R R R K h
( ) L 2 (Ω) H k (Ω) H0 k (Ω) R u h R 2 Φ i Φ i L 2
A : R n R n n N + x x Ax x x 2 A x 2 x 3 x 3 a a n A := a n a nn A x = ( 2 5 9 A = )( x ( ) 2 5 9 x 2 ) ( ) 2x +5x = 2. x +9x 2 Ax = b 2x +5x 2 = b x +9x 2 = b 2.
A A = L U E ij a ij A ( ) 2 A =. 8 7 ( ) 0 E 2 := a 4 2 ( ) ( ) ( 0 2 2 = 4 }{{} E 2 8 7 }{{} A ) A = L U L = E 2 = ( 0 4 ) 0 3 }{{} U Ax = LUx = E2 x = U E 2 b. E 32 E 3 E 2 A = U 0 0 0 0 E 3 = Id, E 32 = 0 0, E 2 = 2 0 0 5 0 0 i,j E i,j Eij i,j E i,j 0 0 0 0 0 0 0 0 2 0 = 2 0 0 5 0 0 0 5 L 0 0 0 0 0 0 2 0 0 0 = 2 0 0 } 0 {{ 0 }} 5 {{ } 0 5 E 2 E 32
L n n n = 00 A 00 2 99 2 98 2 w w 00 2 + 99 2 + 98 2 +...+ 2. w n 2 +(n ) 2 +(n 2) 2 +...+ 2. w w n x 2 x=0 n 0 x 2 dx = 3 x3 n 0 = 3 n3. O ( 3 n3) Au =0 Au = b x 0 A A x A x = λ x, λ x A
P : R 3 R 3 x E P P E Px = x Px 2 = x 2 x x 2 x 3 E Px 3 =0 x. A = ( ) 0 0 x = x 2 = ( ), ( ) λ = λ 2 = 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 3 (A λid) = ( ) 3 λ =(3 λ) 2 = 3 λ = λ 2 6λ +8 A λ =4 λ 2 =2 A 4 Id = ( ) ( ) x =0 λ =4 x = ( ) λ 2 =2 x 2 = ( ) x Ax = λx A(αx) =λ(αx) α y 0 x := {αx} λ n n A C A C n n n C n n C R A R n n λ,...,λ n C A C n n n λ i = (A) i= n λ i = (A). i=
Q α ( ) (α) (α) Q α = (α) (α) 90 o = π 2 ( ) 0 Q π = 2 0 Q π 2 ) ( λ + λ 2 = ( λ λ 2 = Q π 2 Q π 2 =0, ) =. R ( ) Q π λid = 2 ( ) λ λ = λ 2 +=0 λ /2 = ±i. n n n ( ) 3 A = 0 3 A λ /2 =3 (A λid)x = ( 0 0 0 A x = )( x x 2 ) = ( ) 0 ( ) 0. 0 n n A n A n
n n A n x,...,x n S x,...,x n x x n x 2 x 2 n S :=. x n x n n A S x x n λ x x 2 x 2 λ n x n n A S = A = λ x 2 λ n x 2 n x n x n n λ x n λ n x n n x x n λ 0 0 x 2 x 2 n 0 λ 2 0 = x n x n n 0 0 λ n A S = S Λ λ 0 0 0 λ 2 0 Λ:= 0 0 λ n Λ S AS = SΛ S AS = Λ A = SΛS S A A Ax = λx A 2 x A 2 x = Aλx = λax = λ 2 x A 2
A A = SΛS A k =(SΛS ) (SΛS )=SΛ k S. A A 00 A 00 =(SΛS )(SΛS ) (SΛS )=SΛ 00 S. λ i < i =,...,n A k 0 k, n n n A A = λ = λ 2 =...= λ n =. n n 0 0 0 x =,x 2 =,...,x 0 n =. 0 0 ( ) 2 A = λ 0 2 = λ 2 =2 ( ) 0 (A 2Id) x = x =0. 0 0 ( ) A 2Id x = 0
n n A R n n u k+ = Au k, u 0 R n u = Au 0, u 2 = A 2 u 0, u k = A k u 0. u 0 x,x 2,...,x n A u 0 = c x + c 2 x 2 + + c n x n = Sc. Au 0 Au 0 = Ac x + Ac 2 x 2 + + Ac n x n = c λ x + c 2 λ 2 x 2 + + c n λ n x n A k u 0 = c λ k x + c 2 λ k 2x 2 + + c n λ k nx n, 0,,, 2, 3, 5, 8, 3,...
F 00 F k+2 = F k+ + F k u k := A = ( Fk+ F k F k+2 = F k+ + F k F k+ = F k+. ) Au k = u k+ ( ) 0 ( ) (A λid) = λ = λ( λ) =0 λ λ 2 λ = 0 λ /2 = ± +4 2 λ.68 λ 2 0.68. 2 2 ( ) A k =(SΛS ) k.6 k 0 = S 0 0.6 k S F 00 c.6 00,F k c.6 k k u k u k = c x + c 2 x 2 x,x 2 ( λ λ (A λid)x = 0 )( ) x = x 2 ( 0 0 )
( ) λ x = x 2 = ( ) ( ) F u 0 = = 0 F 0 ( ) ( ) λ λ2 c x + c 2 x 2 = c + c 2 = ( ) λ2 c c 2 ( ). 0 2 2 c /2 = ± 5 ±0.447 F 00 0.447.6 99 3.54 0 20. u(0) = du dt = u +2u 2 du 2 dt = u 2u 2 ( ) 0 ( du ) ( dt 2 du 2 = 2 dt )( u u 2 ) = Au. A λ =0 A λ 2 = 3 λ i = (A) = 3 ( ) 2 x =, ( ) x 2 =. u u(t) =c (λ t) x + c 2 (λ 2 t) x 2. c c 2 ( ) ( ) 2 c (λ t) x + c 2 (λ 2 t) x 2 = c + c 2 ( 3t) ( ) u(0) = t =0 0 ( ) ( ) ( ) 2 c + c 2 = c 0 = c 2 = 3
u(t) = 3 x + 3 ( 3t) x 2. t u( ) = ( ) 2. 3 u(t) 0 u(t) v u(t) ± u(t) 0 e λt 0 λ<0. λ C λ = 3+6i e ( 3+6i)t = e 3t e 6it e 6it = (6t)+i (6t) = ( )( ) (6t) (6t) i e 6it ( ) (6t) = = (6t) 2 (6t)+ 2 (6t) =. Re(λ) < 0 λ =0 Re(λ) < 0 Re(λ) > 0 ( ) a b 2 2 A = c d Reλ < 0 Reλ 2 < 0. (A) =a+d = λ ( +λ 2 < ) 0 (A) < 0 2 0 λ 0 2 => 0 (A) > 0.
