SlideShare a Scribd company logo
解析半群を用いた半線形放物型
方程式の解に対する精度保証付
き数値計算法とその応用
高安 亮紀
早稲田大学 基幹理工学部 応用数理学科
第 3 回 数理人セミナー@早稲田大学西早稲田キャンパス
2015 年 1 月 15 日
1/55
自己紹介
高安亮紀
早稲田大学 応用数理学科
助教(大石研究室)
分野:(偏微分方程式の) 数値解析,
精度保証付き数値計算
2/55
計算機を用いた
非線形PDEへの
解析アプローチ
3/55
共同研究者
▶ 水口 信
(早稲田大学 基幹理工学研究科 数学応用数理専攻)
▶ 久保 隆徹
(筑波大学 数理物質系)
▶ 大石 進一
(早稲田大学 基幹理工学部 応用数理学科)
4/55
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
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
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
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
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
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
精度保証付き数値計算
問題 (PJ ) の弱解の存在と一意性を計算機を援用し証明する.
すなわち X を J × Ω 上のある Banach 空間とし,弱解を数
値解 ω を中心とする閉球:
BJ (ω, ρ) := {v ∈ X : ∥v − ω∥X ≤ ρ}.
内に数学的に厳密に包み込む.
9/55
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
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.
11/55
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
Concatenation scheme
13/55
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).
14/55
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).
14/55
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Sketch of proof
Decompose g(z(s)) ∈ H−1
(Ω) into two parts:
g(z(s)) = f(ω(s) + z(s)) − (∂tω(s) + Aω(s))
= g1(s) + g2(s),
g1(s) := f(ω(s) + z(s)) − f(ω(s)),
g2(s) := f(ω(s)) − (∂tω(s) + Aω(s)) .
Put
ν(t) :=
∫ t
t0
(t − s)−1
2 e−1
2
(t−s)λmin
ds,
sup
t∈J
ν(t) ≤ sup
t∈J
∫ t
t0
(t − s)− 1
2 ds = 2
√
τ.
Furthermore, ready an inequality
µ
1
2 ∥u∥L2 ≤ ∥A
1
2 u∥H−1 ≤ M
1
2 ∥u∥L2 .
20/55
Sketch of proof
Decompose g(z(s)) ∈ H−1
(Ω) into two parts:
g(z(s)) = f(ω(s) + z(s)) − (∂tω(s) + Aω(s))
= g1(s) + g2(s),
g1(s) := f(ω(s) + z(s)) − f(ω(s)),
g2(s) := f(ω(s)) − (∂tω(s) + Aω(s)) .
Put
ν(t) :=
∫ t
t0
(t − s)−1
2 e−1
2
(t−s)λmin
ds,
sup
t∈J
ν(t) ≤ sup
t∈J
∫ t
t0
(t − s)− 1
2 ds = 2
√
τ.
Furthermore, ready an inequality
µ
1
2 ∥u∥L2 ≤ ∥A
1
2 u∥H−1 ≤ M
1
2 ∥u∥L2 .
20/55
Sketch of proof
Decompose g(z(s)) ∈ H−1
(Ω) into two parts:
g(z(s)) = f(ω(s) + z(s)) − (∂tω(s) + Aω(s))
= g1(s) + g2(s),
g1(s) := f(ω(s) + z(s)) − f(ω(s)),
g2(s) := f(ω(s)) − (∂tω(s) + Aω(s)) .
Put
ν(t) :=
∫ t
t0
(t − s)−1
2 e−1
2
(t−s)λmin
ds,
sup
t∈J
ν(t) ≤ sup
t∈J
∫ t
t0
(t − s)− 1
2 ds = 2
√
τ.
Furthermore, ready an inequality
µ
1
2 ∥u∥L2 ≤ ∥A
1
2 u∥H−1 ≤ M
1
2 ∥u∥L2 .
20/55
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
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
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
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
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
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
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
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
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
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
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
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
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
Verification scheme
0 t
u0
t1 t2 tk
...
...
28/55
Verification scheme
0 t
u0
t1 t2 tk
...
...
28/55
Verification scheme
0 t
u0
t1 t2 tk
...
...
28/55
Verification scheme
0 t
u0
t1 t2 tk
...
...
28/55
Verification scheme
0 t
u0
t1 t2 tk
...
...
28/55
ここまでのまとめ
▶ 各 t ∈ Jk において対象問題
(PJ )



∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
に対する弱解の存在と一意性を逐次的に数値解の近傍
BJk
(ω, ρk) に包み込む.
▶ (PJ ) の弱解は各 k = 1, 2, ..., n について
B(ω) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: y(t) ∈ BJk
(ω, ρk), t ∈ Jk
}
の中に一意存在する事が計算機援用証明できる.
29/55
ここまでのまとめ
▶ 各 t ∈ Jk において対象問題
(PJ )



∂tu + Au = f(u) in J × Ω,
u(t, x) = 0 on J × ∂Ω,
u(0, x) = u0(x) in Ω,
に対する弱解の存在と一意性を逐次的に数値解の近傍
BJk
(ω, ρk) に包み込む.
▶ (PJ ) の弱解は各 k = 1, 2, ..., n について
B(ω) :=
{
y ∈ L∞
(
J; H1
0 (Ω)
)
: y(t) ∈ BJk
(ω, ρk), t ∈ Jk
}
の中に一意存在する事が計算機援用証明できる.
29/55
Computational results 1
30/55
藤田型方程式
Ω = (0, 1)2
: Square domain



