of solutions for nonlinear problems”, Numer. Math,
Online First, 2013.
10/55
13. 力学系の観点からみた研究
P. Zgliczy´nski and K. Mischaikow,
“Rigorous Numerics for Partial Differential Equations: The
Kuramoto―Sivashinsky Equation”, Foundations of
Computational Mathematics, 1(3), 1615–3375, 2001.
P. Zgliczy´nski,
“Rigorous numerics for dissipative PDEs III. An effective
algorithm for rigorous integration of dissipative PDEs”, Topol.
Methods Nonlinear Anal., 36, 197–262, 2010.
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14. 離散半群を用いた数値スキームの研究
H. Fujita,
“On the semi-discrete finite element approximation for the
evolution equation ut + A(t)u = 0 of parabolic type”, Topics
in numerical analysis III, Academic Press, 143–157, 1977.
H. Fujita and A. Mizutani,
“On the finite element method for parabolic equations, I;
approximation of holomorphic semi-groups”, J. Math. Soc.
Japan, 28, 749–771, 1976.
H. Fujita, N. Saito and T. Suzuki,
“Operator theory and numerical methods”, Elsevier(Holland),
308pages, 2001.
12/55
16. Considered problem
J := (t0, t1] : arbitrary time interval. τ := t1 − t0.
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(t0, x) = u0(x) in Ω,
where u0 is an initial function in H1
0 (Ω).
Vh ⊂ H1
0 (Ω) : a finite dimensional subspace.
Starts from: ˆu0, ˆu1 ∈ Vh
ω(t) = ˆu0ϕ0(t) + ˆu1ϕ1(t), t ∈ J,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
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17. Considered problem
J := (t0, t1] : arbitrary time interval. τ := t1 − t0.
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(t0, x) = u0(x) in Ω,
where u0 is an initial function in H1
0 (Ω).
Vh ⊂ H1
0 (Ω) : a finite dimensional subspace.
Starts from: ˆu0, ˆu1 ∈ Vh
ω(t) = ˆu0ϕ0(t) + ˆu1ϕ1(t), t ∈ J,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
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18. Considered problem
Let the initial function satisfy
∥u0 − ˆu0∥H1
0
≤ ε0.
We rigorously enclose the solution in a Banach space2
,
L∞
(
J; H1
0 (Ω)
)
:=
{
u(t) ∈ H1
0 (Ω) : ess sup
t∈J
∥u(t)∥H1
0
< ∞
}
.
Namely, we compute a radius ρ > 0 of the ball:
BJ (ω, ρ) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: ∥y − ω∥L∞
(J;H1
0 (Ω)) ≤ ρ
}
.
2
∥u∥L∞(J;H1
0 (Ω)) := ess supt∈J ∥u(t)∥H1
0
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19. Considered problem
Let the initial function satisfy
∥u0 − ˆu0∥H1
0
≤ ε0.
We rigorously enclose the solution in a Banach space2
,
L∞
(
J; H1
0 (Ω)
)
:=
{
u(t) ∈ H1
0 (Ω) : ess sup
t∈J
∥u(t)∥H1
0
< ∞
}
.
Namely, we compute a radius ρ > 0 of the ball:
BJ (ω, ρ) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: ∥y − ω∥L∞
(J;H1
0 (Ω)) ≤ ρ
}
.
2
∥u∥L∞(J;H1
0 (Ω)) := ess supt∈J ∥u(t)∥H1
0
15/55
20. Considered problem
Let the initial function satisfy
∥u0 − ˆu0∥H1
0
≤ ε0.
We rigorously enclose the solution in a Banach space2
,
L∞
(
J; H1
0 (Ω)
)
:=
{
u(t) ∈ H1
0 (Ω) : ess sup
t∈J
∥u(t)∥H1
0
< ∞
}
.
Namely, we compute a radius ρ > 0 of the ball:
BJ (ω, ρ) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: ∥y − ω∥L∞
(J;H1
0 (Ω)) ≤ ρ
}
.
2
∥u∥L∞(J;H1
0 (Ω)) := ess supt∈J ∥u(t)∥H1
0
15/55
21. Analytic semigroup
The weak form of A, which is denoted by −A3
, generates the
analytic semigroup {e−tA
}t≥0 over H−1
(Ω). The following
abstract problem has an unique solution:
∂tu + Au = 0, u(0, x) = u0 =⇒ ∃u = e−tA
u0.
Fact
Let x ∈ D(A) and λ0 be a positive number. A satisfies
⟨−Ax, x⟩ ≤ 0, R(λ0I + A) = H−1
(Ω).
