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R
RR
Kh
∞ ( )L2 (Ω)
Hk (Ω) Hk0 (Ω)
R1
uh
R2
Φi
Φi
L2
A : Rn −→ Rn n ∈ N+
x $−→ Ax⎛
⎝x1x2x3
⎞
⎠ $−→ A ·
⎛
⎝x1x2x3
⎞
⎠
A :=
⎛
⎜⎝a11 · · · a1n
an1 · · · ann
⎞
⎟⎠
A =
(2 51 9
)
A · x =
(2 51 9
)(x1x2
)=
(2x1 + 5x21x1 + 9x2
).
Ax = b
2x1 + 5x2 = b1
x1 + 9x2 = b2.
A
A = L · U
Eij aijA
A =
(2 18 7
).
E21 :=
(1 0−4 1
)a21
(1 0−4 1
)
︸ ︷︷ ︸E21
·(2 18 7
)
︸ ︷︷ ︸A
=
(2 10 3
)
︸ ︷︷ ︸U
A = L · U L = E−121 =
(1 04 1
)
Ax = LUx = E−121 Ux = b
⇒ x = U−1E21b.
E32E31E21A = U
E31 = Id, E32 =
⎛
⎝1 0 00 1 00 −5 1
⎞
⎠ , E21 =
⎛
⎝1 0 0−2 1 00 0 1
⎞
⎠
∏i,j Ei,j
E−1ij
∏i,j E
−1i,j
⎛
⎝1 0 00 1 00 −5 1
⎞
⎠
⎛
⎝1 0 0−2 1 00 0 1
⎞
⎠ =
⎛
⎝1 0 0−2 1 010 −5 1
⎞
⎠
L
⎛
⎝1 0 02 1 00 0 1
⎞
⎠
︸ ︷︷ ︸E−1
21
⎛
⎝1 0 00 1 00 5 1
⎞
⎠
︸ ︷︷ ︸E−1
32
=
⎛
⎝1 0 02 1 00 5 1
⎞
⎠
L
n× n
n = 100 A1002 992
982 w
w ≈ 1002 + 992 + 982 + . . .+ 12.
w ≈ n2 + (n− 1)2 + (n− 2)2 + . . .+ 12.
w
w ≈n∑
x=0
x2 ≈ˆ n
0x2dx =
1
3x3|n0 =
1
3n3.
O(13n
3)
Au = 0 Au = b
x = 0 A Ax
Ax = λx,
λ x A
P : R3 → R3
x E PP E
Px1 = 1 · x1Px2 = 1 · x2
x1 ∦ x2 x3 E
Px3 = 0 · x.
A =
(0 11 0
)
x1 =
(11
),
x2 =
(−11
)
λ1 = 1 λ2 = −1
Ax = λx
λ x
Ax = λx ⇔ (A− λ · Id)x = 0
x = 0 A− λ · Id
(A− λId) = 0.
(A− λId)
λ x
A =
(3 11 3
)
(A− λId) =
∣∣∣∣
(3− λ 11 3− λ
)∣∣∣∣ = (3− λ)2 − 1 =
= λ2 − 6λ+ 8
A λ1 = 4 λ2 = 2
⇒ A− 4 · Id =
(−1 11 −1
)
(−1 11 −1
)· x = 0
λ1 = 4 x1 =
(11
)
λ2 = 2 x2 =
(−11
)
x Ax = λx A(αx) = λ(αx)α y = 0 ⟨x⟩ := αxλ
n× n AC A ∈ Cn×n n C
nn C R
A ∈ Rn×n
λ1, . . . ,λn ∈ C A ∈ Cn×n
n∑
i=1
λi = (A)
n∏
i=1
λi = (A).
Qα
Qα =
((α) − (α)(α) (α)
)
90o = π2
Qπ2=
(0 −11 0
)
Qπ2
λ1 + λ2 =(Qπ
2
)= 0,
λ1 · λ2 =(Qπ
2
)= 1.
R
(Qπ
2− λId
)=
∣∣∣∣
(−λ −11 −λ
)∣∣∣∣ = λ2 + 1 = 0
⇒ λ1/2 = ±i.
n×n n
A =
(3 10 3
)
Aλ1/2 = 3
(A− λId)x =
(0 10 0
)(x1x2
)=
(00
).
A x1 =
⟨(10
)⟩
n×n A n
A n
n× n A n x1, . . . , xnS x1, . . . , xn
S :=
⎛
⎜⎜⎜⎝
x11 · · · x1nx21 · · · x2n
xn1 · · · xnn
⎞
⎟⎟⎟⎠.
A · S
A · S = A ·
⎛
⎜⎜⎜⎝
x11 · · · x1nx21 · · · x2n
xn1 · · · xnn
⎞
⎟⎟⎟⎠=
⎛
⎜⎜⎜⎝
λ1x11 · · · λnx1nλ1x21 · · · λnx2n
λ1xn1 · · · λnxnn
⎞
⎟⎟⎟⎠
=
⎛
⎜⎜⎜⎝
x11 · · · x1nx21 · · · x2n
xn1 · · · xnn
⎞
⎟⎟⎟⎠
⎛
⎜⎜⎜⎝
λ1 0 · · · 00 λ2 0 · · ·
0 · · · 0 λn
⎞
⎟⎟⎟⎠
⇒ A · S = S · Λ
Λ :=
⎛
⎜⎜⎜⎝
λ1 0 · · · 00 λ2 0 · · ·
0 · · · 0 λn
⎞
⎟⎟⎟⎠
Λ S−1
AS = SΛ
⇔ S−1AS = Λ
⇔ A = SΛS−1
S−1
A
A
Ax = λx A2x
A2x = Aλx = λAx = λ2x
⇒ A2
A
A = SΛS−1
Ak = (SΛS−1) · · · (SΛS−1) = SΛkS−1.
A A100
A100 = (SΛS−1)(SΛS−1) · · · (SΛS−1) = SΛ100S−1.
Ak → 0 k → ∞,
∥λi∥ < 1 ∀i = 1, . . . , n
nnn
A A =
⎛
⎜⎝1
1
⎞
⎟⎠
λ1 = λ2 = . . . = λn = 1.
n n
x1 =
⎛
⎜⎜⎜⎝
10
0
⎞
⎟⎟⎟⎠, x2 =
⎛
⎜⎜⎜⎝
01
0
⎞
⎟⎟⎟⎠, . . . , xn =
⎛
⎜⎜⎜⎝
00
1
⎞
⎟⎟⎟⎠.
A =
(2 10 2
)λ1 = λ2 = 2
(A− 2Id)x =
(0 10 0
)x = 0.
A − 2Id x1 =
(10
)
n n
A ∈ Rn×n
uk+1 = Auk,
u0 ∈ Rn
u1 = Au0,
u2 = A2u0,
uk = Aku0.
u0x1, x2, . . . , xn A
⇒ u0 = c1x1 + c2x2 + · · ·+ cnxn = Sc.
Au0
Au0 = Ac1x1 +Ac2x2 + · · ·+Acnxn
= c1λ1x1 + c2λ2x2 + · · ·+ cnλnxn
Aku0 = c1λk1x1 + c2λ
k2x2 + · · ·+ cnλ
knxn,
0, 1, 1, 2, 3, 5, 8, 13, . . .
F100
Fk+2 = Fk+1 + Fk
Fk+2 = Fk+1 + Fk
Fk+1 = Fk+1.
uk :=
(Fk+1
Fk
)
Auk = uk+1
A =
(1 11 0
)
(A− λId) =
∣∣∣∣
(1− λ 11 −λ
)∣∣∣∣ = −λ(1− λ)− 1 = 0
⇔ λ2 − λ− 1 = 0
⇔ λ1/2 =1±
√1 + 4
2⇒ λ1 ≈ 1.618
λ2 ≈ −0.618.
2 × 2
Ak = (SΛS−1)k = S
(1.6k 00 −0.6k
)S−1
⇒ F100 ≈ c1 · 1.6100, Fk ≈ c1 · 1.6k
k → ∞uk
uk = c1x1 + c2x2
x1, x2
(A− λId)x = 0
⇔(1− λ 11 −λ
)(x1
x2
)=
(00
)
x1 =
(λ11
)x2 =
(λ21
)c1 c2
u0 =
(F1
F0
)=
(10
)
c1x1 + c2x2 = c1
(λ11
)+ c2
(λ21
)=
(10
).
2×2 c1/2 = ± 1√5≈ ±0.447
F100 ≈ 0.447 · 1.699 ≈ 3.54× 1020.
du1dt
= −u1 + 2u2
du2dt
= u1 − 2u2
u(0) =
(10
)
(du1dtdu2dt
)=
(−1 21 −2
)(u1u2
)= Au.
A λ1 = 0 A λ2 = −3∑λi = (A) =
−3
x1 =
(21
),
x2 =
(1−1
).
u
u(t) = c1 (λ1t)x1 + c2 (λ2t)x2.
c1 c2
c1 (λ1t)x1 + c2 (λ2t)x2 = c1
(21
)+ c2 (−3t)
(1−1
)
u(0) =
(10
)t = 0
c1
(21
)+ c2
(1−1
)=
(10
)⇔ c1 = c2 =
1
3
⇒ u(t) =1
3x1 +
1
3(−3t)x2.
t → ∞
⇒ u(∞) =1
3
(21
).
u(t) → 0
u(t) → v
u(t) → ±∞u(t) → 0
eλt → 0 ⇒ λ < 0.
λ ∈ C
λ = −3 + 6i∣∣∣e(−3+6i)t
∣∣∣ =∣∣e−3t
∣∣ ∣∣e6it∣∣
e6it = (6t) + i (6t) =
((6t)(6t)
)(1i
)
∣∣e6it∣∣ =
∣∣∣∣
((6t)(6t)
)∣∣∣∣ =√
2(6t) + 2(6t) = 1.
Re(λ) < 0
λ = 0 Re(λ) < 0Re(λ) >
0
2× 2 A =
(a bc d
)
Reλ1 < 0
Reλ2 < 0.
(A) = a+d = λ1+λ2 < 0 (A) < 0(−2 00 1
)λ2 = 1 > 0
(A) > 0.
A ∈ Cn×n
du
dt= Au.
A ui uu := Sv S
dv
dt= S−1ASv = Λv
⇒ v(t) = eΛtv(0)
⇔ S−1u(t) = eΛtS−1u(0)
⇔ u(t) = SeΛtS−1u(0).
eAt = SeΛtS−1,
u(t) = eAtu(0) = SeΛtS−1u(0).
eAt = Id+At+(At)2
2+
(At)3
6+ . . .+
(At)n
n!+ . . .
eAt = SeΛtS−1
eAt = Id+At+(At)2
2+
(At)3
6+ . . .
= Id+ (SΛS−1)t+(SΛS−1)(SΛS−1)t2
2 + . . .=
= Id+ (SΛS−1)t+1
2t2(SΛ2S−1) +
1
6t3(SΛ3S−1) + . . .