∂tu − ∆u = u2
in (0, ∞) × Ω,
u(t, x) = 0 on (0, ∞) × ∂Ω,
u(0, x) = u0 in Ω,
Let γ  0 be an parameter of the initial function:
u0(x) = γx1(1 − x1)x2(1 − x2).
h: spatial mesh size (P2 element),
τ: time step of B.E. method.
31/55
Computational results
Table: h = 2−4
, τ = 2−8
, γ = 1.
Tk = (tk−1, tk] εk ρk
(0,0.0039062] 0.020155 0.037646
(0.0039062,0.0078125] 0.030051 0.041554
(0.0078125,0.011719] 0.038089 0.049313
(0.011719,0.015625] 0.044657 0.055699
(0.015625,0.019531] 0.050001 0.060873
...
...
...
(0.48047,0.48438] 0.00013041 0.00014184
(0.48438,0.48828] 0.00012186 0.00013255
(0.48828,0.49219] 0.00011388 0.00012386
(0.49219,0.49609] 0.00010641 0.00011573
(0.49609,0.5] 9.9431E-5 0.00010813
32/55
Computational results
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t
ρk
gamma = 1
gamma = 10
33/55
Computational results
0 0.1 0.2 0.3 0.4 0.5
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
t
ρ
k
gamma = 30
gamma = 50
34/55
半線形放物型方程式
Ω = (0, 1)2
: Square domain



∂tu − ∆u = u − u3
in (0, ∞) × Ω,
u(t, x) = 0 on (0, ∞) × ∂Ω,
u(0, x) = u0 in Ω.
We set the initial function:
u0(x) = x1(1 − x1)x2(1 − x2).
h: spatial mesh size (P2 element),
τ: time step of B.E. method.
35/55
Computational results (h = 2−4
)
0 0.1 0.2 0.3 0.4 0.5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
t
ρ
k
tau = 1/16
tau = 1/32
tau = 1/64
tau = 1/128
tau = 1/256
tau = 1/512
36/55
Computational results (τ ≪ h)
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
t
ρk
h = 1/4
h = 1/8
h = 1/16
h = 1/32
37/55
Global existence proof
using verified computations
38/55
時間大域解の証明
本講演では t ∈ (0, ∞) で存在する (PJ ) の解 u(t) ∈ H1
0 (Ω) を
時間大域解といい,以下の 2 つのステップで証明を試みる.
t′
 0 をある時刻として
▶ (t′
, ∞) で定常解まわりに大域的に存在する範囲を計算
機で導く.(Global existence proof)
▶ ある時刻 t′
までの解を数値解の近傍に包み込む.
(Concatenation scheme)
上記の方法によって,(PJ ) の時間大域解を関数空間
L∞
(
(0, ∞); H1
0 (Ω)
)
で一意存在することが計算機を用いて証明できる.
39/55
時間大域解の証明
本講演では t ∈ (0, ∞) で存在する (PJ ) の解 u(t) ∈ H1
0 (Ω) を
時間大域解といい,以下の 2 つのステップで証明を試みる.
t′
 0 をある時刻として
▶ (t′
, ∞) で定常解まわりに大域的に存在する範囲を計算
機で導く.(Global existence proof)
▶ ある時刻 t′
までの解を数値解の近傍に包み込む.
(Concatenation scheme)
上記の方法によって,(PJ ) の時間大域解を関数空間
L∞
(
(0, ∞); H1
0 (Ω)
)
で一意存在することが計算機を用いて証明できる.
39/55
時間大域解の証明
本講演では t ∈ (0, ∞) で存在する (PJ ) の解 u(t) ∈ H1
0 (Ω) を
時間大域解といい,以下の 2 つのステップで証明を試みる.
t′
 0 をある時刻として
▶ (t′
, ∞) で定常解まわりに大域的に存在する範囲を計算
機で導く.(Global existence proof)
▶ ある時刻 t′
までの解を数値解の近傍に包み込む.
(Concatenation scheme)
上記の方法によって,(PJ ) の時間大域解を関数空間
L∞
(
(0, ∞); H1
0 (Ω)
)
で一意存在することが計算機を用いて証明できる.
39/55
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Computational results 2
48/55
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
Table: 時間大域解の検証例(N = 8, λ = 1/40, τk = 2−7
)
γ n t′
ρ
0.01 5 0.046875 0.01085
0.011 5 0.046875 0.011936
0.0121 6 0.054688 0.011605
0.01331 7 0.0625 0.011274
0.014641 7 0.0625 0.012403
0.016105 8 0.070312 0.012038
0.017716 8 0.070312 0.013244
0.019487 9 0.078125 0.012845
0.021436 10 0.085938 0.012448
0.023579 10 0.085938 0.013695
0.025937 11 0.09375 0.013263
0.028531 11 0.09375 0.014593
0.031384 12 0.10156 0.014123
...
50/55
Table: 時間大域解の検証例(N = 8, λ = 1/40, τk = 2−7
)
γ n t′
ρ
...
2.2876 40 0.32031 0.029945
2.5164 41 0.32812 0.0296
2.768 42 0.33594 0.029446
3.0448 42 0.33594 0.034085
3.3493 43 0.34375 0.034657
3.6842 44 0.35156 0.035917
4.0527 45 0.35938 0.038419
4.4579 45 0.35938 0.050511
4.9037 46 0.36719 0.066455
5.3941 47 0.375 0.17656
このとき
∥u(t)∥H1
0
≤ ρe−
(t−t′)
40 , t ∈ (tn, ∞).
51/55
半線形放物型方程式
Let Ω := (0, 1)2
be an unit square domain in R2
.



∂tu − ∆u = f(u) in (0, ∞) × Ω,
u(t, x) = 0 on (0, ∞) × ∂Ω,
u(0, x) = u0(x) in Ω,
where u0(x) = sin(πx) sin(πy).
▶ f(u) = u2
+ 4
∑
1≤k,l≤3 sin(kπx) sin(lπy);
▶ 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.
52/55
Fig. The numerical solution ˆϕ.
時間大域解の検証は N = 10, λ = 1/40, τk = 2−8
で成功
して,
ρ = 0.04035, t′
= 0.2578125.
53/55
Fig. The numerical solution ˆϕ.
時間大域解の検証は N = 10, λ = 1/40, τk = 2−8
で成功
して,
ρ = 0.04035, t′
= 0.2578125.
53/55
まとめ
▶ 解析半群
{
e−tA
}
t≥0
を用いる精度保証付き数値計算手法
▶ Concatenation scheme(数値解のまわりに包み込む)
▶ 精度保証付き数値計算を用いた時間大域解の存在証明
(定常解のまわりに包み込む)
今後の課題
▶ 方程式の拡張(多種の反応拡散方程式,波動方程式等)
▶ 無限次元力学系との関連
▶ 藤田型方程式の爆発時刻の精度保証付き数値計算
54/55
Thank you for kind attention!
55/55