Then, there exists an analytic semigroup {e−tA
}t≥0 generated by −A.
Proofs are found in several textbooks.
3
A : H1
0 (Ω) → H−1
(Ω) s.t. ⟨Au, v⟩ := a(u, v), ∀v ∈ H1
0 (Ω).
16/55
22. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
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23. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
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24. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
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25. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
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26. Theorem
Assume that the initial function u0 satisfies ∥u0 − ˆu0∥H1
0
≤ ε0;
Assume that ω satisfies the following estimate:
∫ t
t0
e−(t−s)A
(∂tω(s) + Aω(s) − f(ω(s)))ds
L∞
(J;H1
0 (Ω))
≤ δ.
Assume that, for ∀ρ0 ∈ (0, ρ] with a certain ρ 0, f satisfies
∥f(φ) − f(ψ)∥L∞(J;L2(Ω)) ≤ Lρ0 ∥φ − ψ∥L∞
(J;H1
0 (Ω)),
where ∀φ, ψ ∈ BJ (ω, ρ0) ⊂ L∞
(J; H1
0 (Ω)). If
M
µ
ε0 +
2
µ
√
Mτ
e
Lρρ + δ ρ,
then the weak solution u(t), t ∈ J of (PJ ) uniquely exists in
the ball BJ (ω, ρ).
17/55
27. Sketch of proof
Let z(t) ∈ H1
0 (Ω) for t ∈ J. We put u(t) = ω(t) + z(t).
For any v ∈ H1
0 (Ω),
(∂tz(t), v)L2 + a(z(t), v)
= (f(u(t)), v)L2 − ((∂tω(t), v)L2 + ⟨Aω(t), v⟩)
=: ⟨g(z(t)), v⟩ ,
where g(z(t)) = f(u(t)) − (∂tω(t) + Aω(t)). Note that by
the definition of the natural embedding L2
(Ω) → H−1
(Ω),
(ψ, v)L2 = ⟨ψ, v⟩ holds for ψ ∈ L2
(Ω).
Define S : L∞
(J; H1
0 (Ω)) → L∞
(J; H1
0 (Ω)) using the
analytic semigroup e−tA
as
S(z) := e−(t−t0)A
(u0 − ˆu0) +
∫ t
t0
e−(t−s)A
g(z(s))ds.
18/55
28. Sketch of proof
Let z(t) ∈ H1
0 (Ω) for t ∈ J. We put u(t) = ω(t) + z(t).
For any v ∈ H1
0 (Ω),
(∂tz(t), v)L2 + a(z(t), v)
= (f(u(t)), v)L2 − ((∂tω(t), v)L2 + ⟨Aω(t), v⟩)
=: ⟨g(z(t)), v⟩ ,
where g(z(t)) = f(u(t)) − (∂tω(t) + Aω(t)). Note that by
the definition of the natural embedding L2
(Ω) → H−1
(Ω),
(ψ, v)L2 = ⟨ψ, v⟩ holds for ψ ∈ L2
(Ω).
Define S : L∞
(J; H1
0 (Ω)) → L∞
(J; H1
0 (Ω)) using the
analytic semigroup e−tA
as
S(z) := e−(t−t0)A
(u0 − ˆu0) +
∫ t
t0
e−(t−s)A
g(z(s))ds.
18/55
29. Sketch of proof
For ρ 0, Z := {z : ∥z∥L∞
(J;H1
0 (Ω)) ≤ ρ} ⊂ L∞
(J; H1
0 (Ω)).
On the basis of Banach’s fixed-point theorem, we show a
sufficient condition of S having a fixed-point in Z.
S(Z) ⊂ Z Since the analytic semigroup e−tA
is bounded,
the first term of S(z) is estimated4
by
e−(t−t0)A
(ζ − ˆu0) H1
0
≤ µ−1
A e−(t−t0)A
(ζ − ˆu0) H−1
≤
M
µ
e−(t−t0)λmin
ε0.
Then
e−(t−t0)A
(ζ − ˆu0) L∞(J;H1
0 (Ω))
≤
M
µ
ε0.
4
µ∥u∥H1
0
≤ ∥Au∥H−1 ≤ M∥u∥H1
0
is used.
19/55
30. Sketch of proof
For ρ 0, Z := {z : ∥z∥L∞
(J;H1
0 (Ω)) ≤ ρ} ⊂ L∞
(J; H1
0 (Ω)).
On the basis of Banach’s fixed-point theorem, we show a
sufficient condition of S having a fixed-point in Z.
