October 5, 2018 1 / 113
4 ( ) 2 / 113
Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113
Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ p Γ N \ Γ p 9 = Laplace ν ( A.5.4) ν = ν Poisson 4 / 113
Poisson 5.1.1 (Poisson ) b : R, p N : Γ N R, u D : R u = b in, ν u = p N on Γ N, u = u D on Γ D (5.1.1) (5.1.2) (5.1.3) u : R 5 / 113
Poisson 5.1.1 (5.1.1) Poisson b = 0 Laplace Poisson 5.1.1 Laplace (5.1.3) u u D Γ D u u D (5.1.2) ν u Γ N u 5.1.1 ( ) 6 / 113
Poisson u D H 1 (; R) (5.1.3) u U (u D ) = { v H 1 (; R) v = ud on Γ D } 4.6 U (u D ) Hilbert U = { v H 1 (; R) v = 0 on Γ D } (5.1.4) U Hilbert Poisson (5.1.1) v U Gauss-Green ( A.8.2) uv dx = u v dx ν uv dγ = bv dx (5.1.5) Γ N 7 / 113
Poisson Γ D v = 0 (5.1.2) v U Γ N ν uv dγ = Γ N p N v dγ Γ N (5.1.6) (5.1.6) (5.1.5) u v dx = bv dx + p N v dγ (5.1.7) Γ N (5.1.7) Poisson (5.1.7) u v (5.1.7) v 4.6 a (u, v) = u v dx, (5.1.8) l (v) = bv dx + p N v dγ (5.1.9) Γ N 8 / 113
Poisson 5.1.1 9 / 113
Poisson 5.1.2 (Poisson ) U (5.1.4) b L 2 (; R), p N L 2 (Γ N ; R) u D H 1 (; R) a (, ) l ( ) (5.1.8) (5.1.9) v U a (u, v) = l (v) (5.1.10) ũ = u u D U ( u) 5.1.1 5.1.2 5.1.1 (5.1.1) u 2 Γ N ν u 5.1.2 u 2 (5.1.8) u v 1 2 5.1.1 Poisson 5.1.2 Poisson 5.1.2 u 10 / 113
Poisson 5.2 11 / 113
Poisson (5.1.3) Dirichlet 1 Dirichlet Dirichlet 5.1.1 5.1.2 Dirichlet (5.1.2) Neumann 2 Neumann Neumann 5.1.1 5.1.2 Neumann Neumann ( 5.2.6) Dirichlet Neumann Dirichlet Neumann u D = 0 p N = 0 u D 0 p N 0 12 / 113
Poisson Poisson 5.1.1 Poisson Poisson 8 9 5.1.1 Poisson 5.1.3 ( Poisson ) b : R, c : R, p R : R, c : R u + c u = b in, (5.1.11) ν u + c u = p R on, (5.1.12) u : R 13 / 113
Poisson Poisson 5.1.3 (5.1.12) Robin 3 5.1.3 Robin 5.1.3 U = H 1 (; R) (5.1.13) (5.1.11) v U Gauss-Green ( A.8.2) ( u + c u) v dx = ( u v + c uv) dx ν uv dγ = bv dx (5.1.14) (5.1.12) v U ν uv dγ = (p R c u) v dγ (5.1.15) 14 / 113
Poisson Poisson (5.1.15) (5.1.14) ( u v + c uv) dx + c uv dγ = bv dx + p R v dγ (5.1.16) (5.1.16) 5.1.3 15 / 113
Poisson Poisson (5.1.16) u v v a : U U R l : U R a (u, v) = ( u v + c uv) dx + c uv dγ, (5.1.17) l (v) = bv dx + p R v dγ (5.1.18) 5.1.3 5.1.4 ( Poisson ) U (5.1.13) b L 2 (; R), c L (; R), p R L 2 ( ; R), c L ( ; R) a (, ) l ( ) (5.1.17) (5.1.18) v U a (u, v) = l (v) (5.1.19) u U 16 / 113