More Related Content

What's hot

Homotopy perturbation and elzaki transform for solving nonlinear partial diff...
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Homotopy perturbation and elzaki transform for solving nonlinear partial diff...
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...
Alexander Decker
 
Fdtd
FdtdFdtd
Existance Theory for First Order Nonlinear Random Dfferential Equartion
Existance Theory for First Order Nonlinear Random Dfferential EquartionExistance Theory for First Order Nonlinear Random Dfferential Equartion
Existance Theory for First Order Nonlinear Random Dfferential Equartion
inventionjournals
 
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Cristiano Longo
 
Pata contraction
Pata contractionPata contraction
Pata contraction
mohsen rostamian delavar
 
Rousseau
RousseauRousseau
Rousseau
eric_gautier
 
Stochastic calculus
Stochastic calculus Stochastic calculus
Stochastic calculus
Praveen Kumar Gangarapu
 
Perspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problemsPerspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problems
RAJKRISHNA MONDAL
 
Slides smart-2015
Slides smart-2015Slides smart-2015
Slides smart-2015
Arthur Charpentier
 
Introduction to Stochastic calculus
Introduction to Stochastic calculusIntroduction to Stochastic calculus
Introduction to Stochastic calculus
Ashwin Rao
 
Proba stats-r1-2017
Proba stats-r1-2017Proba stats-r1-2017
Proba stats-r1-2017
Arthur Charpentier
 
Numerical solution of boundary value problems by piecewise analysis method
Numerical solution of boundary value problems by piecewise analysis methodNumerical solution of boundary value problems by piecewise analysis method
Numerical solution of boundary value problems by piecewise analysis method
Alexander Decker
 
Slides amsterdam-2013
Slides amsterdam-2013Slides amsterdam-2013
Slides amsterdam-2013
Arthur Charpentier
 
Analysis and numerics of partial differential equations
Analysis and numerics of partial differential equationsAnalysis and numerics of partial differential equations
Analysis and numerics of partial differential equations
Springer
 
International journal of engineering and mathematical modelling vol2 no3_2015_2
International journal of engineering and mathematical modelling vol2 no3_2015_2International journal of engineering and mathematical modelling vol2 no3_2015_2
International journal of engineering and mathematical modelling vol2 no3_2015_2
IJEMM
 
So a webinar-2013-2
So a webinar-2013-2So a webinar-2013-2
So a webinar-2013-2
Arthur Charpentier
 
Quantum modes - Ion Cotaescu
Quantum modes - Ion CotaescuQuantum modes - Ion Cotaescu
Quantum modes - Ion Cotaescu
SEENET-MTP
 
Bayes Independence Test
Bayes Independence TestBayes Independence Test
Bayes Independence Test
Joe Suzuki
 
Koc4(dba)
Koc4(dba)Koc4(dba)
Koc4(dba)
Serhat Yucel
 

What's hot (19)

Homotopy perturbation and elzaki transform for solving nonlinear partial diff...
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Homotopy perturbation and elzaki transform for solving nonlinear partial diff...
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...
 
Fdtd
FdtdFdtd
Fdtd
 
Existance Theory for First Order Nonlinear Random Dfferential Equartion
Existance Theory for First Order Nonlinear Random Dfferential EquartionExistance Theory for First Order Nonlinear Random Dfferential Equartion
Existance Theory for First Order Nonlinear Random Dfferential Equartion
 
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...
 
Pata contraction
Pata contractionPata contraction
Pata contraction
 
Rousseau
RousseauRousseau
Rousseau
 
Stochastic calculus
Stochastic calculus Stochastic calculus
Stochastic calculus
 
Perspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problemsPerspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problems
 
Slides smart-2015
Slides smart-2015Slides smart-2015
Slides smart-2015
 
Introduction to Stochastic calculus
Introduction to Stochastic calculusIntroduction to Stochastic calculus
Introduction to Stochastic calculus
 
Proba stats-r1-2017
Proba stats-r1-2017Proba stats-r1-2017
Proba stats-r1-2017
 
Numerical solution of boundary value problems by piecewise analysis method
Numerical solution of boundary value problems by piecewise analysis methodNumerical solution of boundary value problems by piecewise analysis method
Numerical solution of boundary value problems by piecewise analysis method
 
Slides amsterdam-2013
Slides amsterdam-2013Slides amsterdam-2013
Slides amsterdam-2013
 
Analysis and numerics of partial differential equations
Analysis and numerics of partial differential equationsAnalysis and numerics of partial differential equations
Analysis and numerics of partial differential equations
 
International journal of engineering and mathematical modelling vol2 no3_2015_2
International journal of engineering and mathematical modelling vol2 no3_2015_2International journal of engineering and mathematical modelling vol2 no3_2015_2
International journal of engineering and mathematical modelling vol2 no3_2015_2
 
So a webinar-2013-2
So a webinar-2013-2So a webinar-2013-2
So a webinar-2013-2
 
Quantum modes - Ion Cotaescu
Quantum modes - Ion CotaescuQuantum modes - Ion Cotaescu
Quantum modes - Ion Cotaescu
 
Bayes Independence Test
Bayes Independence TestBayes Independence Test
Bayes Independence Test
 
Koc4(dba)
Koc4(dba)Koc4(dba)
Koc4(dba)
 

Similar to Akitoshi Takayasu

QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...
QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...
QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...
The Statistical and Applied Mathematical Sciences Institute
 
Perspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problemsPerspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problems
RAJKRISHNA MONDAL
 
cmftJYeZhuanTalk.pdf
cmftJYeZhuanTalk.pdfcmftJYeZhuanTalk.pdf
cmftJYeZhuanTalk.pdf
jyjyzr69t7
 