= S(S−1 + ΛS−1t+1
2t2Λ2S−1 + . . .) =
= S(Id+ Λt+ Λ2 t2
2+
Λ3t3
6+ . . .)S−1 =
= SeΛtS−1
⇒ eAt = eSΛS−1t = SeΛtS−1.
eΛt =
⎛
⎜⎝eλ1t 0 · · ·
· · · 0 eλnt
⎞
⎟⎠ .
eAt = SeΛtS−1 → 0,
eΛt → 0 Re(λ) < 0
y′′ + by′ + cy = 0
u =
(y′
y
)
y′′ + by′ + cy = 0
y′ = y′
u′ =
(y′′
y′
)=
(−b −c1 0
)(y′
y
)=
(−b −c1 0
)u.
u′ = f(u, t)
u(t = 0) = u0
u′i = fi(u, t), i = 1, . . . , n
ui(t = 0) = ui,0.
u′ = au, a ≈ ∂f
∂u
u′ = Au, Aij ≈∂fi∂uj
.
un+1 un, un−1, . . . tn, tn−1, . . .un+1 tn+1
n+ 1
u′(t) ≈ Duf (u(t), t) u
u(t) = e−t + e−99t.
u(t)
e−t u(t)e−99t ∆t
A =
(−50 4949 −50
)u = Au
u(t) = e−t + e−99t.
λ1 = −1 λ2 = −99
A
cond(A) :=|λ|max
|λ|min.
cond(A) = 99
a > 0
u′ = f(u, t) = au.
u′ ≈ un+1 − un∆t
.
un+1 − un∆t
= aun
⇔ un+1 = un +∆taun = (1 + a∆t)un
⇒ un = (1 + a∆t)nu0.
1+a∆t > 1 una < 0
|1 + a∆t| ≤ 1.
a ∆ta < 0
0 < ∆t < −2
a.
un+1 un
un+1 − un∆t
= f(un+1, tn+1) = aun+1
⇔ un+1 =1
1− a∆tun
⇒ un =
(1
1− a∆t
)n
u0.
a < 0 ∣∣∣∣1
1− a∆t
∣∣∣∣ < 1 ⇒ ∆t.
a < 0 a > 0
∆t → 0
uk+1 = uk +∆tΦ(uk+1, uk, tk).
Φ
tk = t0 + k ·∆t,
u(tk) : tk,uk : tk.
tk+1
dk+1 := u (tk+1)− u (tk)−∆tΦ (u (tk+1) , u (tk) , . . . , tk) .
u′ = au
un+1 = un +∆taun
u(tn+1) = u(tn) +∆tau(tn) + dn+t
⇒ dn+1 = u(tn−1)− u(tn)−∆tau(tn)
tn
en := u(tn)− un?
en+1 = u(tn+1)− un+1 = u(tn) + a∆tu (tn) + dn+1 − (un + a∆tun)
⇒ en+1 = en + a∆ten + dn+1 = (1 + a∆t)nd1 + . . .+ (1 + a∆t)n+1−kdk + . . .+ dn+1.
|1 + a∆t| ≤ 1 dk+1 =12 (∆t)2 u′′ (tk + θ∆t) , 0 < θ < 1
en+1 ≤ (n+ 1) · 12(∆t)2
∥∥u′′∥∥∞ =
1
2T ·∆t
∥∥u′′∥∥∞
T := (n + 1) · ∆t∆t
gk tk
gk := u(tk)− uk.
gkdk
Φ : B → RL ∈ R 0 < L < ∞
|Φ(x, y1, z,∆t)− Φ(x, y2, z,∆t)| ≤ L |y1 − y2| ,|Φ(x, y, z1,∆t)− Φ(x, y, z2,∆t)| ≤ L |z1 − z2|
(x, y1, z,∆t) , (x, y2, z,∆t) (x, y, z1,∆t) (x, y, z2,∆t) ∈ B
u (tk+1) = u (tk) +∆t · Φ (u (tk+1) , u (tk) , tk) + dk+1.
gk+1 = gk +∆t (Φ (u (tk+1) , u (tk) , tk)− Φ (u (tk+1) , uk, tk)
+Φ (u (tk+1) , uk, tk)− Φ (uk+1, uk, tk)) + dk+1.
∆t · L < 1
|gk+1| ≤ |gk|+∆t (L |u (tk)− uk|+ L |u (tk+1)− uk+1|) + |dk+1|
⇒ |gk+1| ≤1 +∆tL
1−∆tL|gk|+
|dk+1|1−∆tL
.
Φ uk+1
|gk+1| ≤ (1 +∆tL) |gk|+ |dk+1| .
∆tL < 1 K > 0 1+∆tL1−∆tL ≤ 1 +∆tK
|gk+1| ≤ (1 + a) |gk|+ b
a =
∆tK ( )
∆tL ( )b =
K2L |dk+1| ( )
|dk+1| ( )
(gk)k∈N
|gk+1| ≤ (1 + a) |gk|+ b ∀k ∈ N+,
|gk| ≤ (1 + a)k |g0|+(1 + a)k − 1
ab ≤ eka |g0|+
b
a
(eka − 1
)∀k ∈ N.
|gk| ≤ (1 + a) |gk−1|+ b ≤ (1 + a)2 |gk−2|+ ((1 + a) + 1) b
≤ (1 + a)k |g0|+((1 + a)k−1 + . . .+ (1 + a) + 1
)b
= (1 + a)k |g0|+(1 + a)k − 1
a· b.
(1 + t) ≤ et ∀t (1 + a)k ≤ eka
g0 = u(t0)− u0 = 0
D := k |dk| gn tn = t0 + n∆t
|gn| ≤D
∆tL
(en∆tL − 1
)≤ D
∆tL· en∆tL.
|gn| ≤D
2∆tL
(en∆tK − 1
)≤ D
2∆tL· en∆tK .
DL h
dk+1 = u (tk+1)− u (tk)−∆tf (u (tk) , tk) .
u (tk+1) = u (tk) +∆tu′ (tk) +1
2(∆t)2 u′′ (tk + θ∆t)
0 < θ < 1 f (u (tk) , tk) = u′ (tk)
dk+1 = u (tk)+∆tu′ (tk)+1
2(∆t)2 u′′ (tk + θ∆t)−u (tk)−∆tu′ (tk) =
1
2(∆t)2 u′′ (tk + θ∆t) .
M := t0≤ξ≤tn |u′′(ξ)| |dk+1| ≤ 12 (∆t)2M
|gn| ≤∆tM
2Len∆tL.
gn ∆t
pC dk
|dk| ≤ D = C (∆t)p+1 = O((∆t)p+1
).
gn
|gn| ≤C
Len∆tL (∆t)p = O ((∆t)p) .
p
u (tk+1) = u (tk)+∆t
1!u′ (tk)+
(∆t)2
2!u′′ (tk)+
(∆t)3
3!u(3) (tk)+ . . .+
(∆t)p
p!u(p) (tk)+Rp+1
p
dk+1 = Rp+1 =(∆t)p+1
(p+ 1)!u(p+1) (tk + θ∆t) , 0 < θ < 1.
u′ = −2tu2
u(0) = 1 u t
u (tk+1) = u (tk) + c1∆t+ c2 (∆t)2 + c3 (∆t)3 + c4 (∆t)4 + . . .
ci u′ = −2tu2 t = tk +∆t
c1 + 2c2∆t + 3c3 (∆t)2 + 4c4 (∆t)3 + . . .
= −2(tk +∆t)(u (tk) + c1∆t+ c2 (∆t)2 + c3 (∆t)3 + c4 (∆t)4 + . . .
)2
= −2(tk +∆t)(u2 (tk) + 2c1u (tk)∆t+
(c21 + 2c2u (tk)
)(∆t)2+
+ (2c1c2 + 2c3u (tk)) (∆t)3 + . . .)
= −2tku2 (tk) +
(−2u2 (tk)− 4c1tku (tk)
)∆t+
(−4c1u (tk)− 2tk
(c21 + 2c2u (tk)
))(∆t)2 +
+(−2(c21 + 2c2u (tk)
)− 4tk (c1c2 + c3u (tk))
)(∆t)3 + . . .
c1 = −2tku2 (tk) ≈− 2tku
2k
c2 = − (u (tk) + 2c1tk)u (tk) ≈− (uk − 2c1tk)uk
c3 =−(4c1u (tk) + 2tk
(c21 + 2c2u (tk)
))
3≈− 1
3
(4c1uk + 2tk
(c21 + 2c2uk
))
c4 = −1
2c21 − c2u (tk)− tk (c1c2 + c3u (tk)) ≈− 1
2c21 − c2uk − tk (c1c2 + c3uk)
ek := u (tk)− uk
∆t ∆t2
u(1)k+1 = uk +∆tf (uk, tk)
u(2)k+ 1
2
= uk +∆t
2f (uk, tk)
u(2)k+1 = u(2)k+ 1
2
+∆t
2f
(u(2)k+ 1
2
, tk +∆t
2
)
uk+1 := 2u(2)k+1 − u(1)k+1 = 2u(2)k+ 1
2
+∆tf
(u(2)k+ 1
2
, tk +∆t
2
)− uk −∆tf (uk, tk)
= uk +∆tf
(uk +
∆t
2f (uk, tk) , tk +
∆t
2
).
k1 := f (uk, tk) ,
k2 := f
(uk +
∆t
2k1, tk +
∆t
2
),
uk+1 = uk +∆tk2.
u′(t) = f (u(t), t)
[tk, tk+1]
tk+1ˆtk
u′(t)dt =
tk+1ˆtk
f (u(t), t) dt
⇔ u(tk+1)− u(tk) =
tk+1ˆtk
f (u(t), t) dt.
u(t)
tk+1ˆtk
f (u(t), t) dt ≈ ∆t
2(f (uk, tk) + f (uk+1, tk+1)) .
uk+1 = uk +∆t
2(f (uk, tk) + f (uk+1, tk+1))
uk+1
u(0)k+1 = uk +∆tf (uk, tk)
u(n+1)k+1 = uk +
∆t
2
(f (uk, tk) + f
(u(n)k+1, tk+1
))
uk+1 fL ∆tL
2 < 1 ∆t < 2L
Φ (uk, uk+1, tk) :=1
2(f (uk, tk) + f (uk+1, tk+1)) .
dk+1 = u (tk+1)− u (tk)−∆t
2(f (u (tk) , tk) + f (u (tk+1) , tk+1))
= u (tk+1)− u (tk)−∆t
2
(u′ (tk) + u′ (tk+1)
)
= ∆tu′ (tk) +(∆t)2
2u′′ (tk) +
(∆t)3
6u′′′ (tk) +O
((∆t)4
)
−∆t
2
(u′ (tk) + u′ (tk) +∆tu′′ (tk) +
(∆t)2
2u′′′ (tk) +O
((∆t)3
))
= − 1
12(∆t)3 u′′′ (tk) +O
((∆t)4
).