S(Z) ⊂ Z Since the analytic semigroup e−tA
is bounded,
the first term of S(z) is estimated4
by
e−(t−t0)A
(ζ − ˆu0) H1
0
≤ µ−1
A e−(t−t0)A
(ζ − ˆu0) H−1
≤
M
µ
e−(t−t0)λmin
ε0.
Then
e−(t−t0)A
(ζ − ˆu0) L∞(J;H1
0 (Ω))
≤
M
µ
ε0.
4
µ∥u∥H1
0
≤ ∥Au∥H−1 ≤ M∥u∥H1
0
is used.
19/55
31. Sketch of proof
For ρ 0, Z := {z : ∥z∥L∞
(J;H1
0 (Ω)) ≤ ρ} ⊂ L∞
(J; H1
0 (Ω)).
On the basis of Banach’s fixed-point theorem, we show a
sufficient condition of S having a fixed-point in Z.
S(Z) ⊂ Z Since the analytic semigroup e−tA
is bounded,
the first term of S(z) is estimated4
by
e−(t−t0)A
(ζ − ˆu0) H1
0
≤ µ−1
A e−(t−t0)A
(ζ − ˆu0) H−1
≤
M
µ
e−(t−t0)λmin
ε0.
Then
e−(t−t0)A
(ζ − ˆu0) L∞(J;H1
0 (Ω))
≤
M
µ
ε0.
4
µ∥u∥H1
0
≤ ∥Au∥H−1 ≤ M∥u∥H1
0
is used.
19/55
35. Sketch of proof
The term of g1(s):
∫ t
t0
e−(t−s)A
g1(s)ds
H1
0
=
∫ t
t0
e−(t−s)A
(f(ω(s) + z(s)) − f(ω(s)))ds
H1
0
≤ µ−1
∫ t
t0
A e−(t−s)A
(f(ω(s) + z(s)) − f(ω(s)))
H−1
ds
= µ−1
∫ t
t0
A
1
2 e−(t−s)A
A
1
2 (f(ω(s) + z(s)) − f(ω(s)))
H−1
ds
≤ µ−1
e− 1
2
∫ t
t0
(t − s)− 1
2 e− 1
2 (t−s)λmin
A
1
2 (f(ω(s) + z(s)) − f(ω(s)))
H−1
≤ µ−1
M
1
2 e− 1
2
∫ t
t0
(t − s)− 1
2 e− 1
2 (t−s)λmin
∥f(ω(s) + z(s)) − f(ω(s))∥L2 ds
≤ µ−1
M
1
2 e− 1
2 ν(t) ∥f(ω + z) − f(ω)∥L∞(J;L2(Ω)).
21/55
36. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
37. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
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38. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
39. Sketch of proof
Then
∫ t
t0
e−(t−s)A
g1(s)ds
L∞
(J;H1
0 (Ω))
≤
2
µ
√
Mτ
e
Lρρ.
The term of g2(s) is nothing but the residual of the
approximate solution, which is estimated by δ.
Then it follows
∥S(z)∥L∞
(J;H1
0 (Ω)) ≤
M
µ
ε0 +
2
µ
√
Mτ
e
L(ρ)ρ + δ.
∥S(z)∥L∞
(J;H1
0 (Ω)) ρ holds from the condition of the
theorem. It implies that S(z) ∈ Z.
22/55
40. Sketch of proof
For any z1, z2 in Z,
S(z1) − S(z2) =
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
holds. We have
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
H1
0
≤ µ−1
M
1
2 e−1
2 ν(t)∥f(z1 + ω) − f(z2 + ω)∥L∞(J;L2(Ω)).
Here, zi + ω ∈ B(ω, ρ) (i = 1, 2) holds. Then
∥S(z1) − S(z2)∥L∞
(J;H1
0 (Ω)) ≤
2
µ
√
Mτ
e
Lρ∥z1 − z2∥L∞
(J;H1
0 (Ω)).
23/55
41. Sketch of proof
For any z1, z2 in Z,
S(z1) − S(z2) =
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
holds. We have
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
H1
0
≤ µ−1
M
1
2 e−1
2 ν(t)∥f(z1 + ω) − f(z2 + ω)∥L∞(J;L2(Ω)).
Here, zi + ω ∈ B(ω, ρ) (i = 1, 2) holds. Then
∥S(z1) − S(z2)∥L∞
(J;H1
0 (Ω)) ≤
2
µ
√
Mτ
e
Lρ∥z1 − z2∥L∞
(J;H1
0 (Ω)).