Compatible discretizations in our hearts and minds
Compatible discretizations in our hearts and mindsCompatible discretizations in our hearts and minds
Compatible discretizations in our hearts and minds
Marie E. Rognes
 
International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI)
inventionjournals
 
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...
inventionjournals
 
A new axisymmetric finite element
A new axisymmetric finite elementA new axisymmetric finite element
A new axisymmetric finite element
Stefan Duprey
 
Trapezoidal Method IN Numerical Analysis
Trapezoidal Method IN  Numerical AnalysisTrapezoidal Method IN  Numerical Analysis
Trapezoidal Method IN Numerical Analysis
Mostafijur Rahman
 
Kt2418201822
Kt2418201822Kt2418201822
Kt2418201822
IJERA Editor
 
Sawinder Pal Kaur PhD Thesis
Sawinder Pal Kaur PhD ThesisSawinder Pal Kaur PhD Thesis
Sawinder Pal Kaur PhD Thesis
Sawinder Pal Kaur
 
Natalini nse slide_giu2013
Natalini nse slide_giu2013Natalini nse slide_giu2013
Natalini nse slide_giu2013
Madd Maths
 
A numerical method to solve fractional Fredholm-Volterra integro-differential...
A numerical method to solve fractional Fredholm-Volterra integro-differential...A numerical method to solve fractional Fredholm-Volterra integro-differential...
A numerical method to solve fractional Fredholm-Volterra integro-differential...
OctavianPostavaru
 
InfEntr_EntrProd_20100618_2
InfEntr_EntrProd_20100618_2InfEntr_EntrProd_20100618_2
InfEntr_EntrProd_20100618_2
Teng Li
 
05_AJMS_300_21.pdf
05_AJMS_300_21.pdf05_AJMS_300_21.pdf
05_AJMS_300_21.pdf
BRNSS Publication Hub
 
Ma2002 1.19 rm
Ma2002 1.19 rmMa2002 1.19 rm
Ma2002 1.19 rm
Ramakrishna Paduchuri
 
Ichimura 1993: Semiparametric Least Squares (non-technical)
Ichimura 1993: Semiparametric Least Squares (non-technical)Ichimura 1993: Semiparametric Least Squares (non-technical)
Ichimura 1993: Semiparametric Least Squares (non-technical)
The University of Texas at Austin
 
Stability analysis for nonlinear impulsive optimal control problems
Stability analysis for nonlinear impulsive optimal control problemsStability analysis for nonlinear impulsive optimal control problems
Stability analysis for nonlinear impulsive optimal control problems
AI Publications
 
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
ijfls
 
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
Wireilla
 
Comparison Theorems for SDEs
Comparison Theorems for SDEs Comparison Theorems for SDEs
Comparison Theorems for SDEs
Ilya Gikhman
 

Similar to Akitoshi Takayasu (20)

QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...
QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...
QMC: Operator Splitting Workshop, Compactness Estimates for Nonlinear PDEs - ...
 
Perspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problemsPerspectives and application of fuzzy initial value problems
Perspectives and application of fuzzy initial value problems
 
cmftJYeZhuanTalk.pdf
cmftJYeZhuanTalk.pdfcmftJYeZhuanTalk.pdf
cmftJYeZhuanTalk.pdf
 
Compatible discretizations in our hearts and minds
Compatible discretizations in our hearts and mindsCompatible discretizations in our hearts and minds
Compatible discretizations in our hearts and minds
 
International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI)
 
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...
 
A new axisymmetric finite element
A new axisymmetric finite elementA new axisymmetric finite element
A new axisymmetric finite element
 
Trapezoidal Method IN Numerical Analysis
Trapezoidal Method IN  Numerical AnalysisTrapezoidal Method IN  Numerical Analysis
Trapezoidal Method IN Numerical Analysis
 
Kt2418201822
Kt2418201822Kt2418201822
Kt2418201822
 
Sawinder Pal Kaur PhD Thesis
Sawinder Pal Kaur PhD ThesisSawinder Pal Kaur PhD Thesis
Sawinder Pal Kaur PhD Thesis
 
Natalini nse slide_giu2013
Natalini nse slide_giu2013Natalini nse slide_giu2013
Natalini nse slide_giu2013
 
A numerical method to solve fractional Fredholm-Volterra integro-differential...
A numerical method to solve fractional Fredholm-Volterra integro-differential...A numerical method to solve fractional Fredholm-Volterra integro-differential...
A numerical method to solve fractional Fredholm-Volterra integro-differential...
 
InfEntr_EntrProd_20100618_2
InfEntr_EntrProd_20100618_2InfEntr_EntrProd_20100618_2
InfEntr_EntrProd_20100618_2
 
05_AJMS_300_21.pdf
05_AJMS_300_21.pdf05_AJMS_300_21.pdf
05_AJMS_300_21.pdf
 
Ma2002 1.19 rm
Ma2002 1.19 rmMa2002 1.19 rm
Ma2002 1.19 rm
 
Ichimura 1993: Semiparametric Least Squares (non-technical)
Ichimura 1993: Semiparametric Least Squares (non-technical)Ichimura 1993: Semiparametric Least Squares (non-technical)
Ichimura 1993: Semiparametric Least Squares (non-technical)
 
Stability analysis for nonlinear impulsive optimal control problems
Stability analysis for nonlinear impulsive optimal control problemsStability analysis for nonlinear impulsive optimal control problems
Stability analysis for nonlinear impulsive optimal control problems
 
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
 
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...
 