(∆t)3
uk+1 ≈ u(tk+1)
u(p)k+1 = uk +∆tf (uk, tk) ,
uk+1 = uk +∆t
2
(f (uk, tk) + f
(u(p)k+1, tk+1
)).
k1 = f (uk, tk) ,
k2 = f (uk +∆tk1, tk+1) ,
uk+1 = uk +∆t
2(k1 + k2) .
k1 k2 (tk, uk)(tk+1, u
(p)k+1
)
U ⊂ Rn f : U → R fx ∈ U
Dif(x) :=h→0
f(x+ hei)− f(x)
h
ei ∈ Rn (ei)j = δij∂if ∂eif
∂f∂xi
Dif
U ⊂ Rn f : U → Rx ∈ U
f(x) :=
(∂f
∂x1(x), . . . ,
∂f
∂xn(x)
)
f x
(f) ∇f
∇ :=
(∂
∂x1, . . . ,
∂
∂xn
)
f, g : U → R
(f · g) = g · f + f · g
U ⊂ Rn
v = (v1, . . . , vn) : U → Rn
vi
v :=n∑
i=1
∂vi∂xi
v
∇ · v.
U ⊂ R3 v : U →R3
v :=
(∂v3∂x2
− ∂v2∂x3
,∂v1∂x3
− ∂v3∂x1
,∂v2∂x1
− ∂v1∂x2
)
v
v
v = ∇× v
U → Rn f : U → R
∆f := f = ∇f =∂2f
∂x21+ . . .+
∂2f
∂x2n
∆ :=∂2
∂x21+ . . .+
∂2
∂x2n=
n∑
i=1
∂2
∂x2i
u
t= f(t, u)
t f(t, u)f
c(x, t)
F c(x, t)
F = −D · c,
D
Vc V c V
ˆ
V
∂c
∂tdx.
V
−ˆ
∂V
F · ndS =
ˆ
V
∂c
∂tdx.
F : Rn → Rn
V ⊂ Rn
ˆ
V
F (x) dx =
ˆ
∂V
F (x) · n dS.
−ˆ
V
F (x) dx = −ˆ
∂V
F · n dS =
ˆ
V
∂c
∂tdx ∀V
⇒ − F =∂c
∂t
⇒ ∂c
∂t= − (D∇c)
v ρp
∂ρ
∂t= −ρ0 v
ρ0
ρ0∂v
∂t= − p.
p
⇒ p = c2 · ρ
⇒ ∂2
∂t2ρ = −ρ0
(∂v∂t
)= −
(ρ0∂v
∂t
)
⇒ 1
c2∂2
∂t2p = ( p)
⇔ ∂2
∂t2p = c2 · ( p) = c2∆p
R1 R2
R1
utt = uxx.
R2
utt = c2∆u.
Ω ⊂ Rd d ∈ 2, 3 ρ : Ω → R ΩΦ
−∆Φ = ρ Ω.
∆u = 0 Ω ⊂ Rd.
Ω := (x, y) ∈ R2;x2+y2 < 1 x y
x := r · φ
y := r · φ
∆u =∂2u
∂r2+
1
r
∂u
∂r+
1
r2∂2u
∂φ2.
rk (kφ) rk (kφ)r = 1
u|∂Ω = u( φ, φ) = a0 +∞∑
k=1
(ak (kφ) + bk (kφ)) .
r < 1
u(x, y) = a0
∞∑
k=1
rk · (ak (kφ) + bk (kφ)) .
ciΦ
∂ci∂t
= ∇ ·(Di∇ci +Di
ziF
RTci∇Φ
)
−∇(εrε0∇Φ) = ρf +∑
i
ziFci
Di ci zi ρfεr ε0 F
R T
−∆u = f Ω
Ω
Ω
Ω
Ω −∆u = f
Ω = (x, y) : 0 < x < 1, 0 < y < 1.
Ω
ΩΩh h
Ωh =(x, y) ∈ Ω :
x
h,y
h∈ Z
.
Ωh
u(x) uh(x) u(x) uh(x)
h→0
u(x+ h)− u(x)
h≈ u(x+ h)− u(x)
h
Ωh
Ωh
Ω
R
u′′(x) = f(x) Ω = (0, 1),
u(0) = ϕ0,
u(1) = ϕ1.
δ+u(x) = u(x+h)−u(x)h
δ−u(x) = u(x)−u(x−h)h
δ0u(x) = u(x+h)−u(x−h)2h
δ+ δ−
u′′(x) ≈ δ+δ−u(x) =u(x+h)−u(x)
h − u(x)−u(x−h)h
h=
u(x+ h)− 2u(x) + u(x− h)
h2.
[x− h, x+ h] ⊂ Ω
δ±u(x) = u′(x) + hR |R| ≤ 12∥u
′′∥∞
δ0u(x) = u′(x) + h2R |R| ≤ 16∥u
′′′∥∞
δ+δ−u(x) = u′′(x) + h2R |R| ≤ 112∥u
(4)∥∞
u(x± h) = u(x)± hu′(x) +h2
2u′′(x) + . . .
= u(x)± hu′(x) +h2
2u′′(ξ), x ≤ ξ ≤ x+ h
⇔ u(x+ h)− u(x)
h= u′(x) +
h
2u′′(ξ)
≤ u′(x) +h
2∥u′′∥∞
x± h
u(x+ h) = u(x) + hu′(x) +h2
2u′′(x) +
h3
6u′′′(ξ),
u(x− h) = u(x)− hu′(x) +h2
2u′′(x)− h3
6u′′′(ξ).
δ0u(x) =2hu′(x) + h3
6 (u′′′(ξ) + u′′′(ξ))
2h
= u′(x) +h2
12(u′′′(ξ) + u′′′(ξ)
≤ u′(x) +h2
12· 2∥u′′′∥∞ = u′(x) +
h2
6∥u′′′∥∞
x± h
u(x+ h) = u(x) + hu′(x) +h2
2u′′(x) +
h3
6u′′′(x) +
h4
24u(4)(ξ)
u(x− h) = u(x)− hu′(x) +h2
2u′′(x)− h3
6u′′′(x) +
h4
24u(4)(ξ)
2u(x) h2
δ+δ−u(x) =h2u′′(x) + h4
24
(u(4)(ξ) + u(4)(ξ)
)
h2
⇒ δ+δ−u(x) = u′′(x) +h2
24
(u(4)(ξ) + u(4)(ξ)
)
≤ u′′(x) +h2
12∥u(4)∥∞
u′′(x) = ∆u(x) = f(x),
∆ ≈ ∆h = δ+δ−
δ+δ−u(x) = f(x) +O(h2)
δ+δ−u(x) =1
h2
⎛
⎝−2 1 01 −2 10 1 −2
⎞
⎠ ·
⎛
⎝uh(x1)uh(x2)uh(x3)
⎞
⎠ =
⎛
⎝f(x1)− 1
h2u(0)f(x2)f(x3)− 1
h2u(1)
⎞
⎠ .
10 x3x2x1
1
h2·
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
−2 1 0 . . . 0 0 01 −2 1 0 . . . 0 00 1 −2 1 0 . . . 0
0 0 . . . 0 1 −2 10 0 . . . 0 0 1 −2
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
u1
un
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
=
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
f1 − u0h2
f2
fn−1fn − un
h2
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
,
Kh · uh = fh
Kh
(i, j) ∈ 1, . . . , n2 : Ki,j = 0 = O(n).
R
Ω :=(x, y) ∈ R2 : 0 < x < 1, 0 < y < 1
Ωh :=(x, y) ∈ Ω :
x
h,y
h∈ Z
Γ :=(x, y) ∈ R2 : x ∈ 0, 1, y ∈ 0, 1
,
Γh :=(x, y) ∈ Γ :
x
h,y
h∈ Z
.
−∆u = f Ω,
u = ϕ Γ
−∆huh := (−δ−x δ+x − δ−y δ+y )uh(x).
uh uu Ωh
∆h uh
−∆huh = (−δ−x δ+x − δ−y δ+y )uh(x)
= − 1
h2(uh(x− h, y) + uh(x+ h, y) + uh(x, y − h) + uh(x, y + h)− 4uh(x, y)) .
uh
R
R2
1 2
4
3
5 6
7 8 9
1
h2
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
4 −1 0 −1 0 0 0 0 0−1 4 −1 0 −1 0 0 0 00 −1 4 −1 0 −1 0 0 0
−1 0 0 4 −1 0 −1 0 00 −1 0 −1 4 −1 0 −1 00 0 −1 0 −1 4 0 0 −10 0 0 −1 0 0 4 −1 00 0 0 0 −1 0 −1 4 −10 0 0 0 0 −1 0 −1 4
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
· uh = fh
fh f
Kh =1
h2
⎛
⎜⎜⎜⎝
D −I 0 0−I D −I 0
0 0 −I D
⎞
⎟⎟⎟⎠
D =
⎛
⎜⎜⎜⎜⎜⎜⎝
4 −1 0 · · · 0−1 4 −1 0 · · ·
−1
−10 · · · 0 −1 4
⎞
⎟⎟⎟⎟⎟⎟⎠,
−I =
⎛
⎜⎜⎜⎜⎜⎜⎝
−1 0 0 · · · 00 −1 0 · · · 0
0 · · · 0 0 −1
⎞
⎟⎟⎟⎟⎟⎟⎠.
1 2
4
3
5
6
7 8
9
1
h2
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
4 0 0 0 0 −1 −1 0 00 4 0 0 0 −1 0 −1 00 0 4 0 0 −1 −1 −1 −10 0 0 4 0 0 −1 0 −10 0 0 0 4 0 0 −1 −1
−1 −1 −1 0 0 4 0 0 0−1 0 −1 −1 0 0 4 0 00 −1 −1 0 −1 0 0 4 00 0 −1 −1 −1 0 0 0 4
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
· uh = fh
Kh =
(D1 LLT D2
)
Kh Di L LT
R2
−∆huh =1
h2(−u(x− h, y)− u(x+ h, y)− u(x, y − h)− u(x, y + h) + 4u(x, y))
=:1
h2
⎡
⎣−1
−1 4 −1−1
⎤
⎦ = −∆h.
Ωh
Kh
R1
δ+ = 1h ·[0 −1 1
]
δ− = 1h ·[−1 1 0
]
δ0 = 12h ·
[−1 0 1
]
1
hk
⎡
⎢⎢⎢⎢⎢⎢⎣
c−1,1 c0,1 c1,1· · · c−1,0 c0,0 c1,0 · · ·
c−1,−1 c0,−1 c1,−1
⎤
⎥⎥⎥⎥⎥⎥⎦(x, y)
=1
hk·∑
i,j
cijuh(x+ ih, y + jh)
[a b c
] [d e f
]uh =
[a b c
]· (d · uh(x− h) + e · uh(x) + f · uh(x+ h))
= a(duh(x− 2h) + euh(x− h) + fuh(x))
+b(duh(x− h) + euh(x) + fuh(x+ h))
+c(duh(x) + euh(x+ h) + fuh(x+ 2h))
=[ad ae+ bd af + be+ cd bf + ce cf
].
n∑
j=1
aij = 0 ∀i = 1 . . . n.
∆h
aii > 0, aij ≤ 0 (i = j).