23/55
42. Sketch of proof
For any z1, z2 in Z,
S(z1) − S(z2) =
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
holds. We have
∫ t
t0
e−(t−s)A
(f(z1 + ω) − f(z2 + ω))ds
H1
0
≤ µ−1
M
1
2 e−1
2 ν(t)∥f(z1 + ω) − f(z2 + ω)∥L∞(J;L2(Ω)).
Here, zi + ω ∈ B(ω, ρ) (i = 1, 2) holds. Then
∥S(z1) − S(z2)∥L∞
(J;H1
0 (Ω)) ≤
2
µ
√
Mτ
e
Lρ∥z1 − z2∥L∞
(J;H1
0 (Ω)).
23/55
43. Sketch of proof
The condition of theorem also implies
2
µ
√
Mτ
e
L (ρ) 1.
Therefore, S is a contraction mapping. Banach’s fixed point
theorem yields that there uniquely exists a fixed-point in Z.
24/55
44. Theorem (A posteriori error estimate)
Assume that existence and local uniqueness of the weak
solution u(t), t ∈ J, is proved in BJ (ω, ρ). Assume also that
ω satisfies
∫ t1
t0
e−(t1−s)A
(∂tω(s) + Aω(s) − f(ω(s))) ds
H1
0
≤ ˜δ.
Then, the following a posteriori error estimate holds:
∥u(t1) − ˆu1∥H1
0
≤
M
µ
e−τλmin
ε0 +
2
µ
√
Mτ
e
Lρρ + ˜δ =: ε1.
25/55
45. On several intervals
For n ∈ N, 0 = t0 t1 · · · tn ∞.
Jk := (tk−1, tk], τk := tk − tk−1, and J =
∪
Jk. (k=1,2,...,n)
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
where u0 ∈ H1
0 (Ω) is a given initial function satisfies
∥u0 − ˆu0∥H1
0
≤ ε0.
26/55
46. Approximate solution (Backward Euler)
Find {uh
k}k≥0 ⊂ Vh such that
(
uh
k − uh
k−1
τ
, vh
)
L2
+ a(uh
k, vh)L2 = (f(uh
k), vh)L2
and
(
uh
0, vh
)
L2 = (u0, vh)L2 for ∀vh ∈ Vh. Numerically
compute each approximation ˆuk (≈ uh
k) ∈ Vh.
From the data ˆuk(≈ uh
k) ∈ Vh, we construct ω(t):
ω(t) :=
n∑
k=0
ˆukϕk(t), t ∈ T,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
27/55
47. Approximate solution (Backward Euler)
Find {uh
k}k≥0 ⊂ Vh such that
(
uh
k − uh
k−1
τ
, vh
)
L2
+ a(uh
k, vh)L2 = (f(uh
k), vh)L2
and
(
uh
0, vh
)
L2 = (u0, vh)L2 for ∀vh ∈ Vh. Numerically
compute each approximation ˆuk (≈ uh
k) ∈ Vh.
From the data ˆuk(≈ uh
k) ∈ Vh, we construct ω(t):
ω(t) :=
n∑
k=0
ˆukϕk(t), t ∈ T,
where ϕk(t) is a piecewise linear Lagrange basis: ϕk(tj) = δkj
(δkj is a Kronecker’s delta).
27/55
67. 大域解の存在に関する先行研究
S. Cai,
“A computer-assisted proof for the pattern formation on
reaction-diffusion systems”, 学位論文, Graduate School of
Mathematics, Kyushu University (2012) 71 pages.
▶ 反応拡散方程式のあるクラスの定常解に対する精度保
証付き数値計算法を示している．
▶ (t′
, ∞) で定常解まわりに大域的に存在する範囲を
L∞
(Ω) × L∞
(Ω) 上で生成された解析半群を用いて，計
算している．
40/55
68. Considered problem
Let Ω be a bounded polygonal domain in R2
and J := (0, ∞).
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
where A = −∆, u0 ∈ H1
0 (Ω) is an initial function, and
f : R → R is a twice Fr´echet differentiable nonlinear mapping.
41/55
69. Considered problem
Let Ω be a bounded polygonal domain in R2
and J := (0, ∞).
(PJ )
∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
where A = −∆, u0 ∈ H1
0 (Ω) is an initial function, and
f : R → R is a twice Fr´echet differentiable nonlinear mapping.
41/55
70. Aim of this part
Let Ω be a bounded polygonal domain in R2
.
(PG)
∂tu + Au = f(u) in (t′
, ∞) × Ω,
u(t, x) = 0 on (t′
, ∞) × ∂Ω,
u(t′
, x) = η in Ω,
where η ∈ H1
0 (Ω) satisfies ∥η − ˆun∥H1
0
≤ εn for a certain
εn 0.