5.2 5.1 5.1.1 Poisson Poisson (5.1.10) (5.1.19) 2 ( A.7.1) 1 a 1 l U Hilbert U 1 (4.4 ) 17 / 113
5.2.1 ( Hilbert 1 ) a : U U R U 1 v U α > 0 a (v, v) α v 2 U a 18 / 113
U R d 1 x, y R d a (x, y) = x (Ay) A R d d A = A T a A 5.2.2 ( Hilbert 1 ) a : U U R U 1 u, v U β > 0 a (u, v) β u U v U a U = R d 1 a (x, y) = x (Ay) A ( (4.4.3) ) 19 / 113
5.2.3 ( ) a : U U R U 1 l = l ( ) = l, U v U a (u, v) = l (v) u U U = R d 1 a (x, y) = x (Ay) A R d d b R d y R d (Ax) y = b y (5.2.1) x R d 20 / 113
Lax-Milgram 5.2.1 Lax-Milgram 5.2.3 Lax-Milgram 1 a Hilbert Riesz ( 4.4.17) ( [2] p.29 Theorem1.3, [3] p. 297 Theorem 1, [9] p.48 2.6) 5.2.4 (Lax-Milgram ) 5.2.3 a l U 5.2.3 u U 5.2.1 α u U 1 α l U 21 / 113
Lax-Milgram U = R d (5.2.1) x A A x = A 1 b (5.2.2) x R d 1 α b R d α A A A ( A T + A ) /2 x R d x (Ax) α x 2 R d x {( A T + A ) x } 2α x 2 R d Poisson Lax-Milgram 22 / 113
Lax-Milgram 5.2.5 (Poisson ) 5.1.2 Γ D (= Γ D dγ) ũ = u u D U 5.1.2 Lax-Milgram U = { u H 1 (; R) } u = 0 on ΓD Hilbert ˆl (v) = l (v) a (ud, v) (5.2.3) 5.1.2 v U a (ũ, v) = ˆl (v) ũ = u u D U Lax-Milgram 23 / 113
Lax-Milgram 1 a Poincaré ( A.9.4) a (v, v) = v v dx = v 2 L 2 (;R d ) 1 c 2 v 2 H 1 (;R) 1/c 2 α 5.2.1 a 2 a Hölder ( A.9.1) a (u, v) = u v dx u L 2 (;R d ) v L 2 (;R d ) u H 1 (;R) v H 1 (;R) 5.2.2 β = 1 24 / 113
Lax-Milgram 3 ˆl U Lipschitz ( 4.4.2) γ L(H 1 (;R);H 1/2 ( ;R)) = sup v H 1/2 ( ;R) v H 1 (;R)\{0 H 1 (;R) } v H 1 (;R) (5.2.4) c 1 > 0 Hölder ˆl (v) bv dx + p Nv dγ + u D v dx Γ N b L 2 (;R) v L 2 (;R) + pn L 2 (Γ N ;R) v L 2 (Γ N ;R) + u D L2 (;R d ) v L 2 (;R d ) ( ) b L 2 (;R) + c1 pn L 2 (Γ N ;R) + ud H 1 (;R) v H 1 (;R) 5.1.2 b L 2 (; R), p N L 2 (Γ N; R) u D H 1 (; R) ( ) l U ũ = u u D U 25 / 113
Lax-Milgram 5.2.6 (Neumann ) 5.1.2 Γ D = 0 5.1.2 u U 5.2.5 a Γ D > 0 Poincaré ( A.9.4) Neumann Γ D = 0 Poincaré a Lax Milgram u D = 1 u dx (5.2.5) Poincaré ( A.9.3) a (v, v) = v v dx = v 2 L 2 (;R d ) 1 c v 2 ud 2 L 2 (;R d ) 26 / 113
Lax-Milgram a Neumann (5.2.5) u D ũ = u u D U 27 / 113
Lax-Milgram Poisson ( 5.1.3) 5.2.7 ( Poisson ) 5.1.4 1 c L (; R) 2 c L ( ; R) 5.1.4 u U 5.1.4 Lax-Milgram U = H 1 (; R) Hilbert 28 / 113
Lax-Milgram 1 a ess inf x c (x) ess inf x c (x) c 1 > 0 c 2 > 0 γ : H 1 (; R) L 2 ( ; R) γ 1 L(L 2 ( ;R);H 1 (;R)) = sup v H 1 (;R) v L 2 ( ;R)\{0 L 2 ( ;R) } v L 2 ( ;R) (5.2.6) c 3 > 0 a (v, v) v 2 L 2 (;R d ) + c1 v 2 L 2 (;R) + c2 v 2 L 2 ( ;R) ( ) min {1, c 2} + c2 v 2 c 2 H 1 (;R) 3 ( ) α 5.2.1 a 29 / 113