Comparison Theorems for SDEs
Comparison Theorems for SDEs Comparison Theorems for SDEs
Comparison Theorems for SDEs
 

More from Suurist

Hiroki Shigemune
Hiroki  ShigemuneHiroki  Shigemune
Hiroki Shigemune
Suurist
 
Hiroaki Shiokawa
Hiroaki ShiokawaHiroaki Shiokawa
Hiroaki Shiokawa
Suurist
 
Hidetomo Nagai
Hidetomo NagaiHidetomo Nagai
Hidetomo Nagai
Suurist
 
Kensuke Aihara
Kensuke AiharaKensuke Aihara
Kensuke Aihara
Suurist
 
Tatsuhiro Kishi
Tatsuhiro KishiTatsuhiro Kishi
Tatsuhiro Kishi
Suurist
 
Hideitsu Hino
Hideitsu HinoHideitsu Hino
Hideitsu Hino
Suurist
 
Tatsuya Yatagawa
Tatsuya YatagawaTatsuya Yatagawa
Tatsuya Yatagawa
Suurist
 
Hiroyuki Sato
Hiroyuki SatoHiroyuki Sato
Hiroyuki Sato
Suurist
 
Tetsunao Matsuta
Tetsunao MatsutaTetsunao Matsuta
Tetsunao Matsuta
Suurist
 
Naoya Tsuruta
Naoya TsurutaNaoya Tsuruta
Naoya Tsuruta
Suurist
 
Suurist Test Slide
Suurist Test SlideSuurist Test Slide
Suurist Test Slide
Suurist
 
Kohta Suzuno
Kohta SuzunoKohta Suzuno
Kohta Suzuno
Suurist
 
Shunsuke Horii
Shunsuke HoriiShunsuke Horii
Shunsuke Horii
Suurist
 
Yasunori Futamura
Yasunori FutamuraYasunori Futamura
Yasunori Futamura
Suurist
 
Akiyasu Tomoeda
Akiyasu TomoedaAkiyasu Tomoeda
Akiyasu Tomoeda
Suurist
 
Takuya Tsuchiya
Takuya TsuchiyaTakuya Tsuchiya
Takuya Tsuchiya
Suurist
 
Akira Imakura
Akira ImakuraAkira Imakura
Akira Imakura
Suurist
 

More from Suurist (17)

Hiroki Shigemune
Hiroki  ShigemuneHiroki  Shigemune
Hiroki Shigemune
 
Hiroaki Shiokawa
Hiroaki ShiokawaHiroaki Shiokawa
Hiroaki Shiokawa
 
Hidetomo Nagai
Hidetomo NagaiHidetomo Nagai
Hidetomo Nagai
 
Kensuke Aihara
Kensuke AiharaKensuke Aihara
Kensuke Aihara
 
Tatsuhiro Kishi
Tatsuhiro KishiTatsuhiro Kishi
Tatsuhiro Kishi
 
Hideitsu Hino
Hideitsu HinoHideitsu Hino
Hideitsu Hino
 
Tatsuya Yatagawa
Tatsuya YatagawaTatsuya Yatagawa
Tatsuya Yatagawa
 
Hiroyuki Sato
Hiroyuki SatoHiroyuki Sato
Hiroyuki Sato
 
Tetsunao Matsuta
Tetsunao MatsutaTetsunao Matsuta
Tetsunao Matsuta
 
Naoya Tsuruta
Naoya TsurutaNaoya Tsuruta
Naoya Tsuruta
 
Suurist Test Slide
Suurist Test SlideSuurist Test Slide
Suurist Test Slide
 
Kohta Suzuno
Kohta SuzunoKohta Suzuno
Kohta Suzuno
 
Shunsuke Horii
Shunsuke HoriiShunsuke Horii
Shunsuke Horii
 
Yasunori Futamura
Yasunori FutamuraYasunori Futamura
Yasunori Futamura
 
Akiyasu Tomoeda
Akiyasu TomoedaAkiyasu Tomoeda
Akiyasu Tomoeda
 
Takuya Tsuchiya
Takuya TsuchiyaTakuya Tsuchiya
Takuya Tsuchiya
 
Akira Imakura
Akira ImakuraAkira Imakura
Akira Imakura
 

Recently uploaded

aziz sancar nobel prize winner: from mardin to nobel
aziz sancar nobel prize winner: from mardin to nobelaziz sancar nobel prize winner: from mardin to nobel
aziz sancar nobel prize winner: from mardin to nobel
İsa Badur
 
Nucleic Acid-its structural and functional complexity.
Nucleic Acid-its structural and functional complexity.Nucleic Acid-its structural and functional complexity.
Nucleic Acid-its structural and functional complexity.
Nistarini College, Purulia (W.B) India
 
Oedema_types_causes_pathophysiology.pptx
Oedema_types_causes_pathophysiology.pptxOedema_types_causes_pathophysiology.pptx
Oedema_types_causes_pathophysiology.pptx
muralinath2
 
Phenomics assisted breeding in crop improvement
Phenomics assisted breeding in crop improvementPhenomics assisted breeding in crop improvement
Phenomics assisted breeding in crop improvement
IshaGoswami9
 
8.Isolation of pure cultures and preservation of cultures.pdf
8.Isolation of pure cultures and preservation of cultures.pdf8.Isolation of pure cultures and preservation of cultures.pdf
8.Isolation of pure cultures and preservation of cultures.pdf
by6843629
 
Eukaryotic Transcription Presentation.pptx
Eukaryotic Transcription Presentation.pptxEukaryotic Transcription Presentation.pptx
Eukaryotic Transcription Presentation.pptx
RitabrataSarkar3
 
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
University of Maribor
 
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốtmô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
HongcNguyn6
 
ESR spectroscopy in liquid food and beverages.pptx
ESR spectroscopy in liquid food and beverages.pptxESR spectroscopy in liquid food and beverages.pptx
ESR spectroscopy in liquid food and beverages.pptx
PRIYANKA PATEL
 
DMARDs Pharmacolgy Pharm D 5th Semester.pdf
DMARDs Pharmacolgy Pharm D 5th Semester.pdfDMARDs Pharmacolgy Pharm D 5th Semester.pdf
DMARDs Pharmacolgy Pharm D 5th Semester.pdf
fafyfskhan251kmf
 
What is greenhouse gasses and how many gasses are there to affect the Earth.
What is greenhouse gasses and how many gasses are there to affect the Earth.What is greenhouse gasses and how many gasses are there to affect the Earth.
What is greenhouse gasses and how many gasses are there to affect the Earth.
moosaasad1975
 
bordetella pertussis.................................ppt
bordetella pertussis.................................pptbordetella pertussis.................................ppt
bordetella pertussis.................................ppt
kejapriya1
 