A C n n du dt = Au. A u i u u := Sv S dv dt = S ASv =Λv v(t) = e Λt v(0) S u(t) = e Λt S u(0) u(t) = Se Λt S u(0). e At = Se Λt S, u(t) =e At u(0) = Se Λt S u(0). e At = Id + At + (At)2 2 + (At)3 6 +...+ (At)n n! +... e At = Se Λt S e At = Id + At + (At)2 2 + (At)3 6 +... = Id +(SΛS )t + (SΛS )(SΛS )t 2 2+... = Id +(SΛS )t + 2 t2 (SΛ 2 S )+ 6 t3 (SΛ 3 S )+... = S(S +ΛS t + 2 t2 Λ 2 S +...)= = = S(Id +Λt +Λ 2 t2 2 + Λ3 t 3 = Se Λt S 6 +...)S = e At = e SΛS t = Se Λt S.
e λ t 0 e Λt =. 0 e λnt e At = Se Λt S 0, e Λt 0 Re(λ) < 0 ( ) y u = y y + by + cy =0 y + by + cy = 0 y = y ( ) ( )( ) ( ) u y b c y b c = y = = u. 0 y 0
u = f(u, t) u(t = 0) = u 0 u i = f i (u, t), i =,...,n u i (t = 0) = u i,0. u = au, a f u u = Au, A ij f i. u j u n+ u n,u n,... t n,t n,... u n+ t n+ n + u (t) D u f (u(t),t) u
u(t) =e t + e 99t. u(t) e t u(t) e 99t t ( ) 50 49 A = u = Au 49 50 u(t) =e t + e 99t. λ = λ 2 = 99 A cond(a) := λ max λ min. cond(a) = 99 a>0
u = f(u, t) =au. u u n+ u n. t u n+ u n t = au n u n+ = u n + tau n =(+a t)u n u n =(+a t) n u 0. +a t > u n a<0 +a t. a t a<0 0 < t < 2 a. u n+ u n u n+ u n t u n+ = = f(u n+,t n+ )=au n+ a t u n u n = ( ) n u 0. a t a<0 a t < t.
a <0 a >0 t 0 u k+ = u k + tφ(u k+,u k,t k ). Φ t k = t 0 + k t, u(t k ): t k, u k : t k. t k+ d k+ := u (t k+ ) u (t k ) tφ(u (t k+ ),u(t k ),...,t k ). u = au u n+ = u n + tau n u(t n+ ) = u(t n )+ tau(t n )+d n+t d n+ = u(t n ) u(t n ) tau(t n )
t n e n := u(t n ) u n? e n+ = u(t n+ ) u n+ = u(t n )+a tu (t n )+d n+ (u n + a tu n ) e n+ = e n + a te n + d n+ =(+a t) n d +...+(+a t) n+ k d k +...+ d n+. +a t d k+ = 2 ( t)2 u (t k + θ t), 0 <θ< e n+ (n + ) 2 ( t)2 u = 2 T t u T := (n + ) t t g k t k g k := u(t k ) u k. g k d k Φ:B R L R 0 <L< Φ(x, y,z, t) Φ(x, y 2,z, t) L y y 2, Φ(x, y, z, t) Φ(x, y, z 2, t) L z z 2 (x, y,z, t), (x, y 2,z, t)(x, y, z, t)(x, y, z 2, t) B u (t k+ )=u (t k )+ t Φ(u (t k+ ),u(t k ),t k )+d k+.
g k+ = g k + t (Φ (u (t k+ ),u(t k ),t k ) Φ(u (t k+ ),u k,t k ) +Φ (u (t k+ ),u k,t k ) Φ(u k+,u k,t k )) + d k+. t L< g k+ g k + t (L u (t k ) u k + L u (t k+ ) u k+ )+ d k+ g k+ + tl tl g k + d k+ tl. Φ u k+ g k+ ( + tl) g k + d k+. tl < K>0 + tl tl + tk g k+ ( + a) g k + b { { tk () K a = tl () b = 2L d k+ () d k+ () (g k ) k N g k+ ( + a) g k + b k N +, g k ( + a) k g 0 + ( + a)k b e ka g 0 + b ( ) e ka a a k N. g k ( + a) g k + b ( + a) 2 g k 2 +((+a) + ) b ( + a) k g 0 + = (+a) k g 0 + ( + a)k a ( + t) e t t ( + a) k e ka ( ) ( + a) k +...+(+a)+ b b.
g 0 = u(t 0 ) u 0 =0 D := k d k g n t n = t 0 + n t g n D ( e n tl ) D tl tl en tl. g n D ( e n tk ) D 2 tl 2 tl en tk. D L h d k+ = u (t k+ ) u (t k ) tf (u (t k ),t k ). u (t k+ )=u (t k )+ tu (t k )+ 2 ( t)2 u (t k + θ t) 0 <θ< f (u (t k ),t k )=u (t k ) d k+ = u (t k )+ tu (t k )+ 2 ( t)2 u (t k + θ t) u (t k ) tu (t k )= 2 ( t)2 u (t k + θ t). M := t0 ξ t n u (ξ) d k+ 2 ( t)2 M g n tm 2L en tl. g n t p C d k ( d k D = C ( t) p+ = O ( t) p+). g n g n C L en tl ( t) p = O (( t) p ).
p u (t k+ )=u(t k )+ t! u (t k )+ ( t)2 2! u (t k )+ ( t)3 3! u (3) (t k )+...+ ( t)p u (p) (t k )+R p+ p! p d k+ = R p+ = ( t)p+ (p + )! u(p+) (t k + θ t), 0 <θ<. u = 2tu 2 u(0) = u t u (t k+ )=u (t k )+c t + c 2 ( t) 2 + c 3 ( t) 3 + c 4 ( t) 4 +... c i u = 2tu 2 t = t k + t c +2c 2 t + 3c 3 ( t) 2 +4c 4 ( t) 3 +... ( = 2(t k + t) u (t k )+c t + c 2 ( t) 2 + c 3 ( t) 3 + c 4 ( t) 4 +... ( = 2(t k + t) u 2 (t k )+2c u (t k ) t + ( c 2 +2c 2 u (t k ) ) ( t) 2 + ) +(2c c 2 +2c 3 u (t k )) ( t) 3 +... = 2t k u 2 (t k )+ ( 2u 2 (t k ) 4c t k u (t k ) ) t + ( 4c u (t k ) 2t k ( c 2 +2c 2 u (t k ) )) ( t) 2 + + ( 2 ( c 2 +2c 2 u (t k ) ) 4t k (c c 2 + c 3 u (t k )) ) ( t) 3 +... c = 2t k u 2 (t k ) 2t k u 2 k c 2 = (u (t k )+2c t k ) u (t k ) (u k 2c t k ) u k c 3 = ( ( 4c u (t k )+2t k c 2 +2c 2 u (t k ) )) ( ( )) 4c u k +2t k c 2 3 3 +2c 2 u k c 4 = 2 c2 c 2 u (t k ) t k (c c 2 + c 3 u (t k )) 2 c2 c 2 u k t k (c c 2 + c 3 u k ) ) 2
e k := u (t k ) u k t t 2 u () k+ = u k + tf (u k,t k ) u (2) = u k+ k + t 2 2 f (u k,t k ) u (2) k+ = u (2) + t ( k+ 2 2 f u (2),t k+ k + t ) 2 2 u k+ := 2u (2) k+ u() k+ =2u(2) = u k + tf ( u (2) k+ 2 + tf k+ 2 ( u k + t 2 f (u k,t k ),t k + t 2,t k + t 2 ). ) u k tf (u k,t k ) k := f (u k,t k ), ( k 2 := f u k + t 2 k,t k + t ), 2 u k+ = u k + tk 2. u (t) =f (u(t),t) [t k,t k+ ] t k+ t k u (t)dt = u(t k+ ) u(t k ) = t k+ t k f (u(t),t) dt t k+ t k f (u(t),t) dt.