Lax-Milgram 2 a (5.2.4) γ L(H 1 (;R);H 1/2 ( ;R)) c4 c L (; R) c L ( ; R) a (u, v) u L2 (;R d ) v L 2 (;R d ) + c L (;R) u L 2 (;R) v L 2 (;R) + c L ( ;R) u L 2 ( ;R) v L 2 ( ;R) ( ) 1 + c L (;R) + c2 4 c L ( ;R) u H1 (;R d ) v H 1 (;R d ) ( ) β 5.2.2 a 30 / 113
Lax-Milgram 3 l U (5.2.4) γ L(H 1 (;R);H 1/2 ( ;R)) c4 l (v) bv dx + p Rv dγ b L 2 (;R) v L 2 (;R) + pr L 2 ( ;R) v L 2 ( ;R) ( ) b L 2 (;R) + c4 pr L 2 ( ;R) v H 1 (;R) 5.1.4 b L 2 (; R) p R L 2 ( ; R) ( ) l U Lax-Milgram u U 31 / 113
5.2.2 5.2.8 ( ) a : U U R U 1 l = l ( ) = l, U f : U R {f (u) = 12 } a (u, u) l (u) min u U u U U = R d {f (x) = 12 } x (Ax) b x min x R d (5.2.7) x R d 32 / 113
5.2.8 ( [2] p.24 Theorem1.1, [6] p.33 2.1, [9] p.50 2.7) 5.2.9 ( ) 5.2.8 a l U 5.2.8 u U 5.2.3 33 / 113
U = R d A (5.2.7) x R d (5.2.2) Poisson ( 5.1.2) a 5.2.5 5.2.9 5.1.2 5.2.10 (Poisson ) a ˆl (5.1.8) (5.2.3) { f (ũ) = 1 } 2 a (ũ, ũ) ˆl (ũ). min ũ U ũ = u u D U 34 / 113
5.3 Poisson ( 5.1.1) b, p, u D (5.2.3) ˆl U Poisson u u D U = { u H 1 (; R) u = 0 on Γ D } Poisson 8 9 H 1 ( ) C 1 35 / 113
5.3.1 Poisson ( 5.1.1) b, p, u D u = b in, ν u = p N on Γ N, u = u D on Γ D b L 2 (; R), p N H 1 (; R), u D H 2 (; R) Dirichlet Neumann Γ N Γ D B u H 2 ( \ B; R ) b L 2 (; R) Poisson u H 2 ( \ B; R ) 36 / 113
ν C (Γ N ; R) p N H 1 (; R) p N H 1/2 (Γ N ; R) ν u = ν u H 1/2 ( Γ N \ B; R ) u H 2 ( \ B; R ) Sobolev ( 4.3.14) d {2, 3} α (0, 1/2) H 2 (; R) C 0,α ( ; R ) u u u 37 / 113
5.3.2 x 0 Θ α θ Γ1 Γ2 B(x 0,r 0 ) 5.3.1: 2 38 / 113
x 0 u ( ) r (0, r 0 ] u u (r, θ) = i {1,2, } k i u i (r) τ i (θ) + u R (5.3.1) ( [7], [8] p.273 8, [10], [5] p.ix Preface p.182 Chapter 4) u R (5.3.1) 1 k i u i (r) r τ i (θ) θ (0, α) Γ 1 Γ 2 Dirichlet (u = 0) Neumann ( ν u = 0) i {1, 2, } τ i (θ) = sin iπ α θ, τ i (θ) = cos iπ α θ (5.3.2) (5.3.3) 39 / 113
(5.3.2) τ i (0) = τ i (α) = 0 (5.3.3) (dτ i /dθ) (0) = (dτ i /dθ) (α) = 0 Γ 1 Γ 2 Dirichlet Neumann i {1, 2, } τ i (θ) = sin iπ 2α θ (5.3.4) 40 / 113
Laplace ( (r ω 2 sin ωθ) = r 2 + 1 r r + 1 2 r 2 θ 2 ) (r ω sin ωθ) = 0 (5.3.5) ω ω > 1/4 1 ω > 1/4 Γ 1 Γ 2 (α 2π) ω 1/4 ω = 1 (5.3.5) z = x 1 + ix 2 = re iθ C (i ) x = (x 1, x 2 ) R 2 f (z) = z ω u i = Im [z ω ] = r m sin ωθ Cauchy-Riemann (5.3.5) r ω sin ωθ Laplace ( Poisson ) τ i (θ) sin ωθ u i (r) = r ω Laplace x 0 r 0 B (x 0, r 0 ) 41 / 113