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
Wasswaderrick3
 
The debris of the ‘last major merger’ is dynamically young
The debris of the ‘last major merger’ is dynamically youngThe debris of the ‘last major merger’ is dynamically young
The debris of the ‘last major merger’ is dynamically young
Sérgio Sacani
 
Unveiling the Energy Potential of Marshmallow Deposits.pdf
Unveiling the Energy Potential of Marshmallow Deposits.pdfUnveiling the Energy Potential of Marshmallow Deposits.pdf
Unveiling the Energy Potential of Marshmallow Deposits.pdf
Erdal Coalmaker
 
BREEDING METHODS FOR DISEASE RESISTANCE.pptx
BREEDING METHODS FOR DISEASE RESISTANCE.pptxBREEDING METHODS FOR DISEASE RESISTANCE.pptx
BREEDING METHODS FOR DISEASE RESISTANCE.pptx
RASHMI M G
 
Deep Software Variability and Frictionless Reproducibility
Deep Software Variability and Frictionless ReproducibilityDeep Software Variability and Frictionless Reproducibility
Deep Software Variability and Frictionless Reproducibility
University of Rennes, INSA Rennes, Inria/IRISA, CNRS
 
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...
Travis Hills MN
 
Bob Reedy - Nitrate in Texas Groundwater.pdf
Bob Reedy - Nitrate in Texas Groundwater.pdfBob Reedy - Nitrate in Texas Groundwater.pdf
Bob Reedy - Nitrate in Texas Groundwater.pdf
Texas Alliance of Groundwater Districts
 
Orion Air Quality Monitoring Systems - CWS
Orion Air Quality Monitoring Systems - CWSOrion Air Quality Monitoring Systems - CWS
Orion Air Quality Monitoring Systems - CWS
Columbia Weather Systems
 

Recently uploaded (20)

aziz sancar nobel prize winner: from mardin to nobel
aziz sancar nobel prize winner: from mardin to nobelaziz sancar nobel prize winner: from mardin to nobel
aziz sancar nobel prize winner: from mardin to nobel
 
Nucleic Acid-its structural and functional complexity.
Nucleic Acid-its structural and functional complexity.Nucleic Acid-its structural and functional complexity.
Nucleic Acid-its structural and functional complexity.
 
Oedema_types_causes_pathophysiology.pptx
Oedema_types_causes_pathophysiology.pptxOedema_types_causes_pathophysiology.pptx
Oedema_types_causes_pathophysiology.pptx
 
Phenomics assisted breeding in crop improvement
Phenomics assisted breeding in crop improvementPhenomics assisted breeding in crop improvement
Phenomics assisted breeding in crop improvement
 
8.Isolation of pure cultures and preservation of cultures.pdf
8.Isolation of pure cultures and preservation of cultures.pdf8.Isolation of pure cultures and preservation of cultures.pdf
8.Isolation of pure cultures and preservation of cultures.pdf
 
Eukaryotic Transcription Presentation.pptx
Eukaryotic Transcription Presentation.pptxEukaryotic Transcription Presentation.pptx
Eukaryotic Transcription Presentation.pptx
 
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
 
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốtmô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
 
ESR spectroscopy in liquid food and beverages.pptx
ESR spectroscopy in liquid food and beverages.pptxESR spectroscopy in liquid food and beverages.pptx
ESR spectroscopy in liquid food and beverages.pptx
 
DMARDs Pharmacolgy Pharm D 5th Semester.pdf
DMARDs Pharmacolgy Pharm D 5th Semester.pdfDMARDs Pharmacolgy Pharm D 5th Semester.pdf
DMARDs Pharmacolgy Pharm D 5th Semester.pdf
 
What is greenhouse gasses and how many gasses are there to affect the Earth.
What is greenhouse gasses and how many gasses are there to affect the Earth.What is greenhouse gasses and how many gasses are there to affect the Earth.
What is greenhouse gasses and how many gasses are there to affect the Earth.
 
bordetella pertussis.................................ppt
bordetella pertussis.................................pptbordetella pertussis.................................ppt
bordetella pertussis.................................ppt
 
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
 
The debris of the ‘last major merger’ is dynamically young
The debris of the ‘last major merger’ is dynamically youngThe debris of the ‘last major merger’ is dynamically young
The debris of the ‘last major merger’ is dynamically young
 
Unveiling the Energy Potential of Marshmallow Deposits.pdf
Unveiling the Energy Potential of Marshmallow Deposits.pdfUnveiling the Energy Potential of Marshmallow Deposits.pdf
Unveiling the Energy Potential of Marshmallow Deposits.pdf
 
BREEDING METHODS FOR DISEASE RESISTANCE.pptx
BREEDING METHODS FOR DISEASE RESISTANCE.pptxBREEDING METHODS FOR DISEASE RESISTANCE.pptx
BREEDING METHODS FOR DISEASE RESISTANCE.pptx
 
Deep Software Variability and Frictionless Reproducibility
Deep Software Variability and Frictionless ReproducibilityDeep Software Variability and Frictionless Reproducibility
Deep Software Variability and Frictionless Reproducibility
 
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...
 