u(t) t k+ t k f (u(t),t) dt t 2 (f (u k,t k )+f (u k+,t k+ )). u k+ = u k + t 2 (f (u k,t k )+f (u k+,t k+ )) u k+ u (0) k+ = u k + tf (u k,t k ) u (n+) k+ = u k + t ( ( )) f (u k,t k )+f u (n) 2 k+,t k+ u k+ f L tl 2 < t < 2 L Φ(u k,u k+,t k ):= 2 (f (u k,t k )+f (u k+,t k+ )). d k+ = u (t k+ ) u (t k ) t 2 (f (u (t k),t k )+f (u (t k+ ),t k+ )) = u (t k+ ) u (t k ) t ( u (t k )+u (t k+ ) ) 2 = tu (t k )+ ( t)2 u (t k )+ ( t)3 u (t k )+O (( t) 4) ( 2 6 t u (t k )+u (t k )+ tu (t k )+ ( t)2 ( u (t k )+O ( t) 3)) 2 2 = ( 2 ( t)3 u (t k )+O ( t) 4). ( t) 3
u k+ u(t k+ ) u (p) k+ = u k + tf (u k,t k ), u k+ = u k + t ( ( )) f (u k,t k )+f u (p) 2 k+,t k+. k = f (u k,t k ), k 2 = f (u k + tk,t k+ ), u k+ = u k + t 2 (k + k 2 ). ( ) k k 2 (t k,u k ) t k+,u (p) k+
U R n f : U R f x U D i f(x) := h 0 f(x + he i ) f(x) h e i R n (e i ) j = δ ij i f ei f f x i D i f U R n f : U R x U f(x) := f x ( f (x),..., f ) (x) x x n (f) f := ( x,..., f,g : U R x n ) (f g) =g f + f g
U R n v =(v,...,v n ):U R n v i n v i v := x i v i= v. U R 3 v : U R 3 ( v3 v := v 2, v v 3, v 2 v ) x 2 x 3 x 3 x x x 2 v v v = v U R n f : U R f := f = f = 2 f := 2 x 2 +...+ 2 x 2 n = n x 2 2 x 2 i= i +...+ 2 f x 2 n u = f(t, u) t t f(t, u) f
c(x, t) F c(x, t) F = D c, D V c V c V c t d x. V V c F n d S = t d x. V F : R n R n V R n F ( x ) d x = F ( x ) n d S. V V V V F ( x ) d x = V F n d S = V c d x V t F = c t c = (D c) t v ρ p
ρ t = ρ 0 v ρ 0 ρ 0 v t = p. p p = c 2 ρ 2 t 2 ρ = ρ 0 ( ) ) v v t = (ρ 0 t 2 c 2 p = ( p) t2 2 t 2 p = c2 ( p) =c 2 p R R 2 R u tt = u xx. R 2 u tt = c 2 u. Ω R d d {2, 3} ρ :Ω R Ω Φ Φ = ρ Ω.
u =0 Ω R d. Ω:={(x, y) R 2 ; x 2 +y 2 < } x y x := r φ y := r φ u = 2 u r 2 + u r r + 2 u r 2 φ 2. r k (kφ) r k (kφ) r = u Ω = u( φ, φ) =a 0 + r < (a k (kφ)+b k (kφ)). k= u(x, y) =a 0 r k (a k (kφ)+b k (kφ)). k= c i Φ c i t = (ε r ε 0 Φ) = ρ f + i ( ) z i F D i c i + D i RT c i Φ z i Fc i D i c i z i ρ f ε r ε 0 F R T
u = f Ω Ω Ω Ω Ω u = f
Ω={(x, y) :0<x<, 0 <y<}. Ω Ω Ω h h Ω h = {(x, y) Ω: x h, y h Z }. Ω h u(x) u h (x) u(x) u h (x) u(x + h) u(x) u(x + h) u(x) h 0 h h Ω h Ω h Ω R
u (x) = f(x) Ω=(0, ), u(0) = ϕ 0, u() = ϕ. δ + u(x) = u(x+h) u(x) h δ u(x) = u(x) u(x h) h δ 0 u(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) h 2. [x h, x + h] Ω δ ± u(x) =u (x)+hr R 2 u δ 0 u(x) =u (x)+h 2 R R 6 u δ + δ u(x) =u (x)+h 2 R R 2 u(4) u(x ± h) = u(x) ± hu (x)+ h2 2 u (x)+... u(x + h) u(x) h = u(x) ± hu (x)+ h2 2 u (ξ), x ξ x + h = u (x)+ h 2 u (ξ) u (x)+ h 2 u x ± h u(x + h) = u(x)+hu (x)+ h2 2 u (x)+ h3 6 u (ξ), u(x h) = u(x) hu (x)+ h2 2 u (x) h3 6 u ( ξ).
δ 0 u(x) = 2hu (x)+ h3 6 (u (ξ)+u ( ξ)) 2h = u (x)+ h2 2 (u (ξ)+u ( ξ) u (x)+ h2 2 2 u = u (x)+ h2 6 u x ± h u(x + h) = u(x)+hu (x)+ h2 2 u (x)+ h3 6 u (x)+ h4 24 u(4) (ξ) u(x h) = u(x) hu (x)+ h2 2 u (x) h3 6 u (x)+ h4 24 u(4) ( ξ) 2u(x) h 2 δ + δ u(x) = ( ) h 2 u (x)+ h4 24 u (4) (ξ)+u (4) ( ξ) δ + δ u(x) = u (x)+ h2 ( ) u (4) (ξ)+u (4) ( ξ) 24 h 2 u (x)+ h2 2 u(4) u (x) = u(x) =f(x), h = δ + δ δ + δ u(x) =f(x)+o(h 2 ) δ + δ u(x) = h 2 2 0 2 0 2 u h (x ) u h (x 2 ) u h (x 3 ) = f(x ) h 2 u(0) f(x 2 ) f(x 3 ) h 2 u().
