G u + < u, > = f - - PDF document

• ✁✄✂✆☎

Gu + λ < u, · >= f

• ✁✄✂✆☎

Gu + λ < u, · >= f

G

• ✁✄✂✆☎

Gu + λ < u, · >= f

G

• ✁✄✂✆☎

Gu + λ < u, · >= f

G

λ ∈ R

• ✁✄✂✆☎

Gu + λ < u, · >= f

G

λ ∈ R

∈

• ✁✄✂✆☎

Gu + λ < u, · >= f

G

λ ∈ R

∈

a(u, v) + λm(u, v) = f(v)

a(u, v) :=< Gu, v >H′×H and m(u, v) :=< u, v >H×H .

• ✁✄✂✆☎

Gu + λ < u, · >= f

G

λ ∈ R

∈

a(u, v) + λm(u, v) = f(v)

a(u, v) :=< Gu, v >H′×H and m(u, v) :=< u, v >H×H .

a(·, ·)

a(u, v) =

• Ω

v(x)

• Ω

g(x, y)u(y)dydx

g(·, ·)

Ω

Hn

un ∈ Hn

a(un, vn) + λm(un, vn) = f(vn)

Hn

un ∈ Hn

a(un, vn) + λm(un, vn) = f(vn)

a(un, ϕi) + λm(un, ϕi) = f(ϕi) ∀i ∈ I

Hn

un ∈ Hn

a(un, vn) + λm(un, vn) = f(vn)

a(un, ϕi) + λm(un, ϕi) = f(ϕi) ∀i ∈ I

un

un =

• j∈I

xjϕj,

• j∈I

xja(ϕj, ϕi) + λ

• j∈I

xjm(ϕj, ϕi) = f(ϕi)

Gx + λMx = b

Gij := a(ϕj, ϕi),

Gx + λMx = b

Gij := a(ϕj, ϕi), Mij := m(ϕj, ϕi),

Gx + λMx = b

Gij := a(ϕj, ϕi), Mij := m(ϕj, ϕi), bi := f(ϕi).

Gx + λMx = b

Gij := a(ϕj, ϕi), Mij := m(ϕj, ϕi), bi := f(ϕi).

M

Gx + λMx = b

Gij := a(ϕj, ϕi),

Mij := m(ϕj, ϕi),

bi := f(ϕi).

G

• ✸✹➙✴✺★✔✯✲✫✮✶❨✽❍▲✸✧➣❥✿✧✔✳✇✫✸✧✔✹

• ❊✧✔✪✬●✸✹✲✧✩✯✲✫✖▲✸✧✔✳✇✧➑❚➔✴➌⑩➃✫✸✪❬✽▼✳✇✧✌❚❲✧✌❚✥❉✉✧✔✳✢✯❳✽

• ❊✧✔✪✬●✸✹✲✧✩✯✲✫✖▲✸✧✔✳✇✧➑❚➔✴➌⑩➃✫✸✪❬✽▼✳✇✧✌❚❲✧✌❚✥❉✉✧✔✳✢✯❳✽

g(·, ·)

(xν)ν∈K

Rd

(xν)ν∈K

Rd (Lν)ν∈K

Lν(xµ) = δν,µ

(xν)ν∈K

Rd (Lν)ν∈K

Lν(xµ) = δν,µ

˜ g(x, y) :=

• ν∈K

g(xν, y)Lν(x).

(xν)ν∈K

Rd (Lν)ν∈K

Lν(xµ) = δν,µ

˜ g(x, y) :=

• ν∈K

g(xν, y)Lν(x).

G

˜ G

˜ Gij :=

• Ω

ϕi(x)

• Ω

˜ g(x, y)ϕj(y)dydx = (ABT)ij,

(xν)ν∈K

Rd (Lν)ν∈K

Lν(xµ) = δν,µ

˜ g(x, y) :=

• ν∈K

g(xν, y)Lν(x).

G

˜ G

˜ Gij :=

• Ω

ϕi(x)

• Ω

˜ g(x, y)ϕj(y)dydx = (ABT)ij,

Aiν :=

• Ω

ϕi(x)Lν(x)dx

(xν)ν∈K

Rd (Lν)ν∈K

Lν(xµ) = δν,µ

˜ g(x, y) :=

• ν∈K

g(xν, y)Lν(x).

G

˜ G

˜ Gij :=

• Ω

ϕi(x)

• Ω

˜ g(x, y)ϕj(y)dydx = (ABT)ij,

Aiν :=

• Ω

ϕi(x)Lν(x)dx

Bjν :=

• Ω

ϕj(y)g(xν, y)dy.