Bob Reedy - Nitrate in Texas Groundwater.pdf
Bob Reedy - Nitrate in Texas Groundwater.pdfBob Reedy - Nitrate in Texas Groundwater.pdf
Bob Reedy - Nitrate in Texas Groundwater.pdf
 
Orion Air Quality Monitoring Systems - CWS
Orion Air Quality Monitoring Systems - CWSOrion Air Quality Monitoring Systems - CWS
Orion Air Quality Monitoring Systems - CWS
 

Akitoshi Takayasu

  • 1. 解析半群を用いた半線形放物型 方程式の解に対する精度保証付 き数値計算法とその応用 高安 亮紀 早稲田大学 基幹理工学部 応用数理学科 第 3 回 数理人セミナー@早稲田大学西早稲田キャンパス 2015 年 1 月 15 日 1/55
  • 4. 共同研究者 ▶ 水口 信 (早稲田大学 基幹理工学研究科 数学応用数理専攻) ▶ 久保 隆徹 (筑波大学 数理物質系) ▶ 大石 進一 (早稲田大学 基幹理工学部 応用数理学科) 4/55
  • 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
  • 11. 精度保証付き数値計算 問題 (PJ ) の弱解の存在と一意性を計算機を援用し証明する. すなわち X を J × Ω 上のある Banach 空間とし,弱解を数 値解 ω を中心とする閉球: BJ (ω, ρ) := {v ∈ X : ∥v − ω∥X ≤ ρ}. 内に数学的に厳密に包み込む. 9/55
  • 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. 11/55
  • 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). 14/55
  • 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). 14/55
  • 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 15/55
  • 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). 17/55
  • 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). 17/55
  • 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). 17/55
  • 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). 17/55
  • 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
  • 32. Sketch of proof Decompose g(z(s)) ∈ H−1 (Ω) into two parts: g(z(s)) = f(ω(s) + z(s)) − (∂tω(s) + Aω(s)) = g1(s) + g2(s), g1(s) := f(ω(s) + z(s)) − f(ω(s)), g2(s) := f(ω(s)) − (∂tω(s) + Aω(s)) . Put ν(t) := ∫ t t0 (t − s)−1 2 e−1 2 (t−s)λmin ds, sup t∈J ν(t) ≤ sup t∈J ∫ t t0 (t − s)− 1 2 ds = 2 √ τ. Furthermore, ready an inequality µ 1 2 ∥u∥L2 ≤ ∥A 1 2 u∥H−1 ≤ M 1 2 ∥u∥L2 . 20/55
  • 33. Sketch of proof Decompose g(z(s)) ∈ H−1 (Ω) into two parts: g(z(s)) = f(ω(s) + z(s)) − (∂tω(s) + Aω(s)) = g1(s) + g2(s), g1(s) := f(ω(s) + z(s)) − f(ω(s)), g2(s) := f(ω(s)) − (∂tω(s) + Aω(s)) . Put ν(t) := ∫ t t0 (t − s)−1 2 e−1 2 (t−s)λmin ds, sup t∈J ν(t) ≤ sup t∈J ∫ t t0 (t − s)− 1 2 ds = 2 √ τ. Furthermore, ready an inequality µ 1 2 ∥u∥L2 ≤ ∥A 1 2 u∥H−1 ≤ M 1 2 ∥u∥L2 . 20/55
  • 34. Sketch of proof Decompose g(z(s)) ∈ H−1 (Ω) into two parts: g(z(s)) = f(ω(s) + z(s)) − (∂tω(s) + Aω(s)) = g1(s) + g2(s), g1(s) := f(ω(s) + z(s)) − f(ω(s)), g2(s) := f(ω(s)) − (∂tω(s) + Aω(s)) . Put ν(t) := ∫ t t0 (t − s)−1 2 e−1 2 (t−s)λmin ds, sup t∈J ν(t) ≤ sup t∈J ∫ t t0 (t − s)− 1 2 ds = 2 √ τ. Furthermore, ready an inequality µ 1 2 ∥u∥L2 ≤ ∥A 1 2 u∥H−1 ≤ M 1 2 ∥u∥L2 . 20/55
  • 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. 22/55
  • 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
  • 48. Verification scheme 0 t u0 t1 t2 tk ... ... 28/55
  • 49. Verification scheme 0 t u0 t1 t2 tk ... ... 28/55
  • 50. Verification scheme 0 t u0 t1 t2 tk ... ... 28/55
  • 51. Verification scheme 0 t u0 t1 t2 tk ... ... 28/55
  • 52. Verification scheme 0 t u0 t1 t2 tk ... ... 28/55
  • 53. ここまでのまとめ ▶ 各 t ∈ Jk において対象問題 (PJ )    ∂tu + Au = f(u) in J × Ω, u(t, x) = 0 on J × ∂Ω, u(0, x) = u0(x) in Ω, に対する弱解の存在と一意性を逐次的に数値解の近傍 BJk (ω, ρk) に包み込む. ▶ (PJ ) の弱解は各 k = 1, 2, ..., n について B(ω) := { y ∈ L∞ ( J; H1 0 (Ω) ) : y(t) ∈ BJk (ω, ρk), t ∈ Jk } の中に一意存在する事が計算機援用証明できる. 29/55
  • 54. ここまでのまとめ ▶ 各 t ∈ Jk において対象問題 (PJ )    ∂tu + Au = f(u) in J × Ω, u(t, x) = 0 on J × ∂Ω, u(0, x) = u0(x) in Ω, に対する弱解の存在と一意性を逐次的に数値解の近傍 BJk (ω, ρk) に包み込む. ▶ (PJ ) の弱解は各 k = 1, 2, ..., n について B(ω) := { y ∈ L∞ ( J; H1 0 (Ω) ) : y(t) ∈ BJk (ω, ρk), t ∈ Jk } の中に一意存在する事が計算機援用証明できる. 29/55
  • 56. 藤田型方程式 Ω = (0, 1)2 : Square domain    ∂tu − ∆u = u2 in (0, ∞) × Ω, u(t, x) = 0 on (0, ∞) × ∂Ω, u(0, x) = u0 in Ω, Let γ 0 be an parameter of the initial function: u0(x) = γx1(1 − x1)x2(1 − x2). h: spatial mesh size (P2 element), τ: time step of B.E. method. 31/55