∆2h
|aii| ≥n∑
j=1j =i
|aij | ∀i = 1 . . . n
|aii| >n∑
j=1j =i
|aij | ∀i = 1 . . . n
Kh
Kh
Kh
A ∈ Kn×n A ∈Kn×n
A · A = A ·A = En,
En
A ∈ Kn×n
n∑
j=1
λjAj = 0 ⇔ λj = 0 ∀j = 1 . . . n,
Aj A
A ∈ Kn×n
A ∈ Kn×n
A
AT
A n A n
(AT )−1 = (A−1)T
A,B ∈ Kn×n (aij)i,j=1...n (bij)i,j=1...n
A ≥ B aij ≥bij , ∀i, j = 1 . . . n A ≤ B A < B A > B
n×n
aii > 0, aij ≤ 0 ∀i, j = 1 . . . n, i = j
A A−1 ≥ 0
Kh
−∆u = f,
Khuh = fh
Kh AA−1 ≥ 0
A ∈ Kn×n i, j ∈ 1, . . . , n
i j aij = 0
i j (ik)k=1,...,p ⊂1, . . . , n i = i1, ip = j ik−1 ik
k = 2 . . . n
A i ∈ 1, . . . , nj ∈ 1, . . . , n
A ⇔ Π
ΠTAΠ =
(A1 00 A2
).
A = (aij) ∈ Cn×n
Ki :=
⎧⎪⎪⎨
⎪⎪⎩z ∈ C : |z − aii| ≤
n∑
j=1j =i
|aij |
⎫⎪⎪⎬
⎪⎪⎭, i = 1 . . . n
n A
n⋃
i=1
Ki.
v A λ∥v∥∞ := n
j=1 |vj | = 1|vi| = 1
(A− λEn)v = 0.
(aii − λ)vi = −n∑
j=1j =i
aijvj .
|aii − λ| = |(aii − λ)vi| =
∣∣∣∣∣∣∣∣
n∑
j=1j =i
aijvj
∣∣∣∣∣∣∣∣
≤n∑
j=1j =i
|aij ||vj | ≤n∑
j=1j =i
|aij |
⇒ |aii − λ| ≤n∑
j=1j =i
|aij |
⇔ λ ∈ Ki =
⎧⎪⎪⎨
⎪⎪⎩z ∈ C : |z − aii| ≤
n∑
j=1j =i
|aij |
⎫⎪⎪⎬
⎪⎪⎭.
Ki
Kh
∑nj=1,j =i |aij | = 4h−2 ∀j ⇒ 4h−2
aii = 4h−2 4h−2
λ ∈[0, 8h−2
]
Kh
λ ∈(0, 8h−2
).
A
λ ∈(
n⋃
i=1
Ki
)∪(
n⋂
i=1
∂Ki
)
Ki :=
⎧⎨
⎩z ∈ C : |z − aii| <n∑
j=1,j =i
|aij |
⎫⎬
⎭ ,
∂Ki :=
⎧⎨
⎩z ∈ C : |z − aii| =n∑
j=1,j =i
|aij |
⎫⎬
⎭ .
λ A v ∥v∥∞ = 1i ∈
1, . . . , n |vi| = 1 |λ − aii| ≤∑n
j=1,j =i |aij | |λ − aii| <∑nj=1,j =i |aij | λ ∈ Ki
k |vk| = 1
|λ− akk| =n∑
j=1,j =k
|akj | ∀k ∈ k ∈ 1, . . . , n : |vk| = 1 .
|λ−ajj | =∑n
k=1,k =j |ajk| ∀j = 1 . . . n λ ∈⋂n
i=1 ∂Ki
j ∈ 1, . . . , n \ i
i = i0, i1, . . . , il = j aip−1ip = 0.
∣∣λ− aipip∣∣ =
∑nk=1,k =ip
∣∣aipk∣∣ ∣∣vip
∣∣ = 1∣∣λ− aip+1ip+1
∣∣ =∑n
k=1,k =ip+1
∣∣aip+1k
∣∣ ∣∣vip+1
∣∣ = 1 p = 0 . . . l−1
∣∣λ− aipip∣∣ =
∑nj=1,j =ip
∣∣aipj∣∣ ∣∣vip
∣∣ = 1 p ∈ 0, . . . , l − 1
∣∣λ− aipip∣∣ ≤
n∑
k=1,k =ip
∣∣aipk∣∣ |vk|
n∑
k=1,k =ip
∣∣aipk∣∣ |vk| ≥
n∑
k=1,k =ip
∣∣aipk∣∣ .
∥v∥∞ = 1
n∑
k=1,k =ip
∣∣aipk∣∣ |vk| =
n∑
k=1,k =ip
∣∣aipk∣∣ .
|vk| ≤ 1 ∀k |vk| = 1 ∀k aipk = 0∣∣vip+1
∣∣ = 1.|λ− aip+1ip+1 | =
∑nk=1,k =ip+1
|aip+1k|
Kh
aii =4
h2
ri =n∑
j=1,j =i
|aij | =
⎧⎨
⎩
2/h2
3/h2
4/h2.
Kh Kh ∂Kj
Kh
λ ∈(0,
8
h2
).
A ∈ Kn×n
∑
j,j =i
|aij | < |aii| ∀i = 1, . . . , n
A∑
j,j =i
|aij | < |aii| i
∑
j,j =i
|aij | ≤ |aii| ∀i = 1, . . . , n
ϱ(A) A ∈ Kn×n
ϱ(A) := |λ| : λ A.
D−1B D := aii : i = 1, . . . , nB := D −A
A ϱ(D−1B) < 1.
A A ⇔ ϱ(D−1B) < 1
C := D−1B
cij = −aijaii
, (i = j),
cii = 0.
ri :=n∑
j=1j =i
|cij | < 1 ∀i = 1, . . . , n.
λ ∈n⋃
i=1
Kri(cii) =n⋃
i=1
Kri(0)
⇒ |λ| ≤i=1...n
ri < 1
⇒ ϱ(D−1B) < 1.
A ⇒ rj ≤ 1 ∀j = 1, . . . , nri < 1 i
λ ∈n⋃
j=1
Krj (0) ∪
⎛
⎝n⋂
j=1
∂Krj (0)
⎞
⎠ .
i ri < 1 ∂Kri(0) ⊂ K1(0)⋂nj=1 ∂Krj (0) ⊂ K1(0) ϱ(D−1B) < 1
AA A−1 ≥ 0 ⇔ ϱ(D−1B) < 1
⇐ ϱ(D−1B) = ϱ(C) < 1
S :=∞∑
ν=0
Cν = (I − C)−1
⇔ S(I − C) = I
⇔ SD−1(D −B) = SD−1A = I
⇔ A−1 = SD−1.
D−1 ≥ 0, B ≥ 0 ⇒ C ≥ 0 ⇒ Cν ≥ 0 ⇒ S ≥ 0 ⇒ A−1 ≥ 0.⇒ A u = 0 D−1B λ
|u| (|ui|)ni=1
|λ| · |u| = |λu| =∣∣D−1Bu
∣∣ ≤ D−1B |u|
A−1 ≥ 0 D ≥ 0 ⇒ A−1D ≥ 0
⇒ −A−1DD−1B |u| ≤ −A−1D |λ| |u|⇒ |u| = A−1(D −B) |u| = A−1D(I −D−1B) |u|
≤ A−1D |u|−A−1D |λ| |u| = (1− |λ|)A−1D |u|
|λ| ≥ 1 |u|− (1− |λ|)A−1D |u| ≤ 0
I − (1− |λ|)A−1D ≥ 0,
|u| ≤ 0 ⇒ u = 0
A A−1 > 0
B C D α,β ∈ 1, . . . , nA
α = α0,α1, . . . ,αk = β aαpαp+1 < 0 ∀p ∈ 0, . . . , k − 1cαpαp+1 > 0 ∀p ∈ 0, . . . , k − 1
(Ck)αβ =∑
γ1,...,γk−1
cαγ1cγ1γ2 . . . cγk−1β ≥ cαα1cα1α2 · . . . · cαk−1β > 0.
ϱ(C) < 1 S :=∑∞
ν=0Cν
Sαβ ≥ (Ck)αβ > 0 S Ck > 0 S > 0A−1 = SD−1 > 0 A−1 > 0
Kh
V K R C ∥ · ∥ : V → RV u, v ∈ V λ ∈ K
∥u∥ = 0 ⇔ u = 0 ,
∥λu∥ = |λ| ∥u∥ ,
∥u+ v∥ ≤ ∥u∥+ ∥v∥ .
V ∥ · ∥ AV
∥A∥M :=
∥Au∥∥u∥ : u ∈ V \ 0
= ∥Au∥ : u ∈ V, ∥u∥ = 1
∥ · ∥A
∥A∥M ≥ ϱ (A) .
∥A∥M =∥Au∥∥u∥ : u ∈ V \ 0
v A
∥Av∥∥v∥ =
∥λv∥∥v∥ = |λ|
⇒∥Au∥∥u∥
≥
v
∥Av∥∥v∥ = |λ∗| = ϱ(A).
∥·∥∞
∥A∥∞ =i∈1,...,n
⎧⎨
⎩∑
j∈1,...,n
|aij |
⎫⎬
⎭ .
A ≥ B ∥A∥∞ ≥ ∥B∥∞ A ≥ B
A w Aw ≥∥∥A−1
∥∥∞ ≤ ∥w∥∞ .
|u| u
|u| ≤ ∥u∥∞ · ≤ ∥u∥∞ ·Aw.
A A−1 ≥ 0∣∣A−1u
∣∣ ≤ A−1 |u| ≤ A−1 ∥u∥∞Aw
= ∥u∥∞A−1Aw = ∥u∥∞ · w
⇒∣∣A−1u
∣∣∥u∥∞
≤ w
∥∥A−1u∥∥∞
∥u∥∞≤ ∥w∥∞
⇒∥∥A−1
∥∥∞ ≤ ∥w∥∞ .
A B B ≥ A
0 ≤ B−1 ≤ A−1∥∥B−1
∥∥∞ ≤
∥∥A−1∥∥∞
A−1 −B−1 = A−1(B −A)B−1.
A,B A−1 ≥ 0 B−1 ≥ 0 B ≥ A B −A ≥ 0
⇒ A−1 −B−1 = A−1(B −A)B−1 ≥ 0
⇔ A−1 ≥ B−1 ⇒∥∥B−1
∥∥∞ ≤
∥∥A−1∥∥∞ .
∥u∥2 =√∑n
i=1 |ui|2
V∥·∥2
∥A∥2 =√ϱ (ATA)
∥A∥2 =∥Au∥2∥u∥2
: u ∈ V \ 0
=u∈V,∥u∥2=1
∥Au∥2 .
∥A∥22 =u∈V,∥u∥2=1
∥Au∥22 = ∥u∥2=1⟨Au,Au⟩ =
∥u∥2=1⟨ATAu, u⟩.