0 x x 2 x 3 h 2 2 0... 0 0 0 2 0... 0 0 0 2 0... 0 0 0... 0 2 0 0... 0 0 2 u u n = f u 0 h 2 f 2 f n f n u n h 2, K h u h = f h K h R {(i, j) {,...,n} 2 : K i,j 0} = O(n). Ω:= { (x, y) R 2 : 0 <x<, 0 <y< } Ω h := { (x, y) Ω: x h, y } h Z Γ := { (x, y) R 2 : x {0, },y {0, } }, { x Γ h := (x, y) Γ: h, y } h Z. u = f Ω, u = ϕ Γ
h u h := ( δ x δ + x δ y δ + y )u h (x). u h u u Ω h h u h h u h = ( δ x δ + x δ y δ + y )u h (x) = h 2 (u h(x h, y)+u h (x + h, y)+u h (x, y h)+u h (x, y + h) 4u h (x, y)). u h R R 2 7 8 9 4 5 6 2 3
h 2 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 u h = f h f h f K h = h 2 D I 0 0 I D I 0 0 0 I D D = I = 4 0 0 4 0, 0 0 4 0 0 0 0 0 0. 0 0 0
4 9 5 7 3 8 6 2 h 2 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 u h = f h ( D L K h = L T D 2 ) K h D i L L T R 2
h u h = ( u(x h, y) u(x + h, y) u(x, y h) u(x, y + h)+4u(x, y)) h2 =: h 2 4 = h. Ω h K h R δ + = h [ 0 ] δ = h [ 0 ] δ 0 = 2h [ 0 ] h k c, c 0, c, c,0 c 0,0 c,0 c, c 0, c, = h k c ij u h (x + ih, y + jh) i,j (x, y) [ ][ ] a b c d e f uh = [ a b c ] (d u h (x h)+e u h (x)+f u h (x + h)) = a(du h (x 2h)+eu h (x h)+fu h (x)) +b(du h (x h)+eu h (x)+fu h (x + h)) +c(du h (x)+eu h (x + h)+fu h (x +2h)) = [ ad ae + bd af + be + cd bf + ce cf ].
n a ij =0 j= i =...n. h a ii > 0, a ij 0 (i j). 2 h K h a ii a ii > n a ij i =...n j= j i n a ij i =...n j= j i K h K h
A K n n à K n n A à = à A = E n, E n A K n n n λ j A j =0 λ j =0 j =...n, j= A j A A K n n A K n n A A T A n A n (A T ) =(A ) T A, B K n n (a ij ) i,j=...n (b ij ) i,j=...n A B a ij b ij, i, j =...n A B A<B A>B n n a ii > 0, a ij 0 i, j =...n, i j A A 0 K h u = f, K h u h = f h
K h A A 0 A K n n i, j {,...,n} i j a ij 0 i j (i k ) k=,...,p {,...,n} i = i, i p = j i k i k k =2...n A i {,...,n} j {,...,n} A Π ( ) Π T A 0 AΠ=. 0 A 2 A =(a ij ) C n n K i := z C : z a ii n a ij, i =...n j= j i n A n K i. i= v A λ v := n j= v j = v i = (A λe n )v =0.
(a ii λ)v i = n a ij v j. j= j i a ii λ = n (a ii λ)v i = a ij v j j= j i n n a ij v j a ij j= j i a ii λ λ K i = j= j i n a ij j= j i z C : z a ii n a ij. j= j i K i K h n j=,j i a ij =4h 2 j 4h 2 a ii =4h 2 4h 2 λ [ 0, 8h 2] K h λ ( 0, 8h 2). A ( n ) ( n ) λ K i K i i= i=
K i := K i := z C : z a ii < z C : z a ii = n j=,j i n j=,j i a ij, a ij. λ A v v = i {,...,n} v i = λ a ii n j=,j i a ij λ a ii < n j=,j i a ij λ K i k v k = λ a kk = n j=,j k a kj k {k {,...,n} : v k =}. λ a jj = n k=,k j a jk j =...n λ n i= K i j {,...,n}\{i} i = i 0,i,...,i l = j a ip i p 0. λ aipip = n aip k=,k i p k vip = λ aip+ i p+ = n aip+ k=,k i p+ k vip+ = p =0...l λ a ip i p = n j=,j i p a ip j v ip = p {0,...,l } n λ aipip aipk vk n aipk k=,k i p vk n aipk. k=,k i p k=,k i p v = n n aipk vk = aipk. k=,k i p k=,k i p
v k k v k = k a ipk 0 vip+ =. λ a ip+ i p+ = n k=,k i p+ a ip+ k K h a ii = 4 h 2 n 2/h 2 r i = a ij = 3/h 2 j=,j i 4/h 2. K h K h K j K h λ (0, 8h ) 2. A K n n a ij < a ii i =,...,n j,j i A a ij < a ii i j,j i a ij a ii i =,...,n j,j i ϱ(a) A K n n ϱ(a) := { λ : λ A}.
D B D := {a ii : i =,...,n} B := D A A ϱ(d B) <. A A ϱ(d B) < C := D B c ij = a ij, a ii c ii = 0. (i j), r i := n c ij < i =,...,n. j= j i n λ K ri (c ii ) = i= n K ri (0) i= λ i < i=...n ϱ(d B) <. A r j j =,...,n r i < i n n λ K rj (0) K rj (0). j= i r i < K ri (0) K (0) n j= K r j (0) K (0) ϱ(d B) < A A A 0 ϱ(d B) < ϱ(d B)=ϱ(C) < S := j= C ν =(I C) ν=0 S(I C) =I SD (D B) =SD A = I A = SD.
D 0, B 0 C 0 C ν 0 S 0 A 0. A u 0 D B λ u ( u i ) n i= λ u = λu = D Bu D B u A 0 D 0 A D 0 A DD B u A D λ u u = A (D B) u = A D(I D B) u A D u A D λ u = ( λ )A D u λ u ( λ )A D u 0 I ( λ )A D 0, u 0 u =0 A A > 0 B C D α, β {,...,n} A α = α 0,α,...,α k = β a αpαp+ < 0 p {0,...,k } c αp α p+ > 0 p {0,...,k } (C k ) αβ = γ,...,γ k c αγ c γ γ 2...c γk β c αα c α α 2... c αk β > 0. ϱ(c) < S := ν=0 Cν S αβ (C k ) αβ > 0 S C k > 0 S>0 A = SD > 0 A > 0 K h V K R C : V R V u, v V λ K u = 0 u =0, λu = λ u, u + v u + v.
V A V { } Au A M := u : u V \{0} = { Au : u V, u =} A A M ϱ (A). { } A M = Au u : u V \{0} v A Av v { } Au u = λv v = λ v Av v = λ = ϱ(a). A = a ij. i {,...,n} j {,...,n} A B A B A B A w Aw A w.
u u u u u Aw. A A 0 A u A u A u Aw = u A Aw = u w A u A u w w u u A w. A B B A 0 B A B A A B = A (B A)B. A, B A 0 B 0 B A B A 0 A B = A (B A)B 0 A B B A. u 2 = n i= u i 2 V 2 A 2 = ϱ (A T A) { } Au 2 A 2 = : u V \{0} = u Au 2. 2 u V, u 2 = A 2 2 = u V, u 2 = Au 2 2 = Au, Au = u 2 = u 2 = AT Au, u.
A T A P T A T AP = (λ i )=: D PP T = P T u = u u V λi > 0 A T A Au 2 2 = u 2 = AT Au, u = u 2 = AT AP P T u, P P T u = }{{} ũ=p T u ũ 2 = AT AP ũ, P ũ = P T A T AP ũ, ũ = Dũ, ũ ũ 2 = ũ 2 = ( n ) λ i ũ i 2 = λ max = ϱ ( A T A ) = ũ 2 = A 2 = i= ϱ (A T A). A A T A = A 2 ρ ( A 2) = ρ 2 (A) A 2 = ϱ(a) A K n n A Au, u > 0 u K n \{0}. A A A P : P T AP = (λ i ) λ i A P Au, u = AP ũ, P ũ = P T AP ũ, ũ = n λ i u i 2 i= Au, u > 0 u K n \{0} λ i > 0 i.