• ✁✄✂✆☎

• ❊✪❙✯✼✫✾✽❍✭r❧✟✦

• ✁✄✂✆☎

Γ := γ([0, 1])

0 = x0 < x1 < ... < xn = 1

Ix :=

n

• i=1

u(γ(xi))γ(xi) − γ(xi−1).

Γ := γ([0, 1])

0 = x0 < x1 < ... < xn = 1

Ix :=

n

• i=1

u(γ(xi))γ(xi) − γ(xi−1).

1, ..., n)

Ix −

1

• u(γ(y))γ′(y)dy ≤ ǫ.

Γ := γ([0, 1])

0 = x0 < x1 < ... < xn = 1

Ix :=

n

• i=1

u(γ(xi))γ(xi) − γ(xi−1).

1, ..., n)

Ix −

1

• u(γ(y))γ′(y)dy ≤ ǫ.

i=1

C1([0, 1], R2)

γi([0, 1])

Γ := ∪m

i=1Γi

• Γ

u(x)dx :=

m

• i=1

1

• u(γi(y))γ′

i(y)dy.

• ❃✴❬✳✰✴✺❚➍✧❞✽❀✳✇✯✼➁❱✴❄✽✰✯❅✪✬✫✮✭

γi : [0, 1] → R2, y → pi−1(1 − y) + piy,

i = j

i=1γi([0, 1])

• ❃✴❬✳✰✴✺❚➍✧❞✽❀✳✇✯✼➁❱✴❄✽✰✯❅✪✬✫✮✭

γi : [0, 1] → R2, y → pi−1(1 − y) + piy,

i = j

i=1γi([0, 1])

−1 1 2 3 4 5 6 7 8 9 1.5 2 2.5 3 3.5 4 4.5 5 5.5 p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p11 p12 p13 p14 p15 p16 p17 p10

−1 1 2 3 4 5 6 7 8 1.5 2 2.5 3 3.5 4 4.5 5 5.5 p0 p1 p2 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 γ4([0,1]) p3=γ4(0) p4

−1 1 2 3 4 5 6 7 8 1.5 2 2.5 3 3.5 4 4.5 5 5.5 p0 p1 p2 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 γ4([0,1]) p4 p3 γ4(0.4)

−1 1 2 3 4 5 6 7 8 1.5 2 2.5 3 3.5 4 4.5 5 5.5 p0 p1 p2 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p3 p4 = γ4(1) γ4([0,1])

Gslp[u](x) :=

• Γ

log(x − y)u(y)dy

Gslp[u](x) :=

• Γ

log(x − y)u(y)dy

aslp(u, y) :=

• Γ

v(x)

• Γ

log(x − y)u(y)dydx.

Gslp[u](x) :=

• Γ

log(x − y)u(y)dy

aslp(u, y) :=

• Γ

v(x)

• Γ

log(x − y)u(y)dydx.

x = y

Gslp[u](x) :=

• Γ

log(x − y)u(y)dy

aslp(u, y) :=

• Γ

v(x)

• Γ

log(x − y)u(y)dydx.

x = y

i=1

ϕi ◦ γj ≡ δij

Gij = aslp(ϕi, ϕj) =

• Γ

ϕi(x)

• Γ

log(x − y)ϕj(y)dydx = pi − pi−1pj − pj−1

1

• 1
• log(γi(x) − γj(y))dydx.

Gslp[u](x) :=

• Γ

log(x − y)u(y)dy

aslp(u, y) :=

• Γ

v(x)

• Γ

log(x − y)u(y)dydx.

x = y

i=1

ϕi ◦ γj ≡ δij

Gij = aslp(ϕi, ϕj) =

• Γ

ϕi(x)

• Γ

log(x − y)ϕj(y)dydx = pi − pi−1pj − pj−1

1

• 1
• log(γi(x) − γj(y))dydx.

pi = pj

˜ (g)(x, y) =

• ν∈K

log(xt

ν − y)(L)t ν(x)

˜ (g)(x, y) =

• ν∈K

log(xt

ν − y)(L)t ν(x)

At,s

iν

=

• Γ

ϕi(x)(L)t

ν(x)dx = pi − pi−1 1

• (L)t

ν(γi(x))dx,

˜ (g)(x, y) =

• ν∈K

log(xt

ν − y)(L)t ν(x)

At,s

iν =

• Γ

ϕi(x)(L)t

ν(x)dx = pi − pi−1 1

• (L)t

ν(γi(x))dx,

Bt,s

jν =

• Γ

ϕj(x) log(xt

ν − y)dy = pj − pj−1 1

• log(xt

ν − γj(y))dy.