ATA
P TATAP = (λi) =: D
PP T =∥∥P Tu
∥∥ = ∥u∥ ∀u ∈ V λi > 0 ATA
∥Au∥22 =∥u∥2=1
⟨ATAu, u⟩ =∥u∥2=1
⟨ATAPP Tu, PP Tu⟩
=︸︷︷︸u=PTu
∥u∥2=1⟨ATAPu, P u⟩ =
∥u∥2=1⟨P TATAPu, u⟩ =
∥u∥2=1⟨Du, u⟩
=∥u∥2=1
(n∑
i=1
λi|ui|2)
= λmax = ϱ(ATA
)
⇒ ∥A∥2 =√ϱ (ATA).
A ATA = A2 ρ(A2)= ρ2 (A)
∥A∥2 = ϱ(A)
A ∈ Kn×n A
⟨Au, u⟩ > 0 ∀u ∈ Kn \ 0 .
A A
A ⇒ ∃P : P TAP = (λi) λi AP
⇒ ⟨Au, u⟩ = ⟨APu, P u⟩ = ⟨P TAPu, u⟩ =n∑
i=1
λi |ui|2
⟨Au, u⟩ > 0 ∀u ∈ Kn \ 0 ⇔ λi > 0 ∀i.
Aaii > 0 A
n∑
j=1,j =i
|aij | < aii ⇒ (0,∞)
⇒ λi ⇒ A
λ λ A
∥A∥2 = λ
∥A−1∥2 =1
λ
A ∥A∥2 = ϱ (A) = λ A−1
∥A−1∥2 = ϱ(A−1
)= 1
λ
Kh
Ω Ω = (0, 1)× (0, 1)
⎡
⎣1
1 −4 11
⎤
⎦ ,
Kh
Kh
Kh
∥Kh∥∞ ≤ 8h2
∥∥K−1h
∥∥∞ ≤ 1
8
∥Kh∥2 ≤8h2
2(πh2
)< 8
h2
∥∥K−1h
∥∥2≤ 1
8h2 −2 (πh
2
)= 1
2π2+O(h2) <116 h
Kh
Kh
Kh Kh
∥Kh∥∞ = i=1,...,n
∑nj=1 |Kij |
= 1
h2 6, 7, 8 = 8h2
∥∥K−1h
∥∥∞ ≤ 1
8
w(x, y) =x(1− x)
2
Khwh (x, y) = −(x− h)(1− (x− h))
2h2−(x+ h)(1− (x+ h))
2h2+2 · x(1− x)
2h2= 1
Khwh ≥ wh ∥wh∥∞ ≤ x,y w (x, y) = 18∥∥K−1h
∥∥∞ ≤ ∥w∥∞ ≤ 1
8
Kh uν,µ (1 ≤ ν, µ ≤ n− 1)
uν,µj,k = (νπjh) (µπkh)
λν,µ =2
h2
(2
(νπh
2
)+ 2
(µπh
2
)).
(j, k)
(Khuν,µ)j,k =
1
h2(4 (νπjh) (µπkh)
− (νπ (j − 1)h) (µπkh)− (νπ (j + 1)h) (µπkh)
− (νπjh) (µπ (k − 1)h)− (νπjh) (µπ (k + 1)h)) .
(a± b) = a b± b a
a := νπjh b := νπh a := µπkh b := µπh
(Khuν,µ)j,k =
1
h2(4− 2 (νπh)− 2 (µπh)) (νπjh) (µπkh) .
1− (a) = 2 2(a2
)
(Khuν,µ)j,k =
4
h2
(2
(νπh
2
)+ 2
(µπh
2
))uν,µj,k .
⇒ ∥Kh∥2 = ϱ(A) = λmax ≤ 8h2
2(π(n−1)h
2
)= 8
h22(πh2
)< 8
h2
∥∥K−1h
∥∥2
= ϱ(K−1h ) =1
λmin
λmin =8
h22
(πh
2
)=
8
h2
(πh
2−O(h3)
)2
=8
h2
((πh
2
)2
−O(h4)
)= 2π2 −O(h2)
⇒∥∥K−1h
∥∥2
≤ 1
2π2 −O(h2).
uh −∆huh = fh fh = 0 uh|∂Ωh= ϕh
uh ∂Ωh
−∆huh = 0
uh(x, y) =1
4(uh(x− h, y) + uh(x+ h, y) + uh(x, y − h) + uh(x, y + h))
uh(x, y) (x, y) ∈ Ωh
uh(x ± h, y) uh(x, y ± h)Kh Ωh uh
uh, vh
−∆huh = fh uh|∂Ωh= ϕu
h
−∆hvh = fh vh|∂Ωh= ϕv
h
∥uh − vh∥∞ ≤ x∈∂Ωh|ϕu(x)− ϕv(x)|
uh ≤ vh ϕu ≤ ϕv ∂Ωh
wh := vh−uh wh −∆hwh = 0wh ≥ 0 ∂Ωh
ϕu ≤ ϕv.
wh > 0 ⇒uh = ϕu vh = ϕv ∂Ωh
|wh| ≤∂Ωh
|ϕu − ϕv| ∂Ωh.
Ωh ⇒
−∆u = f Ω,
u|Γ = ϕ
−∆huh = fh Ωh,
uh|Γh = ϕh.
h ∈ H ⊂ R+ H H =1n : n ∈ N
Uh Ωh
Rh : C(Ω)
−→ Uh
u $→ Rhu
(Rhu)(x) = u(x) x ∈ Ωh Ω Ωh
K
Kh h ∈ H ⊂ R+
h∈H
∥∥K−1h
∥∥ ≤ C < ∞
Kh(uh) = fh,
Kh(uh) = fh + ε.
uh = K−1h (fh)
uh = K−1h (fh + ϵ)
⇒ ∥uh − uh∥ ≤ C · ∥ε∥
Kh
∥∥K−1h
∥∥∞ ≤ 1
8,
Khuh = fh Ku = f Km Rh Rh u f
(Kh, Rh, Rh) Kk
∥∥∥KhRhu− RhKu∥∥∥ ≤ C · hk · ∥u∥Ck+m(Ω) ∀u ∈ Ck+m(Ω).
Rh = Rh
(Rhu)(x) = u(x) ∀x ∈ Ωh.
(Kh, Rh, Rh) = (∆h, Rh, Rh)
(∂−∂+u)(x) = u′′(x) + h2R, |R| ≤ 1
12
∥∥∥u(4)∥∥∥C0(Ω)
.
R2 x y
−∆hRu(x, y) = −∆u(x, y) + h2(Rx +Ry)
|Rx| , |Ry| ≤ 1
12
∥∥∥u(4)∥∥∥C0(Ω)
≤ 1
12∥u∥C4(Ω) .
∥KhRhu−RhKu∥ ≤ C · h2∥u∥C4(Ω) C = 16
Khuh = fh Ku = f Km uh ∈ Uh (h ∈ H)
k u
∥uh −Rhu∥ ≤ C · hk · ∥u∥Ck+m(Ω)
uh −Rhu
K m (Kh, Rh, Rh) Kk
k u ∈ Ck+m(Ω)
wh = uh −Rhu wh → 0 h → 0.
Khwh = Khuh −KhRhu = fh −KhRhu = Rhf −KhRhu = RhKu−KhRhu
⇒ wh = K−1h (RhKu−KhRhu)
⇒ ∥wh∥ ≤∥∥K−1h
∥∥∥∥∥RhKu−KhRhu
∥∥∥
⇒ ∥uh −Rhu∥ ≤ Chk ∥u∥Ck+m(Ω) .
u ∈ C4(Ω)
∥uh −Rhu∥∞ ≤ h2
48· ∥u∥C4(Ω) .
u|Γ = ϕ u
−∆u = f Ω = (0, 1)× (0, 1)∂u
∂n= ϕ Γ
∂u∂n
u u+ c´Ω u dx = 0
f ϕ
uˆ
Ω
f dx+
ˆ
∂Ω
ϕ ds = 0.
−ˆ
Ω
f dx =
ˆ
Ω
∆u dx =
ˆ
Ω
(∇u) dx
=
ˆ
∂Ω
∇u · n ds =
ˆ
∂Ω
∂u
∂nds =
ˆ
∂Ω
ϕ ds.
Ω = [0, 1] × [0, 1]
−∆huh(x, y) =1h2 (4uh(x, y)− uh(x− h, y)− uh(x+ h, y)− uh(x, y − h)− uh(x, y + h))
Ω−∆huh(x, y) =
1h2 (3uh(x, y)− uh(x− h, y)− uh(x, y − h)− uh(x, y + h))
−∆huh(x, y) =1h2 (2uh(x, y)− uh(x− h, y)− uh(x, y + h))
u n
∂uh∂n
(x) ≈ (∂−n uh)(x) =1
h(uh(x)− u(x− hn)) = ϕ(x)
1
h(uh(x, 0)− uh(x, h)) = ϕ(x, 0) ,
1
h(uh(x, 1)− uh(x, 1− h)) = ϕ(x, 1)
1
h(uh(0, y)− uh(h, y)) = ϕ(0, y) ,
1
h(uh(1, y)− uh(1− h, y)) = ϕ(1, y)
uh
Khuh = fh = fh +1
hϕh,
ϕh =∑
ϕ (x, y)
h2∑
x∈Ωh
f(x) + h∑
x∈Γ′h
ϕ(x) = 0,
Γ′h
Khuh = fh
c
Kh
Kh · c · = 0 ⇒ c · ∈ (Kh) .
Kh ( (Kh)) = 1.Khuh = fh fh ∈ (Kh) (Kh) =
(KTh )⊥ = (Kh)⊥ = span( )⊥ Khuh = fh∑x∈Ωh
fh(x) = 0 fh ϕh∑x∈Ωh
fh(x) =∑
x∈Ωhfh(x) +
1h
∑x∈Γ′
hϕ(x)
fh fh− 1hϕh fh
x0 ∈ Ωh uh
uh(x0) = 0.
Khuh = fh
Kh Kh Kh
Kh
Kh
xi
Khuh = fh
(fh)j=
(fh)j , j = i
−∑
k =i (fh)k , j = i
(uh)i = 0.
(fh)i= (fh)i
ii
f ϕ(fh)i− (fh)i = O
(h−1
)xi
Khuh = fh
Kh =
(KhT 0
), uh =
(uhλ
), fh =
(fhσ
)
σ
Khuh = fh
/∈ (Kh) (Kh, ) = (Kh)+1Kh Kh
uh Khuh = fh uhKhuh = fh fh = fh − λ ·
λ
λ = 0 uh
T · uh =∑
x∈Ωh
uh(x) = σ
λ =
∑x∈Ωh
fh(x)T
f ϕ λ = O (h)
Ku = f Ω,
K =n∑
i,j=1
aij(x)∂2
∂xixj+
n∑
i=1
bi(x)∂
∂xi+ c(x).
aij(x) = aji(x)∂2
∂xixj= ∂2
∂xjxi
A(x) = (aij(x))i,j=1...naij(x)
A
n∑
i,j=1
aij(x)ξiξj > 0 ∀x ∈ Ω, 0 = ξ ∈ Rn
n∑
i,j=1
aij(x)ξiξj < 0 ∀x ∈ Ω, 0 = ξ ∈ Rn.