A a ii > 0 A n j=,j i λ i A a ij <a ii (0, ) λ λ A A 2 = λ A 2 = A A 2 = ϱ (A) =λ A A 2 = ϱ ( A ) = λ λ K h Ω Ω=(0, ) (0, ) 4, K h K h K h K h 8 h 2 K h K h 2 8 2 ( πh h 2 2 K h 8 ) < 8 h 2 2 8 h2 2 ( πh 2 ) = 2π 2 +O(h 2 ) < 6 h K h K h K h K h
{ n } K h = i=,...,n j= K ij = {6, 7, 8} = 8 h 2 h 2 K h 8 K h w h (x, y) = w(x, y) = x( x) 2 (x h)( (x h)) (x + h)( (x + h)) 2 x( x) 2h 2 2h 2 + 2h 2 = K h w h w h w h x,y w (x, y) = 8 K h w 8 K h u ν,µ ( ν, µ n ) u ν,µ j,k = (νπjh) (µπkh) λ ν,µ = 2 ( ( ) ( )) νπh µπh h 2 2 + 2. 2 2 (j, k) (K h u ν,µ ) j,k = (4 (νπjh) (µπkh) h2 (νπ (j ) h) (µπkh) (νπ (j + ) h) (µπkh) (νπjh) (µπ (k ) h) (νπjh) (µπ (k + ) h)). (a ± b) = a b ± b a a := νπjh b := νπh a := µπkh b := µπh (K h u ν,µ ) j,k = (4 2 (νπh) 2 (µπh)) (νπjh) (µπkh). h2 (a) =2 2 ( a 2) (K h u ν,µ ) j,k = 4 ( ( ) ( )) νπh µπh h 2 2 + 2 u ν,µ 2 2 j,k. ( ) K h 2 = ϱ(a) =λ max 8 2 π(n )h h 2 2 = 8 2 ( ) πh h 2 2 < 8 h 2
K h 2 = ϱ(k h )= λ min λ min = 8 ( ) πh h 2 2 = 8 ( ) πh 2 2 h 2 2 O(h3 ) ( (πh = 8 ) 2 h 2 O(h )) 4 =2π 2 O(h 2 ) 2 K h 2 2π 2 O(h 2 ). u h h u h = f h f h =0 u h Ωh = ϕ h u h Ω h h u h =0 u h (x, y) = 4 (u h(x h, y)+u h (x + h, y)+u h (x, y h)+u h (x, y + h)) u h (x, y) (x, y) Ω h u h (x ± h, y) u h (x, y ± h) K h Ω h u h
u h,v h h u h = f h u h Ωh = ϕ u h h v h = f h v h Ωh = ϕ v h u h v h x Ωh ϕ u (x) ϕ v (x) u h v h ϕ u ϕ v Ω h w h := v h u h w h h w h =0 w h 0 Ω h ϕ u ϕ v. w h > 0 u h = ϕ u v h = ϕ v Ω h w h Ω h ϕ u ϕ v Ω h. Ω h u = f Ω, u Γ = ϕ h u h = f h Ω h, u h Γh = ϕ h. h H R + H H = { n : n N} U h Ω h R h : C ( Ω ) U h u R h u (R h u)(x) =u(x) x Ω h Ω Ω h
K K h h H R + K h C< h H K h (u h ) = f h, K h (ũ h ) = f h + ε. u h = K h (f h) ũ h = K h (f h + ϵ) ũ h u h C ε K h K 8, h K h u h = f h Ku = f K m R h R h u f (K h,r h, R h ) K k K h R h u R h Ku C h k u C k+m (Ω) u C k+m (Ω). R h = R h (R h u)(x) =u(x) x Ω h. (K h,r h, R h )=( h,r h,r h )
( + u)(x) =u (x)+h 2 R, R u (4) C 2. 0 ( Ω) R 2 x y h Ru(x, y) = u(x, y)+h 2 (R x + R y ) R x, R y u (4) C 2 0 ( Ω) 2 u C 4 ( Ω). K h R h u R h Ku C h 2 u C 4 ( Ω) C = 6 K h u h = f h Ku = f K m u h U h (h H) k u u h R h u C h k u C k+m (Ω) u h R h u K m (K h,r h, R h ) K k k u C k+m (Ω) w h = u h R h u w h 0 h 0. K h w h = K h u h K h R h u = f h K h R h u = R h f K h R h u = R h Ku K h R h u w h = K h ( R h Ku K h R h u) w h K Rh Ku K h R h u u h R h u Ch k u C k+m (Ω). h u C 4 (Ω) u h R h u h2 48 u C 4 (Ω).
u Γ = ϕ u u = f Ω=(0, ) (0, ) u n = ϕ Γ u n u u + c Ω udx=0 f ϕ u fdx+ ϕds = 0. Ω Ω fdx = udx= ( u) dx Ω = Ω Ω Ω u n d s = Ω u n ds = Ω ϕds. Ω=[0, ] [0, ]
h u h (x, y) = h 2 (4u h (x, y) u h (x h, y) u h (x + h, y) u h (x, y h) u h (x, y + h)) Ω h u h (x, y) = h 2 (3u h (x, y) u h (x h, y) u h (x, y h) u h (x, y + h)) h u h (x, y) = h 2 (2u h (x, y) u h (x h, y) u h (x, y + h)) u n u h n (x) ( n u h )(x) = h (u h(x) u(x h n)) = ϕ(x) h (u h(x, 0) u h (x, h)) = ϕ(x, 0), h (u h(x, ) u h (x, h)) = ϕ(x, ) h (u h(0,y) u h (h, y)) = ϕ(0,y), h (u h(,y) u h ( h, y)) = ϕ(,y) u h K h u h = f h = f h + h ϕ h, ϕ h = ϕ (x, y) h 2 f(x)+h ϕ(x) = 0, x Ω h x Γ h Γ h
K h u h = f h c K h K h c =0 c (K h ). K h ( (K h )) =. K h u h = f h f h (K h ) (K h )= (Kh T ) = (K h ) = span( ) K h u h = f h x Ω h x Ω fh h (x) = x Ω fh h (x)+ h fh (x) = 0 f h ϕ h x Γ ϕ(x) h f h f h h ϕ h f h x 0 Ω h u h u h (x 0 )=0. K h û h = f h Kh K h K h K h K h x i (u h ) i =0. K h u h = ˇf h ( { ˇfh )j = (f h ) j, j i k i (f h) k, j = i
( ˇfh )i =(f h) i i i f ϕ ( ˇfh ) i (f h) i = O ( h ) x i K h ū h = f h σ K h = ( Kh T 0 ) ( uh, ū h = λ ) (, f fh h = σ ) K h ū h = f h / (K h ) (K h, )=(K h )+ K h K h ū h K h ū h = f h u h K h u h = ˇf h ˇf h = f h λ λ λ =0 u h T u h = x Ω h u h (x) =σ λ = x Ω h f h (x) T f ϕ λ = O (h)
Ku = f Ω, n 2 K = a ij (x) + x i x j i,j= n i= b i (x) x i + c(x). a ij (x) =a ji (x) 2 x i x j = 2 x j x i A(x) =(a ij (x)) i,j=...n a ij (x) A n a ij (x)ξ i ξ j > 0 x Ω, 0 ξ R n i,j= n a ij (x)ξ i ξ j < 0 x Ω, 0 ξ R n. i,j= u = f ( ) 0 A =. 0 K Ω n a ij (x)ξ i ξ j c(x) ξ 2,c(x) > 0 x Ω, 0 ξ R n i,j= Ω=(0, ) (0, ) a (x, y) x + x +2a 2 (x, y) x 0 y 0 + a 22 (x, y) y + y + b (x, y) x 0 + b 2 (x, y) y 0 + c(x, y) = h 2 2 a 2(x, y) a 22 (x, y) 2 a 2(x, y) a (x, y) 2(a (x, y)+a 22 (x, y)) a (x, y) + 2 a 2(x, y) a 22 (x, y) 2 a 2(x, y) 0 b 2 (x, y) 0 0 0 0 +(2h) b (x, y) 0 b (x, y) + 0 c(x, y) 0. 0 b 2 (x, y) 0 0 0 0
X R C X : X [0, ) (X, X ) Ω Ω R n C 0 ( Ω) () (2) X 0 <C< C x () x (2) C x () x X.