˜ (g)(x, y) =

• ν∈K

log(xt

ν − y)(L)t ν(x)

At,s

iν =

• Γ

ϕi(x)(L)t

ν(x)dx = pi − pi−1 1

• (L)t

ν(γi(x))dx,

Bt,s

jν =

• Γ

ϕj(x) log(xt

ν − y)dy = pj − pj−1 1

• log(xt

ν − γj(y))dy.

γi

iν

• ➷

i=0

• ➷

i=0

• ➷

• ➷

i=0

• ➷

• ➷

• ❢✽✰✭✈❧

• Ω

f(ξ)dξ ≈

• K∈M

|K|

PK

• l=1

wK

l f(πK l ).

l

l

−1 f(x)dx

1

• −1

f(ξ)dξ = w1f(x1) + w2f(x2)

−1 f(x)dx

1

• −1

f(ξ)dξ = w1f(x1) + w2f(x2)

−1 f(x)dx

1

• −1

f(ξ)dξ = w1f(x1) + w2f(x2)

1

• −1

1dx = 2 = w1 + w2

−1 f(x)dx

1

• −1

f(ξ)dξ = w1f(x1) + w2f(x2)

1

• −1

1dx = 2 = w1 + w2

1

• −1

xdx = 0 = w1x1 + w2x2

−1 f(x)dx

1

• −1

f(ξ)dξ = w1f(x1) + w2f(x2)

1

• −1

1dx = 2 = w1 + w2

1

• −1

xdx = 0 = w1x1 + w2x2

1

• −1

x2dx = 2 3 = w1x2

1 + w2x2 2

−1 f(x)dx

1

• −1

f(ξ)dξ = w1f(x1) + w2f(x2)

1

• −1

1dx = 2 = w1 + w2

1

• −1

xdx = 0 = w1x1 + w2x2

1

• −1

x2dx = 2 3 = w1x2

1 + w2x2 2 1

• −1

x3dx = 0 = w1x3

1 + w2x3 2.

x1 x2

w1x3

1 − w1

x1 x2 x3

2 = 0 ⇒ x2 1 = x2 2

x1 x2

w1x3

1 − w1

x1 x2 x3

2 = 0 ⇒ x2 1 = x2 2

x1 x2

w1x3

1 − w1

x1 x2 x3

2 = 0 ⇒ x2 1 = x2 2

x1 x2

w1x3

1 − w1

x1 x2 x3

2 = 0 ⇒ x2 1 = x2 2

2 = w1 + w2 ⇒ 2w1 = 2 ⇒ w1 = w2 = 1

x1 x2

w1x3

1 − w1

x1 x2 x3

2 = 0 ⇒ x2 1 = x2 2

2 = w1 + w2 ⇒ 2w1 = 2 ⇒ w1 = w2 = 1

2 3 = (w1 + w2)x2 1 = 2x2 1 ⇒ x2 1 = 1 3

x1 x2

w1x3

1 − w1

x1 x2 x3

2 = 0 ⇒ x2 1 = x2 2

2 = w1 + w2 ⇒ 2w1 = 2 ⇒ w1 = w2 = 1

2 3 = (w1 + w2)x2 1 = 2x2 1 ⇒ x2 1 = 1 3

w1 = w2 = 1 x1 = − 1 √ 3 x2 = 1 √ 3

ν

pi−1

pi

ν

pi−1

pi

γ

R2

γ(t) := sx + tdx sy + tdy

• and

c := cx cy

• .

∈ γ([0, 1])

ν

pi−1

pi

γ

R2

γ(t) := sx + tdx sy + tdy

• and

c := cx cy

• .

∈ γ([0, 1])

b =

1

• log(γ(t) − c2)γ′(t)2dt.

ν

pi−1

pi

γ

R2

γ(t) := sx + tdx sy + tdy

• and

c := cx cy

• .

∈ γ([0, 1])

b =

1

• log(γ(t) − c2)γ′(t)2dt.

γ(t) := 1 + t 1 + 2t

• and

c :=

• ,

ν

pi−1

pi

γ

R2

γ(t) := sx + tdx sy + tdy

• and

c := cx cy

• .

∈ γ([0, 1])

b =

1

• log(γ(t) − c2)γ′(t)2dt.

γ(t) := 1 + t 1 + 2t

• and

c :=

• ,

γ(t) := 1 + 2t 1 + 4t

• ②