−∆u = f
A =
(−1 00 −1
).
K Ωn∑
i,j=1
aij(x)ξiξj ≥ c(x) ∥ξ∥2 , c(x) > 0 ∀x ∈ Ω, 0 = ξ ∈ Rn
Ω = (0, 1)× (0, 1)
a11(x, y)∂+x ∂−x + 2a12 (x, y) ∂
0x∂
0y + a22(x, y)∂
+y ∂−y + b1(x, y)∂
0x + b2(x, y)∂
0y + c(x, y)
= h−2
⎡
⎣−1
2a12(x, y) a22(x, y)12a12(x, y)
a11(x, y) −2(a11(x, y) + a22(x, y)) a11(x, y)12a12(x, y) a22(x, y) −1
2a12(x, y)
⎤
⎦+
+(2h)−1
⎡
⎣0 b2(x, y) 0
−b1(x, y) 0 b1(x, y)0 b2(x, y) 0
⎤
⎦+
⎡
⎣0 0 00 c(x, y) 00 0 0
⎤
⎦ .
X R C ∥·∥X : X → [0,∞)
(X, ∥·∥X)
Ω Ω ⊂ Rn
C0(Ω) ∥·∥∞∥·∥(1) ∥·∥(2) X
0 < C < ∞1
C∥x∥(1) ≤ ∥x∥(2) ≤ C ∥x∥(1) ∀x ∈ X.
X Y ∥·∥X ∥·∥YT : X → Y
∥T∥ :=x∈X
∥Tx∥Y∥x∥X
: x = 0
.
∥T∥ T
L (X,Y )(T1 + T2)x = T1x+ T2x
(X, ∥·∥) A ⊂ Xx ∈ A ε > 0
Kε (x) := y ∈ X : ∥x− y∥ < ε
A
(X, ∥·∥) xn ∈ X : n ≥ 1
∥xn − xm∥ : n,m ≥ k → 0, k → ∞
∀ ε > 0 ∃n0 (ε) ∈ N : ∀n,m ≥ n0 (ε) : ∥xn − xm∥ < ε.
(X, ∥·∥) X
(·, ·) : X ×X → K X
(x, x) > 0 ∀x ∈ X,x = 0,
(λx+ y, z) = λ(x, z) + (y, z) ∀λ ∈ K, x, y, z ∈ X,
(x, y) = (y, x) ∀x, y ∈ X.
∥x∥ :=√
(x, x).
X (·, ·) XX (·, ·)
X A Xσ X
∅ ∈ A
A ∈ A ⇒ Ac ∈ A
(An)n∈N ⊂ A ⇒⋃
n∈NAn ∈ A
(X,A) A
X Oσ σ O
X
(X,A) f : X → RX =
⋃n
k=1Ak Ak ∈ A k = 1, . . . n f |Ak
k = 1 . . . n
(X,A) (Y,B) f : X → Y
f−1 (B) ∈ A ∀B ∈ B.
Y = R σ f(tn)n∈N f
(X,A) µ : A → [0,∞]
µ (∅) = 0
(An)n∈N ⊂ A An ∩Am = ∅ n = m σ
µ
(⋃
n∈NAn
)=∑
n∈Nµ (An) .
(X,A, µ)
In
n∏n
k=1(ak, bk] ⊂ Rn ak ≤ bk
µIn : In → [0,∞] µIn
⎛
⎝m⋃
j=1
n∏
k=1
(ajk, bjk]
⎞
⎠ =m∑
j=1
n∏
j=1
(bjk − ajk)
In
In σ σ B Rn µIn
µ (Rn,B)
(X,A, µ) A ∈ Aµ(A) = 0
X Y f, g : X → YN
f (x) = g (x) ∀ x ∈ X \N.
(X,A, µ) Y f : X → Y(Ak)
nk=1 ∈ A
µ (Ak) < ∞ f |Ak f |A = 0 A =⋃n
k=1Ak
X Y T (X,Y )t ∈ T (X,Y )
ˆ
X
t dµ :=n∑
k=1
t (Ak)µ (Ak) .
(X,A, µ)L1 (X) f : X → R (tn)n∈N ⊂ T (X,R)
T (X,R)
|t|1 :=
ˆ
X
|t| dµ
ff ∈ L1 (X)
ˆ
X
f dµ :=k→∞
ˆ
X
tk dµ.
L1 (X) L1 (X)
In µ
f, g ∈ L1 (X) f = g ⇔ f(x) = g(x)L1 (X)
∥f∥1 :=
ˆ
X
|f | dµ
µ (X) < ∞
L1 (X)f : X → Rf : X → R ⇔ f |f |
X ⊂ Rn σ Bµ
L1 (X)
∞ ( )
D ⊂ Rn (Rn,B, µ)σ µ L∞ (D)
D L∞ (D)
f = g, f = g ;
∥u∥L∞(D) :=A∈B
µ(A)=0
x∈D\A|u (x)|
.
L2 (Ω)
Ω Rn L2 (Ω)
L2 (Ω) :=f : Ω → R : f , |f |2 ∈ L1 (Ω)
.
f g A µ(A) = 0
(u, v)0 = (u, v)L2(Ω) :=
ˆ
Ω
uv dµ ∀u, v ∈ L2(Ω)
∥u∥0 = ∥u∥L2(Ω) =
√√√√ˆ
Ω
|u|2 dµ
L2 (Ω)
L2(Ω) f ∈ L2(Ω)
ˆΩf ′(x)ϕ(x)dx = −
ˆΩf(x)ϕ′(x)dx
f,ϕ ϕ|∂Ω = 0
L2 (Ω)
Ω ⊂ Rn
C∞c (Ω)ϕ
C∞c (Ω) :=ϕ ∈ C∞ (Ω) : x ∈ Ω : ϕ (x) = 0
.
f ∈ L2(Ω) g ∈ L2(Ω)ˆ
Ω
g(x)ϕ(x) dx = −ˆ
Ω
f(x)ϕ′(x) dx ∀ϕ ∈ C∞c (Ω) ,
g f
α = (α1, . . . ,αn)
|α| :=n∑
i=1
αi,
Dα :=∂|α|
∂α1x1 . . . ∂αnxn
f ∈ L2(Ω) g α fˆ
Ω
g(x)ϕ(x) dx = (−1)|α|ˆ
Ω
f(x)Dαϕ(x) dx ∀ϕ ∈ C∞c (Ω) .
Hk (Ω) Hk0 (Ω)
u L2 (Ω) Dαu ∈ L2 (Ω)
Hk (Ω) :=u ∈ L2 (Ω) : Dαu ∈ L2 (Ω) , |α| ≤ k
k ∈ N0 Hk (Ω) W k2 (Ω) W k,2 (Ω)
Hk (Ω)
(u, v)k := (u, v)Hk(Ω) :=
⎛
⎝∑
|α|≤k
∥Dαu∥2L2(Ω)
⎞
⎠
12
.
Hs (Rn) ⊂ Ck (Rn) , s > k +n
2, k ∈ N0.
X R X ′
X R
X ′ = L (X,R) .
∥∥x′∥∥ :=
∥∥x′∥∥R←X
:=
|x′ (x)|∥x∥X
: 0 = x ∈ X
.
X ′ x′ ∈ X ′ X
⟨x, x′
⟩X×X′ := x′ (x) .
X Y T ∈ L (X,Y ) y′ ∈ Y ′
⟨Tx, y′
⟩Y×Y ′ =
⟨x, x′
⟩X×X′ ∀x ∈ X
x′ ∈ X ′
T ′ : Y ′ −→ X ′
y′ $−→ x′
⟨Tx, y′⟩Y×Y ′ = ⟨x, T ′y′⟩X×X′
∥∥T ′∥∥X′←Y ′ = ∥T∥Y←X .
∥∥T ′∥∥X′←Y ′ =
y′ =0
∥T ′y′∥X′
∥y′∥Y ′
=
x,y′ =0
⟨x, T ′y′⟩X×X′
∥x∥X ∥y′∥Y ′
=x,y′ =0
⟨Tx, y′⟩Y×Y ′
∥x∥X ∥y′∥Y ′
≤
x,y′ =0
∥T∥Y←X ∥x∥X ∥y′∥Y ′
∥x∥X ∥y′∥Y ′
= ∥T∥Y←X
∥T∥Y←X =x =0
∥Tx∥Y∥x∥X
≤
x,y′ =0
⟨Tx, y′⟩Y×Y ′
∥x∥X ∥y′∥Y ′
=x,y′ =0
⟨x, T ′y′⟩X×X′
∥x∥X ∥y′∥Y ′
≤
x,y′ =0
∥T ′∥X′←Y ′ ∥y′∥Y ′ ∥x∥X
∥x∥X ∥y′∥Y ′
= ∥T ′∥X′←Y ′ ,
y′ ∈ Y ′ ⟨Tx, y′⟩Y×Y ′ = ∥Tx∥Y ∥y′∥Y ′ = 1x ∈ X
X R y ∈ X
fy (x) := (x, y)X
fy ∈ X ′ ∥fy∥X′ = ∥y∥Xfy y ∈ X
X f ∈ X ′ yf ∈ X
f (x) = (x, yf )X ∀x ∈ X ∥yf∥X = ∥f∥X′ .
N = x ∈ X : f(x) = 0 f N = X,yf = 0 N = X
w ∈ X \N d := d (w,N) = x∈N ∥w − x∥ w N(xn)n∈N N d = n→∞ ∥w − xn∥
(xn)n∈NX
∥(w − xm) + (w − xn)∥2 + ∥(w − xm)− (w − xn)∥2 = 2(∥w − xm∥2 + ∥w − xn∥2
)
⇔∥xm − xn∥2 = 2(∥w − xm∥2 + ∥w − xn∥2
)− 4
∥∥∥∥w − 1
2(xm + xn)
∥∥∥∥2
.
12 (xm + xn) ∈ N 4
∥∥w − 12 (xm + xn)
∥∥2 ≥ 4d2 ε > 0 m,n
2(∥w − xm∥2 + ∥w − xn∥2
)< 4d2 + ε
∥xm − xn∥2 < 4d2 + ε− 4d2 = ε.
(xn)n∈N f NX (xn)n∈N x∗ ∈ N ∥w − x∗∥ = d
λ ∈ R x ∈ N
d2 ≤ ∥w − (x∗ + λx)∥2 = ∥w − x∗∥2 + λ2 ∥x∥2 − 2λ (w − x∗, x)
⇒λ2 ∥x∥2 − 2λ (w − x∗, x) ≥ 0.
λ ∈ R
(w − x∗, x) = 0 ∀x ∈ N
λ = ∥x∥−2 (w − x∗, x)z = w − x∗ x ∈ X
f
(x− f (x) z
f (z)
)= f (x)− f
(f (x) z
f (z)
)
= f (x)− f (x)
f (z)f (z) = 0
⇒ x− f (x) z
f (z)∈ N.
x(z, x− f (x)
f (z)z
)= 0
⇔ (z, x)− f (x)
f (z)(z, z) = 0
⇔f(x) =(x, z)
∥z∥2f (z) =
(x,
f (z) z
∥z∥2
).
yf = f(z)
∥z∥2 z
yf
f (x) = (x, yf ) = (x, yf ) ∀x ∈ X
⇒ (x, yf − yf ) = 0 ∀x ∈ X
⇒ yf − yf = 0.