X Y X Y T : X Y T := x X { Tx Y x X T T } : x 0. L (X, Y ) (T + T 2 ) x = T x + T 2 x (X, ) A X x A ε>0 A K ε (x) := {y X : x y <ε} (X, ) {x n X : n } { x n x m : n, m k} 0, k ε>0 n 0 (ε) N: n, m n 0 (ε) : x n x m <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 (, ) X X (, )
X A X σ X A A A A c A (A n ) n N A n N A n A (X, A) A X O σ σ O X (X, A) f : X R n X = k= A k A k A k =,...n f Ak k =...n (X, A) (Y,B) f : X Y f (B) A B B. Y = R σ f (t n ) n N f (X, A) µ: A [0, ] µ ( ) =0 (A n ) n N A A n A m = n m σ ( ) µ A n = µ (A n ). n N n N (X, A,µ) I n n n k= (a k,b k ] R n a k b k m n m n µ I n : I n [0, ] µ I n (a jk,b jk ] = (b jk a jk ) j= k= j= j=
I n I n σ σ B R n µ I n µ (R n, B) (X, A,µ) A A µ(a) =0 X Y f,g: X Y N f (x) = g (x) x X \ N. (X, A,µ) Y f : X Y (A k ) n k= A n µ (A k ) < f Ak f A =0 A = k= A k X Y T (X, Y ) t T (X, Y ) X tdµ := n t (A k ) µ (A k ). k= (X, A,µ) L (X) f : X R (t n ) n N T (X, R) T (X, R) t := X t dµ f f L (X) X fdµ := k X t k dµ. L (X) L (X) I n µ
f,g L (X) f = g f(x) =g(x) L (X) f := X f dµ µ (X) < L (X) f : X R f : X R f f X R n σ B µ L (X) ( ) D R n (R n, B,µ) σ µ L (D) D L (D) f = g, f = g ; { } u L (D) := A B µ(a)=0 { u (x) } x D\A. L 2 (Ω) Ω R n L 2 (Ω) L 2 (Ω) := { } f :Ω R: f, f 2 L (Ω).
f g A µ(a) =0 (u, v) 0 =(u, v) L 2 (Ω) := uv dµ u, v L 2 (Ω) Ω u 0 = u L 2 (Ω) = Ω u 2 dµ L 2 (Ω) L 2 (Ω) f L 2 (Ω) f (x)ϕ(x)dx = f(x)ϕ (x)dx Ω f,ϕ ϕ Ω =0 L 2 (Ω) Ω R n Cc (Ω) ϕ { } Cc (Ω) := ϕ C (Ω) : {x Ω: ϕ (x) 0}. Ω f L 2 (Ω) g L 2 (Ω) g(x)ϕ(x) dx = f(x)ϕ (x) dx ϕ Cc (Ω), Ω g f Ω
α =(α,...,α n ) α := D α := n α i, i= α α x... αn x n f L 2 (Ω) g α f g(x)ϕ(x) dx = ( ) α f(x)d α ϕ(x) dx ϕ Cc (Ω). Ω H k (Ω) H k 0 (Ω) Ω u L 2 (Ω) D α u L 2 (Ω) H k (Ω) := { u L 2 (Ω) : D α u L 2 (Ω), α k } k N 0 H k (Ω) W k 2 (Ω) W k,2 (Ω) H k (Ω) (u, v) k :=(u, v) H k (Ω) := D α u 2 L 2 (Ω) α k 2. H s (R n ) C k (R n ), s>k+ n 2,k N 0. 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 { T y } { X = X x, T y } Y y 0 y = X X Y x,y 0 x X y Y { Tx,y } { = Y Y T Y x,y 0 x X y X x X y } Y Y x,y 0 x X y = T Y X Y { } { Tx Y Tx,y } T Y X = Y Y x 0 x X x,y 0 x X y Y { x, T y } { = X X T } x,y 0 x X y X Y y Y x X Y x,y 0 x X y = T X Y, Y y Y Tx,y Y Y = Tx Y y Y = x X X R y X f y (x) := (x, y) X f y X f y X = y X f y y X X f X y f X f (x) =(x, y f ) X x X y f X = f X.
N = {x X : f(x) =0} f N = X, y f =0 N X w X \ N d := d (w, N) = x N w x w N (x n ) n N N d = n w x n (x n ) n N X (w x m )+(w x n ) 2 + (w x m ) (w x n ) 2 =2 ( w x m 2 + w x n 2) ( x m x n 2 =2 w x m 2 + w x n 2) 4 w 2 (x m + x n ) 2 (x m + x n ) N 4 w 2 (x m + x n ) 2 ( 4d 2 ε>0 m, n 2 w x m 2 + w x n 2) < 4d 2 + ε x m x n 2 < 4d 2 + ε 4d 2 = ε. (x n ) n N f N X (x n ) n N x N w x = d λ R x N d 2 w (x + λ x) 2 = w x 2 + λ 2 x 2 2λ (w x, x) λ 2 x 2 2λ (w x, x) 0. λ R λ = x 2 (w x, x) z = w x x X ( f x f (x) z ) = f (x) f f (z) x ( z, x f (x) ) f (z) z =0 (w x, x) =0 x N (z,x) f (x) (z, z) =0 f (z) (x, z) f(x) = z 2 f (z) = ( ) f (x) z f (z) = f (x) f (x) f (z) =0 f (z) x f (x) z f (z) N. ( x, f (z) z z 2 ). 2.
y f = f(z) z 2 z ỹ f f (x) = (x, ỹ f )=(x, y f ) x X (x, ỹ f y f )=0 x X ỹ f y f =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 (, ) C s a (x, y) C s x V y V x, y V. A L (V,V ) x V a (x, y) = Ax, y V V x, y V, A V V C s. ϕ x (y) := a (x, y) ϕ x V ϕ x V C s x V A: V V Ax := ϕ x Ax V C s x V { } A V V = Ax V C s. 0 x V x V a (, ) C E > 0 a (x, x) C E x 2 V x V.