  • 57. Computational results Table: h = 2−4 , τ = 2−8 , γ = 1. Tk = (tk−1, tk] εk ρk (0,0.0039062] 0.020155 0.037646 (0.0039062,0.0078125] 0.030051 0.041554 (0.0078125,0.011719] 0.038089 0.049313 (0.011719,0.015625] 0.044657 0.055699 (0.015625,0.019531] 0.050001 0.060873 ... ... ... (0.48047,0.48438] 0.00013041 0.00014184 (0.48438,0.48828] 0.00012186 0.00013255 (0.48828,0.49219] 0.00011388 0.00012386 (0.49219,0.49609] 0.00010641 0.00011573 (0.49609,0.5] 9.9431E-5 0.00010813 32/55
  • 58. Computational results 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t ρk gamma = 1 gamma = 10 33/55
  • 59. Computational results 0 0.1 0.2 0.3 0.4 0.5 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 t ρ k gamma = 30 gamma = 50 34/55
  • 60. 半線形放物型方程式 Ω = (0, 1)2 : Square domain    ∂tu − ∆u = u − u3 in (0, ∞) × Ω, u(t, x) = 0 on (0, ∞) × ∂Ω, u(0, x) = u0 in Ω. We set the initial function: u0(x) = x1(1 − x1)x2(1 − x2). h: spatial mesh size (P2 element), τ: time step of B.E. method. 35/55
  • 61. Computational results (h = 2−4 ) 0 0.1 0.2 0.3 0.4 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 t ρ k tau = 1/16 tau = 1/32 tau = 1/64 tau = 1/128 tau = 1/256 tau = 1/512 36/55
  • 62. Computational results (τ ≪ h) 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 t ρk h = 1/4 h = 1/8 h = 1/16 h = 1/32 37/55
  • 63. Global existence proof using verified computations 38/55
  • 64. 時間大域解の証明 本講演では t ∈ (0, ∞) で存在する (PJ ) の解 u(t) ∈ H1 0 (Ω) を 時間大域解といい,以下の 2 つのステップで証明を試みる. t′ 0 をある時刻として ▶ (t′ , ∞) で定常解まわりに大域的に存在する範囲を計算 機で導く.(Global existence proof) ▶ ある時刻 t′ までの解を数値解の近傍に包み込む. (Concatenation scheme) 上記の方法によって,(PJ ) の時間大域解を関数空間 L∞ ( (0, ∞); H1 0 (Ω) ) で一意存在することが計算機を用いて証明できる. 39/55
  • 65. 時間大域解の証明 本講演では t ∈ (0, ∞) で存在する (PJ ) の解 u(t) ∈ H1 0 (Ω) を 時間大域解といい,以下の 2 つのステップで証明を試みる. t′ 0 をある時刻として ▶ (t′ , ∞) で定常解まわりに大域的に存在する範囲を計算 機で導く.(Global existence proof) ▶ ある時刻 t′ までの解を数値解の近傍に包み込む. (Concatenation scheme) 上記の方法によって,(PJ ) の時間大域解を関数空間 L∞ ( (0, ∞); H1 0 (Ω) ) で一意存在することが計算機を用いて証明できる. 39/55
  • 66. 時間大域解の証明 本講演では t ∈ (0, ∞) で存在する (PJ ) の解 u(t) ∈ H1 0 (Ω) を 時間大域解といい,以下の 2 つのステップで証明を試みる. t′ 0 をある時刻として ▶ (t′ , ∞) で定常解まわりに大域的に存在する範囲を計算 機で導く.(Global existence proof) ▶ ある時刻 t′ までの解を数値解の近傍に包み込む. (Concatenation scheme) 上記の方法によって,(PJ ) の時間大域解を関数空間 L∞ ( (0, ∞); H1 0 (Ω) ) で一意存在することが計算機を用いて証明できる. 39/55
  • 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
  • 85. Table: 時間大域解の検証例(N = 8, λ = 1/40, τk = 2−7 ) γ n t′ ρ 0.01 5 0.046875 0.01085 0.011 5 0.046875 0.011936 0.0121 6 0.054688 0.011605 0.01331 7 0.0625 0.011274 0.014641 7 0.0625 0.012403 0.016105 8 0.070312 0.012038 0.017716 8 0.070312 0.013244 0.019487 9 0.078125 0.012845 0.021436 10 0.085938 0.012448 0.023579 10 0.085938 0.013695 0.025937 11 0.09375 0.013263 0.028531 11 0.09375 0.014593 0.031384 12 0.10156 0.014123 ... 50/55
  • 86. Table: 時間大域解の検証例(N = 8, λ = 1/40, τk = 2−7 ) γ n t′ ρ ... 2.2876 40 0.32031 0.029945 2.5164 41 0.32812 0.0296 2.768 42 0.33594 0.029446 3.0448 42 0.33594 0.034085 3.3493 43 0.34375 0.034657 3.6842 44 0.35156 0.035917 4.0527 45 0.35938 0.038419 4.4579 45 0.35938 0.050511 4.9037 46 0.36719 0.066455 5.3941 47 0.375 0.17656 このとき ∥u(t)∥H1 0 ≤ ρe− (t−t′) 40 , t ∈ (tn, ∞). 51/55
  • 87. 半線形放物型方程式 Let Ω := (0, 1)2 be an unit square domain in R2 .    ∂tu − ∆u = f(u) in (0, ∞) × Ω, u(t, x) = 0 on (0, ∞) × ∂Ω, u(0, x) = u0(x) in Ω, where u0(x) = sin(πx) sin(πy). ▶ f(u) = u2 + 4 ∑ 1≤k,l≤3 sin(kπx) sin(lπy); ▶ 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. 52/55
  • 88. Fig. The numerical solution ˆϕ. 時間大域解の検証は N = 10, λ = 1/40, τk = 2−8 で成功 して, ρ = 0.04035, t′ = 0.2578125. 53/55
  • 89. Fig. The numerical solution ˆϕ. 時間大域解の検証は N = 10, λ = 1/40, τk = 2−8 で成功 して, ρ = 0.04035, t′ = 0.2578125. 53/55
  • 90. まとめ ▶ 解析半群 { e−tA } t≥0 を用いる精度保証付き数値計算手法 ▶ Concatenation scheme(数値解のまわりに包み込む) ▶ 精度保証付き数値計算を用いた時間大域解の存在証明 (定常解のまわりに包み込む) 今後の課題 ▶ 方程式の拡張(多種の反応拡散方程式,波動方程式等) ▶ 無限次元力学系との関連 ▶ 藤田型方程式の爆発時刻の精度保証付き数値計算 54/55
  • 91. Thank you for kind attention! 55/55