V a (·, ·) : V × V −→ R
a (x+ λy, z) = a (x, z) + λa (y, z) ,
a (x, y + λz) = a (x, y) + λa (x, z) ∀λ ∈ R, x, y, z ∈ V.
a (·, ·) Cs
|a (x, y)| ≤ Cs ∥x∥V ∥y∥V ∀x, y ∈ V.
A ∈L (V, V ′)
a (x, y) = ⟨Ax, y⟩V ′×V ∀x, y ∈ V,
∥A∥V ′←V ≤ Cs.
x ∈ Vϕx (y) := a (x, y)
ϕx ∈ V ′ ∥ϕx∥V ′ ≤ Cs ∥x∥VA : V → V ′
Ax := ϕx
∥Ax∥V ′ ≤ Cs ∥x∥V
⇒∥A∥V ′←V =0=x∈V
∥Ax∥V ′
∥x∥V
≤ Cs.
a (·, ·) CE > 0
a (x, x) ≥ CE ∥x∥2V ∀x ∈ V.
V
a : V × V −→ R
f : V −→ R
J (v) :=1
2a (v, v)− f (v)
V u
a (u, v) = f (v) ∀v ∈ V.
u
u, v ∈ V, t ∈ R
J (u+ tv) =1
2a (u+ tv, u+ tv)− f (u+ tv)
=1
2
(a (u, u) + 2ta (u, v) + t2a (v, v)
)− f (u)− tf (v)
= J (u) + t (a (u, v)− f (v)) +1
2t2a (v, v)
u ∈ V a(u, v) = f(v) ∀v ∈ V
t=1⇒ J (u+ v) = J (u) + (f (v)− f (v)) +1
2a (v, v)
= J (u) +1
2a (v, v) > J (u) ∀v ∈ V.
u
u ∈ V t $→ J (u+ tv) v ∈ Vt = 0
⇒ 0 =dJ (u+ tv)
dt|t=0 = a (u, v)− f (v)
⇔ a(u, v) = f(v).
u1, u2
a (u1, v) = f (v) ∧ a (u2, v) = f (v) ∀v ∈ V
⇒ a (u1 − u2, v) = 0 ∀v ∈ V
⇒ u1 − u2 = 0.
Lu := −n∑
i,k=1
∂i (aik∂ku) + a0u
a0 (x) ≥ 0 (x ∈ Ω) A = (aik)i,k
Lu = f Ω
u = g ∂Ω
f ∈ L2 (Ω) g ∈ H12 (∂Ω) :=
v ∈ L2 (∂Ω) : ∃w ∈ H1 (Ω) , γ (w) = v
γg ∈ H1 (Ω) γ (g) = g
w := u− g
⇒ Lw = f − Lg =: f1 Ω
w = 0 ∂Ω.
−∑
i,k
∂i (aik∂ku) + a0u = f Ω
u = 0 ∂Ω
J (v) :=
ˆ
Ω
⎛
⎝1
2
∑
i,k
aik∂iv∂kv +1
2a0v
2 − fv
⎞
⎠ dx →
C2(Ω) ∩ C0(Ω)
ˆ
Ω
v (∇ · w) +∇v · wdx =
ˆ
∂Ω
v (w · n) ds.
v|∂Ω = 0
−ˆ
Ω
v (∇ · w) dx =
ˆ
Ω
∇v · wdx
wi =∑
k aik∂ku
−ˆ
Ω
v∑
i
∂i
(∑
k
aik∂ku
)dx =
ˆ
Ω
∑
i,k
aik∂iv∂kudx.
a (u, v) :=
ˆ
Ω
∑
i,k
aik∂iv∂ku+ a0uv dx,
f (v) :=
ˆ
Ω
fv dx
v
a (u, v)− f (v) =
ˆ
Ω
v(−∑
∂i (aik∂ku) + a0u− f)dx
=
ˆ
Ω
v(Lu− f)dx =Lu=f
0.
a (·, ·)f
u
u ∈ C2(Ω) ∩ C0(Ω)
⇓
u
J (u) :=
ˆ
Ω
|∇u|2 dx.
J (u) → ⇐⇒ −∆u = 0 Ω,
u = ϕ Γ.
J (u) ≥ 0 u⇒
−∆u = 0 Ω,
u = ϕ Γ.
J (u) =
1ˆ
0
u2 (x) dx −→ u ∈ C0 ([0, 1])
u(0) = 1, u(1) = 0
1
1
u1(x)
1n
un(x)
un(x) =
1− nx 0 ≤ x ≤ 1
n ,0 x > 1
n .
n→0 un = 0 J (u)
J (u) = 0,
V H a : H ×H −→ Rl ∈ H ′
J (v) :=1
2a (v, v)− ⟨l, v⟩ −→
V
J
J (v) ≥ 1
2CE ∥v∥2 − ∥l∥ ∥v∥
=1
2CE
(C2E ∥v∥2 − 2CE ∥l∥ ∥v∥+ ∥l∥2
)− 1
2CE∥l∥2
=1
2CE(CE ∥v∥ − ∥l∥)2 − 1
2CE∥l∥2 ≥ −∥l∥2
2CE.
c1 := J (v) : v ∈ V (vn)n∈N
n→∞J (vn) = c1.
(vn)n∈Na (·, ·)
CE ∥vn − vm∥2 ≤ a (vn − vm, vn − vm)
= 2a (vn, vn) + 2a (vm, vm)− a (vn + vm, vn + vm) .
a (v, v) a (v, v) = 2J (v) + 2 ⟨l, v⟩
CE ∥vn − vm∥2 ≤ 4J (vn) + 4 ⟨l, vn⟩+ 4J (vm) + 4 ⟨l, vm⟩
−(8J
(vn + vm
2
)+ 4 ⟨l, vn + vm⟩
)
= 4J (vn) + 4J (vm)− 8J
(vn + vm
2
)
≤ 4J (vn) + 4J (vm)− 8c1.
V vn+vm2 ∈ V
J(vn+vm
2
)> c1
J (vn) → c1 (n → ∞) J (vm) → c1 (m → ∞)
CE ∥vn − vm∥2 −→ 0 n,m → ∞,
(vn)n∈N H Hu ∈ H V u ∈ V n→∞ vn = u J (u) =
n→∞ J (vn) = v∈V J (v)⇒ J (v) = 1
2a (v, v)− ⟨l, v⟩ −→ u ∈ V
u1 u2u1, u2, u1, u2, . . .
u1, u2, u1, u2, . . . u1 = u2
V = H l ∈ H ′ u ∈ H
a (u, v) = ⟨l, v⟩ ∀v ∈ H.
a (u, v) := (u, v)l ∈ H ′ u ∈ H
(u, v) = ⟨l, v⟩ ∀v ∈ H.
H ′ −→ H
l $−→ u.
u ∈ H10 (Ω)
Lu = f Ω,
u = 0 Γ
L
a (u, v) = (f, v)0 ∀v ∈ H10 (Ω)
a (u, v) :=
ˆ
Ω
∑
i,k
aik∂iu∂kv + a0uv dx.
L
Lu = f Ω,
u = 0 Γ
f ∈ L2 (Ω) H10 (Ω)
1
2a (v, v)− (f, v)0 −→ H1
0 (Ω) .
−∆u = f Ω,
u = 0 Γ
a (u, v) =
ˆ
Ω
∇u∇v dx.
u ∈ H10 (Ω)
(∇u,∇v)0 = (f, v)0 ∀v ∈ H10 (Ω) .
u
1
2
ˆ
Ω
∇u∇v dx− (f, v)0 −→ H10 (Ω) .
Lu = f Ω,∑
i,k
niaik∂ku = g Γ,
ni i f ∈ L2 (Ω) g ∈ L2 (Γ)
⟨l, v⟩ :=ˆ
Ω
fv dx+
ˆ
Γ
gv dx
u ∈ H1 (Ω)
1
2a (u, v) = (f, v)0,Ω + (g, v)0,Γ ∀v ∈ H1 (Ω) .
Ω
ˆ
Ω
f (x) dx+
ˆ
∂Ω
g (x) ds = 0
V :=
⎧⎨
⎩v ∈ H1 (Ω) :
ˆ
Ω
v (x) dx = 0
⎫⎬
⎭ .
J (v) :=1
2a (u, v)− (f, v)0,Ω − (g, v)0,Γ −→ V
u
Lu = f Ω,∑
i,k
niaik∂ku = g Γ,
u ∈ C2 (Ω) ∩ C1(Ω)
H1 (Ω)C2 (Ω) ∩ C1
(Ω)
−∆u = f Ω,
u = 0 Γ
u ∈ H10 (Ω)
a (u, v) = (f, v) ∀v ∈ H10 (Ω)
a (u, v) :=
ˆ
Ω
∇u∇v dx,
(f, v) :=
ˆ
Ω
fv dx.
H10 (Ω)
Vh
(⊂ H1
0 (Ω))
J (v) :=1
2a (v, v)− ⟨l, v⟩ −→ Vh.
uh ∈ Vh
a (uh, v) = ⟨l, v⟩ ∀v ∈ Vh.
ψ1,ψ2, . . . ,ψN Vh
a (uh, v) = ⟨l, v⟩ ∀v ∈ Vh
a (·, ·) l (·)
a (uh,ψi) = ⟨l,ψi⟩ ∀i = 1, 2, . . . , N.
uh ∈ Vh ψi
uh =N∑
k=1
zkψk
zk
N∑
k=1
a (ψk,ψi) zk = ⟨l,ψi⟩ i = 1, 2, . . . N.
Aik := a (ψk,ψi) bi := ⟨l,ψi⟩
Az = b.
a (·, ·) A
zTAz =∑
i,k
ziAikzk = a
(∑
k
zkψk,∑
i
ziψi
)
= a (uh, uh) ≥ CE ∥uh∥2V .
uh ∈ Vh
u ∈ V
V a : V × V → R CS
CE l ∈ V ′ Vh ⊂ Vu ∈ V
a (u, v) = l (v) ∀v ∈ V
uh ∈ Vh
a (uh, v) = l (v) ∀v ∈ Vh.
∥u− uh∥ ≤ CS
CE vh∈Vh
∥u− vh∥ .
Vh ⊂ V v ∈ Vh
a (u, v)− a (uh, v) = a (u− uh, v) = 0 ∀v ∈ Vh.
a (·, ·)
CE ∥u− uh∥2 ≤ a (u− uh, u− uh) .
a (u− uh, uh − vh) vh ∈Vh
CE ∥u− uh∥2 ≤ a (u− uh, u− uh) + a (u− uh, uh − vh)
= a (u− uh, u− vh)
≤ CS ∥u− uh∥ ∥u− vh∥ ∀vh ∈ Vh.