V a: V V R f : V R J (v) := a (v, v) f (v) 2 V u u u, v V, t R a (u, v) =f (v) v V. J (u + tv) = a (u + tv, u + tv) f (u + tv) 2 = ( a (u, u)+2ta (u, v)+t 2 a (v, v) ) f (u) tf (v) 2 = J (u)+t (a (u, v) f (v)) + 2 t2 a (v, v) u V a(u, v) =f(v) v V u t= J (u + v) = J (u)+(f (v) f (v)) + a (v, v) 2 = J (u)+ a (v, v) >J(u) v V. 2 u V t J (u + tv) v V t =0 dj (u + tv) 0 = t=0 = a (u, v) f (v) dt a(u, v) = f(v).
u,u 2 a (u,v)=f (v) a (u 2,v)=f (v) v V a (u u 2,v)=0 v V u u 2 =0. n Lu := i (a ik k u)+a 0 u i,k= a 0 (x) 0(x Ω) A =(a ik ) i,k Lu = f Ω u = g Ω f L 2 (Ω) g H 2 ( Ω) := { v L 2 ( Ω) : w H (Ω),γ(w) =v } γ g H (Ω) γ (g) =g w := u g Lw = f Lg =: f Ω w = 0 Ω. i,k i (a ik k u)+a 0 u = f Ω u = 0 Ω J (v) := a ik i v k v + 2 2 a 0v 2 fv dx Ω i,k C 2 (Ω) C 0 ( Ω)
v ( w )+ v w d x = Ω Ω v ( w n ) ds. v Ω =0 w i = k a ik k u Ω v i Ω i ( k v ( w ) dx = Ω ) a ik k u dx = v w d x Ω a ik i v k udx. i,k a (u, v) := f (v) := Ω a ik i v k u + a 0 uv dx, i,k fv dx Ω v ( a (u, v) f (v) = v ) i (a ik k u)+a 0 u f dx = Ω Ω v(lu f)dx = Lu=f 0. a (, ) f u u C 2 (Ω) C 0 ( Ω) u
J (u) := u 2 dx. Ω J (u) u = 0 Ω, u = ϕ Γ. J (u) 0 u u = 0 Ω, u = ϕ Γ. u(0) =, u() = 0 J (u) = 0 u 2 (x) dx u C 0 ([0, ]) u (x) n u n (x) { nx 0 x u n (x) = n, 0 x> n.
n 0 u n =0 J (u) J (u) =0, V H a: H H R l H V J J (v) := a (v, v) l, v 2 J (v) 2 C E v 2 l v = = ( CE 2 v 2 2C E l v + l 2) l 2 2C E 2C E (C E v l ) 2 l 2 l 2. 2C E 2C E 2C E c := {J (v) :v V } (v n ) n N J (v n)=c. n (v n ) n N a (, ) C E v n v m 2 a (v n v m,v n v m ) = 2a (v n,v n )+2a (v m,v m ) a (v n + v m,v n + v m ). a (v, v) a (v, v) =2J (v) +2 l, v C E v n v m 2 4J (v n )+4 l, v n +4J (v m )+4 l, v m ( ( ) ) vn + v m 8J +4 l, v n + v m 2 ( ) vn + v m =4J (v n )+4J (v m ) 8J 2 4J (v n )+4J (v m ) 8c.
V v n+v m 2 V J ( ) v n +v m 2 >c J (v n ) c (n ) J (v m ) c (m ) C E v n v m 2 0 n, m, (v n ) n N H H u H V u V n v n = u J (u) = n J (v n )= v V J (v) J (v) = 2a (v, v) l, v u V u u 2 u,u 2,u,u 2,... u,u 2,u,u 2,... u = u 2 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 H0 (Ω) L Lu = f Ω, u = 0 Γ a (u, v) = (f,v) 0 v H0 (Ω) a (u, v) := a ik i u k v + a 0 uv dx. Ω i,k
L Lu = f Ω, u = 0 Γ f L 2 (Ω) H0 (Ω) 2 a (v, v) (f,v) 0 H 0 (Ω). u = f Ω, a (u, v) = u = 0 Γ Ω u v dx. u H0 (Ω) ( u, v) 0 =(f,v) 0 v H 0 (Ω). u u v dx (f,v) 2 0 H0 (Ω). Ω Lu = f Ω, n i a ik k u = g Γ, i,k n i i f L 2 (Ω) g L 2 (Γ) l, v := fv dx+ gv dx u H (Ω) Ω 2 a (u, v) =(f,v) 0,Ω +(g, v) 0,Γ v H (Ω). Γ
Ω f (x) dx + g (x) ds =0 Ω Ω V := v H (Ω) : Ω v (x) dx =0. J (v) := 2 a (u, v) (f,v) 0,Ω (g, v) 0,Γ V u u C 2 (Ω) C ( Ω) Lu = f Ω, n i a ik k u = g Γ, i,k H (Ω) C 2 (Ω) C ( Ω) u = f Ω, u = 0 Γ u H 0 (Ω) a (u, v) =(f,v) v H 0 (Ω) a (u, v) := (f,v) := Ω Ω u v dx, fv dx.