We enclose a solution for t ∈ (t′
, ∞) in a neighborhood of a
stationary solution ϕ ∈ D(A) of (PJ ) such that
{
Aϕ = f(ϕ) in Ω,
ϕ = 0 on ∂Ω.
42/55
71. 記号
For ρ 0, v ∈ L∞
((t′
, ∞); H1
0 (Ω)), define a ball
B(v, ρ) :=
{
y ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥y − v∥L∞
((t′,∞);H1
0 (Ω)) ≤ ρ
}
.
The Fr´echet derivative of f at w is denoted by
f′
[w] : L∞
((t′
, ∞); H1
0 (Ω)) → L∞
((t′
, ∞); L2
(Ω)).
For y ∈ B(v, ρ), we assume that there exists a non-decreasing
function L : R → R such that
∥f′
[y]u∥L∞(J;L2(Ω)) ≤ L(ρ)∥u∥H1
0
, u ∈ H1
0 (Ω).
43/55
72. 記号
Define a function space Xλ: for a fixed λ 0,
Xλ :=
{
u ∈ L∞
((t′
, ∞); H1
0 (Ω)) : ess sup
t∈(t′,∞)
e(t−t′)λ
∥u(t)∥H1
0
∞
}
which becomes a Banach space with the norm
∥ · ∥Xλ
:= ess sup
t∈(t′,∞)
e(t−t′)λ
∥u(t)∥H1
0
.
44/55
73. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
74. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
75. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
76. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
77. Theorem (Global existence)
Assume that
▶ a solution of (P) is enclosed until t′
0,
▶ a stationary solution ϕ ∈ D(A) uniquely exists around a
numerical solution ˆϕ,
▶ For a ˆun ∈ Vh, εn 0, the initial function satisfies
∥η − ˆun∥H1
0
εn.
For a fixed λ satisfying 0 ≤ λ λmin/2, if ρ 0 satisfies
∥η − ϕ∥H1
0
+ L(ρ)ρ
√
2π
e(λmin − 2λ)
ρ.
Then a solution u(t) for t ∈ (t′
, ∞) uniquely exists in
Uϕ :=
{
u ∈ L∞
(
(t′
, ∞); H1
0 (Ω)
)
: ∥u − ϕ∥Xλ
≤ ρ
}
.
45/55
78. Sketch of proof
Let z ∈ Uϕ. A nonlinear operator
S : L∞
((t′
, ∞); H1
0 (Ω)) → L∞
((t′
, ∞); H1
0 (Ω)) is defined by
S(z) := e−(t−t′)A
(η − ϕ) +
∫ t
t′
e−(t−s)A
(f(z(s)) − f(ϕ)) ds.
On the basis of Banach’s fixed-point theorem, we show a
condition of S having a fixed-point in Uϕ.
For s ∈ (t′
, ∞) and ψ1, ψ2 ∈ Uϕ, the mean-value theorem
states that there exists y ∈ Uϕ such that
∥f(ψ1(s)) − f(ψ2(s))∥L2 = ∥f′
[y(s)](ψ1(s) − ψ2(s))∥L2 .
Since y ∈ Uϕ ⊂ B(ϕ, ρ) holds, we obtain
∥f(ψ1(s)) − f(ψ2(s))∥L2 ≤ L(ρ)∥ψ1(s) − ψ2(s)∥H1
0
.
46/55
79. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
80. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
81. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
82. How to get ∥η − ϕ∥H1
0
?
Since ∥η − ˆun∥H1
0
εn and a stationary solution ϕ encloses in
a neighborhood of a numerical solution, it follows
∥η − ϕ∥H1
0
≤ ∥η − ˆun∥H1
0
+ ∥ˆun − ˆϕ∥H1
0
+ ∥ˆϕ − ϕ∥H1
0
≤ εn + ∥ˆun − ˆϕ∥H1
0
+ ρ′
.
We need to estimate
∥u(t′
) − ˆun∥H1
0
≤ εn.
This can be obtained by the concatenation scheme!
47/55
84. 藤田型方程式
Let Ω := (0, 1)2
be an unit square domain in R2
.
(F)
∂tu − ∆u = u2
in (0, ∞) × Ω,
u(t, x) = 0 on (0, ∞) × ∂Ω,
u(0, x) = u0(x) in Ω,
where u0(x) = γ sin(πx) sin(πy).
▶ Vh :=
{∑N
k,l=1 ak,l sin(kπx) sin(lπy) : ak,l ∈ R
}
;
▶ Crank-Nicolson scheme is employed;
▶ we fixed λ = 1/40 in the global existence theorem.
49/55