∥u− uh∥ ≤ CS
CE∥u− vh∥ ∀vh ∈ Vh
⇒ ∥u− uh∥ ≤ CS
CE vh∈Vh
∥u− vh∥ .
u − uha Vh
CSCE
Vh
a ∥v∥a := (a (v, v))12 V
CE CS
∥·∥a uh u Vh
Vh
−∆u = f Ω = (0, 1)× (0, 1),
u = 0 Γ.
Ω
I
II
III
IV
V
VI
VII
VIII
L R
O
U
LO
RU
Z
∂1ψZ−1h
1h
1h
−1h
∂2ψZ−1h
−1h
1h
1h
ψZ
Vh
Vh =v ∈ C
(Ω): v v|Γ = 0
.
v
v (x, y) = a+ bx+ cy.
a b cN Vh = N N
Vh ψiNi=1
ψi (Kj) = δij , Kj j , ∀i, j = 1, . . . , N.
ψZ
Au = b
Aij = a (ψi,ψj) .
a (ψZ ,ψZ) a (ψZ ,ψO) a (ψZ ,ψU ) a (ψZ ,ψL)a (ψZ ,ψR) a (ψZ ,ψLO) a (ψZ ,ψRU )
a (ψZ ,ψZ)
a (ψZ ,ψZ) =
ˆ
Ω
(∇ψZ)2 dxdy =
ˆ
−
(∇ψZ)2 dxdy
= 2
ˆ
, ,
((∂1ψZ)
2 + (∂2ψZ)2)dxdy
= 2
ˆ
,
(∂1ψZ)2 dxdy + 2
ˆ
,
(∂2ψZ)2 dxdy
=2
h2
ˆ
,
dxdy +2
h2
ˆ
,
dxdy
= 4.
a (ψZ ,ψO)
a (ψZ ,ψO) =
ˆ
−
∇ψZ∇ψOdxdy
=
ˆ
,
∇ψZ∇ψOdxdy =
ˆ
,
∂1ψZ∂1ψO + ∂2ψZ∂2ψOdxdy
=
ˆ
,
∂2ψZ∂2ψOdxdy =
ˆ
,
−1
h· 1hdxdy
= − 1
h2
ˆ
,
dxdy = −1.
a (ψZ ,ψO) = a (ψZ ,ψU ) = a (ψZ ,ψL) = a (ψZ ,ψR) = −1.
a (ψZ ,ψRU ) = a (ψZ ,ψLO) = 0.
⎡
⎢⎢⎢⎢⎢⎢⎣
0 −1 0
−1 4 −1
0 −1 0
⎤
⎥⎥⎥⎥⎥⎥⎦
h−2
T = T1, T2, . . . , TM Ω
Ω =⋃M
i=1 Ti
Ti ∩ Tj Ti
Tj
Ti ∩ Tj (i = j) Ti ∩ Tj
Ti Tj
A B
Ωh Ω ⊂ Rd
V ph (T ) =
u ∈ H1 : T ∈ T : u|T ∈ Pp
p
A DCB
Rd (d = 1, 2, 3)
R1
−∆u+ u = f a (u, v) =
ˆ
Ω
(∇u∇v + uv) dx.
N = a = x0, x1, x2, . . . , xN+1 = b [a, b]hi = xi+1 − xi ϕi
ϕi (xj) = δij .
uh
uh =N∑
i=1
aiϕi
ai = uh (xi) .
uh
ϕi Φi
[xi, xi+1][0, 1]
[0, 1] Φi
1
1
1
x i x i+1x i− 1
A B
Ii = [xi, xi+1] ξ ∈ [0, 1]
xIi : [0, 1] −→ Ii,
ξ $−→ xi + hiξ;
ξIi : Ii −→ [0, 1] ,
x $−→ (x− xi)
hi
[0, 1] [xi, xi+1]
uh (ξ) = α1 + α2ξ.
ui = uh (0) = α1 ui+1 = uh (1) = α1 + α2 ξ ∈ [0, 1]
uh (ξ) = α1 + α2ξ = ui + (ui+1 − ui) ξ
= (1− ξ)ui + ξui+1 =: uiΦ1 (ξ) + ui+1Φ2 (ξ) .
∀ξ ∈ [0, 1] : Φ1 (ξ) + Φ2 (ξ) = 1.
ϕi (x) =
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
Φ2(ξIi−1 (x)
), x ∈ Ii−1
Φ1 (ξIi (x)) , x ∈ Ii
0, .
a (ϕi,ϕj) A
a (ϕi,ϕj) =N∑
k=1
ˆ
Ik
∇ϕi(x)∇ϕj(x) + ϕi(x)ϕj(x) dx.
ϕi,ϕj Φn Φm
n,m ∈ 1, 2x ξ
Ik [0, 1]
(AIk)nm =
ˆ
Ik
∇xΦn (ξIk (x))∇xΦm (ξIk (x)) + Φn (ξIk (x))Φm (ξIk (x)) dx
= hk
1ˆ
0
∇ξΦn (ξ) ξ′Ik (xIk (ξ))∇ξΦm (ξ) ξ′Ik (xIk (ξ)) + Φn (ξ)Φm (ξ) dξ
= hk
1ˆ
0
1
h2k∇Φn∇Φm + ΦnΦmdξ.
(AIk)mn Ikm n AIk
(AIk)11 =
1ˆ
0
1
hk∇Φ1∇Φ1 + Φ1Φ1 · hkdξ
=
1ˆ
0
1
hk+ hk(1− ξ)2dξ =
1
hk+
1
3hk
(AIk)12 =
1ˆ
0
1
hk∇Φ1∇Φ2 + Φ1Φ2 · hkdξ
=
1ˆ
0
− 1
hk+ ξ(1− ξ) · hkdξ
= − 1
hk+
1
6hk
(AIk)21 = − 1
hk+
1
6hk
(AIk)22 =1
hk+
1
3hk
⇒ AIk =1
hk
⎡
⎢⎢⎣1 −1
−1 1
⎤
⎥⎥⎦+ hk
⎡
⎢⎢⎣1/3 1/6
1/6 1/3
⎤
⎥⎥⎦ .
uh (ξ)
uh (ξ) = α1 + α2ξ + α3ξ2 I = [0, 1] .
ui ui+1
ui+ 12= uh
(xi + xi+1
2
).
ui = uh (0) = α1,
ui+1 = uh (1) = α1 + α2 + α3,
ui+ 12
= uh
(1
2
)= α1 +
1
2α2 +
1
4α3.
αi, i = 1, . . . , 3
uh (ξ) = uiΦ1 (ξ) + ui+1Φ2 (ξ) + ui+ 12Φ3 (ξ)
Φi (ξ) = 2
(ξ − 1
2
)(ξ − 1)
Φ2 (ξ) = 2ξ
(ξ − 1
2
)
Φ3 (ξ) = 4ξ (1− ξ )
1
1
Φ3(ξ)
Φ2(ξ)Φ1(ξ)
AIk 3× 3
∇Φi (ξ)
R2
R1
y
x
1
1
x = x1 + (x2 − x1) ξ + (x3 − x1) η,
y = y1 + (y2 − y1) ξ + (y3 − y1) η
⇔
⎛
⎜⎜⎝x
y
⎞
⎟⎟⎠ =
⎛
⎜⎜⎝x1
y1
⎞
⎟⎟⎠+
⎛
⎜⎜⎝x2 − x1 x3 − x1
y2 − y1 y3 − y1
⎞
⎟⎟⎠
⎛
⎜⎜⎝ξ
η
⎞
⎟⎟⎠ ,
(xi, yi) i
⎛
⎜⎜⎝ξ
η
⎞
⎟⎟⎠ =
⎛
⎜⎜⎝x2 − x1 x3 − x1
y2 − y1 y3 − y1
⎞
⎟⎟⎠
−1
︸ ︷︷ ︸A−1
⎛
⎜⎜⎝x− x1
y − y1
⎞
⎟⎟⎠
=1
A
⎛
⎜⎜⎝y3 − y1 x1 − x3
y1 − y2 x2 − x1
⎞
⎟⎟⎠
⎛
⎜⎜⎝x− x1
y − y1
⎞
⎟⎟⎠
A = (x2 − x1) (y3 − y1)− (x3 − x1) (y2 − y1)
ux = uξξx + uηηx,
uy = uξξy + uηηy
ξx =y3 − y1
A,
ηx =y1 − y2
A,
ξy =x1 − x2
A,
ηy =x2 − x1
A.
A
dxdy = Adξdη.
∇Φi, i = 1, 2, 3 ξx, ξy, ηx, ηy AIk
Φi
uh (ξ, η) = α1 + α2ξ + α3η,
uj := uh(Pj), j = 1, 2, 3.
Pj
u1 = uh(0, 0) = α1,
u2 = uh(1, 0) = α1 + α2,
u3 = uh(0, 1) = α1 + α3.
uh (ξ, η) = u1 + (u2 − u1) ξ + (u3 − u1) η = (1− ξ − η)u1 + ξu2 + ηu3.
Φ1 = 1− ξ − η, Φ2 = ξ, Φ3 = η
uh (ξ, η) = u1Φ1 + u2Φ2 + u3Φ3
Φi(Pj)
= δij i, j = 1, 2, 3,3∑
i=1
Φi (ξ, η) = 1 ξ, η ∈ T .
Φi
uh
uh (ξ, η) = α1 + α2ξ + α3η + α4ξ2 + α5ξη + α6η
2.
Φ1 = (1− ξ − η)(1− 2ξ + 2η)
Φ2 = ξ(2ξ − 1)
Φ3 = η(2η − 1)
Φ4 = 4ξ(1− ξ − η)
Φ5 = 4ξη
Φ6 = 4η(1− ξ − η)
R3
uh (ξ, η, ζ) = α1 + α2ξ + α3η + α4ζ.
α1, . . . ,α4
a a (u, v) =´Ω
∇u∇vdx
∥u∥a := (a (u, u))12 .
∥u− uh∥a =vh∈Vh
∥u− vh∥a .
Ω ⊂ Rd, d ≤ 3 u ∈ H2 (Ω)uh ∈ Vh
a (uh, vh) = ⟨f, vh⟩ ∀vh ∈ Vh ⊂ H10 (Ω)
∥u− uh∥a ≤ c · h · ∥f∥L2(Ω) , f ∈ L2 (Ω) .
f ∈ L2 (Ω)
∥u∥H2(Ω) ≤ c ∥f∥L2(Ω) ,
H2
L2
Ω ⊂ Rd, d ≤ 3 u ∈ H2 (Ω) H2
∥u− uh∥L2(Ω) ≤ c · h ∥u− uh∥a ,
∥u− uh∥L2(Ω) ≤ c · h2 ∥f∥L2(Ω) .
Ωh
ˆ
B
∂u
∂tdx =
ˆ
B
− F dx =
ˆ
∂B
F · n ds
Ωh
Ω Bi
uh
KFVh uFV
h = fFVh .
Kuh = b,
uh
uh = K−1b,
K
A