H 0 (Ω) V h ( H 0 (Ω) ) J (v) := 2 a (v, v) l, v V h. u h V h a (u h,v)= l, v v V h. {ψ,ψ 2,...,ψ N } V h a (u h,v)= l, v v V h a (, ) l ( ) a (u h,ψ i )= l, ψ i i =, 2,...,N. u h V h ψ i N u h = z k ψ k k= z k N a (ψ k,ψ i ) z k = l, ψ i i =, 2,...N. k= A ik := a (ψ k,ψ i ) b i := l, ψ i Az = b. a (, ) A z T Az = i,k ( z i A ik z k = a z k ψ k, i = a (u h,u h ) C E u h 2 V. k z i ψ i ) u h V h u V
V a: V V R C S C E l V V h V u V a (u, v) =l (v) v V u h V h a (u h,v)=l (v) v V h. u u h C S u v h. C E v h V h V h V v V h a (u, v) a (u h,v)=a (u u h,v)=0 v V h. a (, ) C E u u h 2 a (u u h,u u h ). a (u u h,u h v h ) v h V h C E u u h 2 a (u u h,u u h )+a (u u h,u h v h ) = a (u u h,u v h ) C S u u h u v h v h V h. u u h C S C E u v h v h V h u u h C S u v h. C E v h V h u u h a V h C S C E V h a v a :=(a(v, v)) 2 V C E C S a u h u V h
V h u = f Ω=(0, ) (0, ), u = 0 Γ. Ω LO O IV II L III Z I R VI VIII V VII U RU
ψ Z h h h h 2 ψ Z h h h h ψ Z V h V h = { v C ( Ω) : v v Γ =0 }. v v (x, y) =a + bx + cy. a b c N V h = N N V h {ψ i } N i= ψ i (K j )=δ ij, K j j, i, j =,...,N. ψ Z Au = b A ij = a (ψ i,ψ j ). a (ψ Z,ψ Z ) a (ψ Z,ψ O ) a (ψ Z,ψ U ) a (ψ Z,ψ L ) a (ψ Z,ψ R ) a (ψ Z,ψ LO ) a (ψ Z,ψ RU )
a (ψ Z,ψ Z ) a (ψ Z,ψ O ) a (ψ Z,ψ Z ) = a (ψ Z,ψ O ) = = =,, Ω = 2 = 2 ( ψ Z ) 2 dxdy =,,, = 2 h 2 = 4. = h 2, ( ψ Z ) 2 dxdy ( ( ψ Z ) 2 +( 2 ψ Z ) 2) dxdy ( ψ Z ) 2 dxdy +2 dxdy + 2 h 2 ψ Z ψ O dxdy ψ Z ψ O dxdy = 2 ψ Z 2 ψ O dxdy =, dxdy =.,,,, dxdy ( 2 ψ Z ) 2 dxdy ψ Z ψ O + 2 ψ Z 2 ψ O dxdy h h dxdy a (ψ Z,ψ O )=a (ψ Z,ψ U )=a (ψ Z,ψ L )=a (ψ Z,ψ R )=. a (ψ Z,ψ RU )=a (ψ Z,ψ LO )=0. 0 0 4 0 0 h 2
T = {T,T 2,...,T M } Ω Ω = M i= T i T i T j T i T j T i T j (i j) T i T j T i T j A B Ω h Ω R d V p h (T )={ u H : T T: u T P p } p
A B C D R d (d =, 2, 3) R u + u = f a (u, v) = ( u v + uv) dx. Ω N = {a = x 0,x,x 2,...,x N+ = b} [a, b] h i = x i+ x i ϕ i u h ϕ i (x j )=δ ij. u h = N a i ϕ i i= a i = u h (x i ). u h ϕ i Φ i [x i,x i+ ] [0, ] [0, ] Φ i
A B x i x i x i + I i =[x i,x i+ ] ξ [0, ] x Ii :[0, ] I i, ξ x i + h i ξ; ξ Ii : I i [0, ], x (x x i) h i [0, ] [x i,x i+ ] u h (ξ) =α + α 2 ξ. u i = u h (0) = α u i+ = u h () = α + α 2 ξ [0, ] u h (ξ) = α + α 2 ξ = u i +(u i+ u i ) ξ = ( ξ) u i + ξu i+ =: u i Φ (ξ)+u i+ Φ 2 (ξ). ξ [0, ] : Φ (ξ)+φ 2 (ξ) =. ϕ i (x) = Φ 2 ( ξii (x) ), Φ (ξ Ii (x)), x I i x I i 0,.
a (ϕ i,ϕ j ) A N a (ϕ i,ϕ j )= k= I k ϕ i (x) ϕ j (x)+ϕ i (x)ϕ j (x) dx. ϕ i,ϕ j Φ n Φ m n, m {, 2} x ξ I k [0, ] (A Ik ) nm = x Φ n (ξ Ik (x)) x Φ m (ξ Ik (x)) + Φ n (ξ Ik (x)) Φ m (ξ Ik (x)) dx I k = h k ξ Φ n (ξ) ξ I k (x Ik (ξ)) ξ Φ m (ξ) ξ I k (x Ik (ξ)) + Φ n (ξ)φ m (ξ) dξ = h k 0 h 2 k 0 Φ n Φ m +Φ n Φ m dξ.
(A Ik ) mn I k m n A Ik (A Ik ) = = (A Ik ) 2 = = 0 0 0 0 h k Φ Φ +Φ Φ h k dξ h k + h k ( ξ) 2 dξ = h k + 3 h k h k Φ Φ 2 +Φ Φ 2 h k dξ h k + ξ( ξ) h k dξ = h k + 6 h k (A Ik ) 2 = h k + 6 h k (A Ik ) 22 = h k + 3 h k A Ik = h k + h k /3 /6 /6 /3. u h (ξ) u h (ξ) =α + α 2 ξ + α 3 ξ 2 I =[0, ]. u i u i+ ( ) xi + x i+ u i+ = u h. 2 2 u i = u h (0) = α, u i+ = u h () = α + α 2 + α 3, ( ) u i+ = u h = α + 2 2 2 α 2 + 4 α 3.
α i,i=,...,3 u h (ξ) =u i Φ (ξ)+u i+ Φ 2 (ξ)+u i+ Φ 3 (ξ) 2 ( Φ i (ξ) = 2 ξ ) (ξ ) 2 ( Φ 2 (ξ) = 2ξ ξ ) 2 Φ 3 (ξ) = 4ξ ( ξ ) Φ 3 (ξ) Φ (ξ) Φ 2 (ξ) A Ik 3 3 Φ i (ξ) R 2 R
y x x = x +(x 2 x ) ξ +(x 3 x ) η, y = y +(y 2 y ) ξ +(y 3 y ) η x y = x y + x 2 x x 3 x y 2 y y 3 y ξ η, (x i,y i ) i ξ η = = x 2 x x 3 x y 2 y y 3 y }{{} A A x x y y y 3 y x x 3 x x y y 2 x 2 x y y A =(x 2 x )(y 3 y ) (x 3 x )(y 2 y ) u x = u ξ ξ x + u η η x, u y = u ξ ξ y + u η η y
ξ x = y 3 y A, η x = y y 2 A, ξ y = x x 2 A, η y = x 2 x A. A dxdy = Adξdη. Φ i,i=, 2, 3 ξ x,ξ y,η x,η y A Ik Φ i u h (ξ,η) = α + α 2 ξ + α 3 η, ( ) u j := u h Pj, j =, 2, 3. P j u = u h (0, 0) = α, u 2 = u h (, 0) = α + α 2, u 3 = u h (0, ) = α + α 3. u h (ξ,η) =u +(u 2 u ) ξ +(u 3 u ) η =( ξ η) u + ξu 2 + ηu 3. Φ = ξ η, Φ 2 = ξ, Φ 3 = η u h (ξ,η) =u Φ + u 2 Φ 2 + u 3 Φ 3 ( ) Φ i Pj = δij i, j =, 2, 3, 3 Φ i (ξ,η) = ξ,η T. i=
Φ i u h u h (ξ,η) =α + α 2 ξ + α 3 η + α 4 ξ 2 + α 5 ξη + α 6 η 2. Φ = ( ξ η)( 2ξ +2η) Φ 2 = ξ(2ξ ) Φ 3 = η(2η ) Φ 4 = 4ξ( ξ η) Φ 5 = 4ξη Φ 6 = 4η( ξ η) R 3 u h (ξ,η,ζ) =α + α 2 ξ + α 3 η + α 4 ζ. α,...,α 4 a a (u, v) = u a :=(a (u, u)) 2. u u h a = v h V h u v h a. Ω u vdx Ω R d,d 3 u H 2 (Ω) u h V h a (u h,v h )= f,v h v h V h H 0 (Ω) u u h a c h f L 2 (Ω), f L2 (Ω). f L 2 (Ω) H 2 u H 2 (Ω) c f L 2 (Ω),
L 2 Ω R d,d 3 u H 2 (Ω) H 2 u u h L 2 (Ω) c h u u h a, u u h L 2 (Ω) c h 2 f L 2 (Ω).
Ω h u t d x = Fd x= F n d s B B Ω h Ω B i u h Kh FV ufv h = f FV h. Ku h = b, u h u h = K b, K A B