Теоретические основы методов компьютерного моделирования. Часть 1. Яревский Е.А. - 77 стр.

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СПбГУ | Санкт-Петербург

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u(t) ∈ V u(0) = u
0
d
dt
(u(t), v) + a(u(t), v) = (f(t), v) v ∈ V.
(•, •) L
2
(Ω) a(u(t), v) = (∇u(t), ∇v)
V H
1
0
(Ω) ⊂ V ⊂
H
1
(Ω) a(•, •) : V × V → R
f ∈ L
2
v = u(t)
1
2
d
dt
||u(t)||
2
+ a(u(t), u(t)) = (f(t), u(t)),
||•|| = ||•||
L
2
(Ω)
V α a(•, •)
1
2
d
dt
||u(t)||
2
+ α||u(t)||
2
≤ ||f(t)||||u(t)||,
d
dt
||u(t)|| + α||u(t)|| ≤ ||f(t)||.
||u(t)|| ≤ ||u
0
||e
−αt
+
Z
t
0
e
−α(t−s)
||f(s)||ds 0 ≤ t ≤ T.
t t = 0
∂
∂t
u(x, t) − ∆u(x, t) = f(x, t) Ω × (0, T )
u(•, 0) = u
0
Ω
u(x, t) = 0 ∂Ω × (0, T ).
u
0
∈ L
2
(Ω) t ≥ δ > 0 k H
k
(Ω)
u(t)
||u(t)||
H
k
(Ω)
≤ Ct
−
1
2
k
||u
0
|| t > 0.

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