1 Γ 1 Γ 2 Dirichlet (u = 0) u (r, θ) = kr π/α sin π α θ + u R (5.3.6) 2 Γ 1 Γ 2 Neumann ( ν u = 0) u (r, θ) = kr π/α cos π α θ + u R (5.3.7) 3 Γ 1 Dirichlet Γ 2 Neumann u (r, θ) = kr π/(2α) sin π 2α θ + u R (5.3.8) k α 42 / 113
5.3.1 ( ) 2 x 0 α (0, 2π) u x 0 B (x 0, r 0 ) u = r ω τ (θ) τ (θ) C ((0, α), R) k {0, 1, 2, } p (1, ) ω > k 2 p (5.3.9) u W k,p (B (x 0, r 0 ) ; R) 43 / 113
u = r ω τ (θ) k r ω k τ (θ) τ (θ) C ((0, α), R) u k B (x 0, r 0 ) p Lebesgue r0 α 0 0 r p(ω k) r τ (θ) dθdr < p (ω k) + 1 > 1 (5.3.9) 44 / 113
Poisson u r ω 5.3.1 5.3.2 5.3.2 ( ) 2 x 0 Θ α (0, 2π) Poisson ( 5.1.1) u x 0 u H s (B (x 0, r 0 ) ; R) 1 x 0 Γ 1 Γ 2 α [π, 2π) s (3/2, 2] 2 Γ 1 Γ 2 α [π/2, π) s (3/2, 2] α [π, 2π) s (5/4, 3/2] 45 / 113
Γ 1 Γ 2 (5.3.6) (5.3.7) ω = π/α α [π, 2π) ω (1/2, 1] (5.3.9) s 1 2 p = 3 2 2 2 = 1 2 < ω s 2 2 p = 2 2 2 = 1 ω (1/2, 1] s (1) Γ 1 Γ 2 (5.3.8) ω = π/ (2α) α [π/2, π) ω (1/2, 1] (5.3.9) s (2) α [π, 2π) ω (1/4, 1/2] (5.3.9) s 1 2 p = 5 4 2 2 = 1 4 < ω s 2 2 p = 3 2 2 2 = 1 2 ω (1/4, 1/2] s (2) 46 / 113
B(x 0,r 0 ) Γ N B(x 0,r 0 ) Γ D (a) α > π (b) α > π/2 5.3.2: 2 47 / 113
5.3.2 (α = 2π) x 0 ϵ > 0 u H 3/2 ϵ (B (x 0, r 0 ) ; R) (5.3.10) x 0 x 0 (α = π) (5.3.10) u W 1, 5.3.3 ( ) 2 x 0 Θ α (0, 2π) x 0 1 Γ 1 Γ 2 α < π 2 Γ 1 Γ 2 α < π/2 Poisson ( 5.1.1) u W 1, (B (x 0, r 0 ) ; R2) 48 / 113
Γ 1 Γ 2 (5.3.6) (5.3.7) ω = π/α α < π ω > 1 (5.3.9) (1) Γ 1 Γ 2 (5.3.8) ω = π/ (2α) α < π/2 ω > 1 (5.3.9) (2) 5.3.2 (2) Γ 1 Γ 2 5.1.3 Poisson c : R Dirichlet Neumann 49 / 113
5.4 Stokes p N Γ p u D Γ D b 5.4.1: 50 / 113
R d d {2, 3} Lipschitz Γ D (Dirichlet ) Γ N = \ Γ D (Neumann ) Γ p Γ N Γ p Γ N \ Γ p 9 5.4.1 2 5.2.6 Γ D > 0 b : R d p N : Γ N R d u D : R d u : R d 51 / 113
5.4.1 u u 2 + 2 x dx 2 2 u u 1 + 1 x dx 2 2 dx 2 u u 2 u dx 2 + 2 dx 1 1 x 1 u 1 u u 1 + 1 x dx 1 1 5.4.2: 2 u ( u T) T 52 / 113
1 1 d {2, 3} 1 u (0, l) du/dx d {2, 3} u d ( u T) T = ( ui / x j ) ij R d d 2 ( ) 5.4.2 u ( u T) T ( u T ) T = E (u) + R (u) (5.4.1) E (u) = E T (u) = (ε ij (u)) ij = 1 2 { u T + ( u T) T } (5.4.2) R (u) = R T (u) = (r ij (u)) ij = 1 2 { ( u T ) T u T } (5.4.3) 53 / 113
E (u) 2 5.4.3 (a) (c) d R (u) 2 5.4.3 (d) d ε 22 ε 12 r 12 1 ε 11 1 1 1 (a) ε 11 (b) ε 22 (c) ε 12 = ε 21 (d) r 12 5.4.3: 2 E (u) R (u) 54 / 113
(5.4.2) (5.4.3) u 0 R d ( ) u u 2 Almansi Green 55 / 113
Cauchy 5.4.2 Cauchy d = 2 5.4.4 (b) 3 d = 3 5.4.5 3 ν (d = 2 d = 3 ) p R d p i, j {1,, d} σ ij x i x j S = (σ ij ) R d d Cauchy 56 / 113
Cauchy x2 u D Γ D dγ p N Γ p x1 b (a) dγ σ11 σ12 dγ1 dγ σ21 dγ2 σ22 ν p (b) Cauchy df =pdγ =S(u)νdγ 5.4.4: 2 Cauchy S p 57 / 113
Cauchy x2 ν df=pdγ=sνdγ σ11 dγ σ33 p x3 σ22 x1 5.4.5: 3 Cauchy S p 58 / 113
Cauchy Cauchy S p 5.4.1 (Cauchy ) p S Cauchy S T ν = Sν = p (5.4.4) d = 2 i {1, 2} x i σ 1i dγ 1 + σ 2i dγ 2 = p i dγ ( 5.4.4 (b)) ν 1 = dγ 1 /dγ ν 2 = dγ 2 /dγ σ 1i ν 1 + σ 2i ν 2 = p i (5.4.5) 59 / 113
Cauchy (5.4.5) (5.4.4) σ 21 = σ 12 ( 5.4.6) d = 3 σ22 σ21 σ11 σ12 (0,ε) 2 σ12 σ11 σ21 σ22 5.4.6: 2 (ϵ 1) 60 / 113
5.4.3 1 1 d S (u) = S T (u) = (σ ij (u)) ij = CE (u) = c ijkl ε kl (u) (k,l) {1,,d} 2 ij (5.4.6) C = (c ijkl ) ijkl : R d d d d 4 S (u) E (u) c ijkl = c jikl, c ijkl = c ijlk (5.4.7) 61 / 113
C L A = (a ij ) ij R d d B = (b ij ) ij R d d A (CA) α A 2, A (CB) β A B (5.4.8) (5.4.9) α β A B = i,j{1,,d} a ijb ij (5.4.8) C (5.4.9) C C u ( ) (5.4.6) Hooke C u 62 / 113
C d = 3 1 C 3 4 = 81 2 (5.4.7) 36 3 w w = 1 w E (u) (CE (u)), S (u) = 2 E (u) 2 c ijkl = c klij (5.4.10) 21 4 9 5 2 63 / 113
λ L µ L S (u) = 2µ L E (u) + λ L tr (E (u)) I λ L µ L Lamé tr (E (u)) = i {1,,d} e ii (u) µ L e Y ν P E (u) = 1 + ν P S (u) ν P tr (S (u)) I e Y e Y e Y ν P (Young ) Poisson k b k b = λ L + 2µ L 3, e Y = 2µ L (1 + ν P ), λ L = 2µ Lν P 1 2ν P 64 / 113
5.4.4 Cauchy (5.4.6) Hooke σ11 σ σ 22 22 + ε σ σ 21 x2 21 + ε x 2 b σ 12 2 σ 12 + ε x 1 b 1 σ σ 11 11 + σ12 (0,ε) 2 ε σ21 σ22 x 1 5.4.7: (ϵ 1) 65 / 113
2 4 5.4.7 x 1 x 2 σ 11 + σ 21 + b 1 = 0, x 1 x 2 σ 12 + σ 22 + b 2 = 0 x 1 x 2 d {2, 3} T S (u) = b T [ (5.4.11) T S (u) = C (5.4.11) { ( 1 2 u T + ( u T) )}] T u 2 C (5.4.11) 66 / 113
(5.4.11) 5.4.2 ( ) b : R d, p N : Γ N R d u D : R d T S (u) = b T in, S (u) ν = p N on Γ N, u = u D on Γ D (5.4.12) (5.4.13) (5.4.14) u : R d 67 / 113
5.4.4 5.4.2 u U = { v H 1 ( ; R d) v = 0R d on Γ D } (5.4.15) (5.4.12) v U Gauss-Green (A.8.2) ( T S (u) ) v dx = (S (u) ν) v dγ + S (u) E (v) dx Γ N = b v dx (5.4.16) (5.4.13) v U Γ N 68 / 113
(S (u) ν) v dγ = Γ N p N v dγ Γ N (5.4.17) (5.4.16) 2 1 (5.4.17) S (u) E (v) dx = b v dx + p N v dγ Γ N v U a(u, v) = S (u) E (v) dx, (5.4.18) l (v) = b v dx + p N v dγ (5.4.19) Γ N 69 / 113
5.4.3 ( ) U (5.4.15) b L 2 ( ; R d), p N L 2 ( Γ N ; R d), u D H 1 ( ; R d) C L ( ; R d d d d) a (, ) l ( ) (5.4.18) (5.4.19) v U a (u, v) = l (v) ũ = u u D U 70 / 113
5.4.6 v l (v) a (u, v) 5.4.4 ( ) 5.4.3 Γ D > 0 ũ = u u D U 71 / 113
Lax-Milgram U Hilbert v U ˆl (v) = l (v) a (ud, v) 5.4.3 a (ũ, v) = ˆl (v) ũ = u u D U Lax-Milgram 72 / 113
1 a Γ D > 0 Korn 2 ( A.9.6) c v 2 H 1 (;R d ) c E (v) 2 L 2 (;R d d ) (5.4.8) C v U a (v, v) = E (v) (CE (v)) dx c 1 E (v) 2 L 2 (;R d d ) c1 c v 2 H 1 (;R d ) c 1 (5.4.8) α c 1/c α 5.2.1 a 2 a (5.4.9) β β 5.2.2 a 73 / 113
3 ˆl U Lipschitz ( 4.4.2) γ L(H 1 (;R d );H 1/2 ( ;R d )) c2 > 0 Hölder ˆl (v) b v dx + p N v dγ + Γ N β E (u D) E (v) dx b L2 (;R d ) v L 2 (;R d ) + pn L 2 (Γ N ;R d ) v L 2 (Γ N ;R d ) + β E (u D) L2 (;R d d ) E (v) L 2 (;R d d ) ( b L2 (;R d ) + c2 pn L 2 (Γ N ;R d ) ) + β E (u D) L2 (;R d d ) v H1 (;R d ) Lax-Milgram 5.4.3 ũ = u u D U 74 / 113
Stokes 5.5 Stokes Stokes Stokes u D b 5.5.1: Stokes 75 / 113
Stokes R d d {2, 3} Lipschitz b : R d Dirichlet u D : R d u D = 0 in (5.5.1) µ 5.5.1 2 Stokes Stokes u : R d p : R (ν ) u = ( u T) T ν ν u 76 / 113
Stokes 5.5.1 (Stokes ) b : R d, u D : R d µ R T ( µ u T) + T p = b T in, (5.5.2) u = 0 in, u = u D on, p dx = 0 (u, p) : R d+1 (5.5.3) (5.5.4) (5.5.5) 5.5.1 (5.5.2) Stokes (5.5.3) Newton (5.5.2) T ( µ u T pi ) = b T in (5.5.6) 77 / 113
Stokes I d (5.4.2) E (u) Cauchy S (u, p) = pi + 2µE (u) (5.5.7) (5.5.2) T S (u, p) = b T in, (5.5.8) (5.5.3) 5.6 (5.5.2) 78 / 113
Stokes 5.5.1 u U = { u H 1 ( ; R d) u = 0R d on } (5.5.9) (5.5.2) v U Gauss-Green (A.8.2) { T ( µ u T) T p + b T} v dx = (µ ν u pν) v dγ { ( + µ u T ) ( v T) + p v + b v } dx { ( = µ u T ) ( v T) + p v + b v } dx = 0 79 / 113
Stokes v U Stokes p { } P = q L 2 (; R) q dx = 0 (5.5.10) (5.5.3) q P q u dx = 0 q P Stokes a (u, v) = µ ( u T) ( v T) dx, (5.5.11) 80 / 113
Stokes b (v, q) = q v dx, l (v) = b v dx (5.5.12) (5.5.13) Stokes 5.5.2 (Stokes ) U P (5.5.9) (5.5.10) u D H 1 ( ; R d) (5.5.1) µ a (, ), b (, ) l ( ) (5.5.11), (5.5.12) (5.5.13) (v, q) U P a (u, v) + b (v, p) = l (v), b (u, q) = 0 (5.5.14) (5.5.15) (ũ, p) = (u u D, p) U P 81 / 113
5.6 Stokes u Stokes u p Stokes 82 / 113
U P Hilbert a : U U R b : U P R U U U P (4.4.4 ) a = a L(U,U;R) = b = b L(U,P ;R) = a (u, v) sup, u,v U\{0 U } u U v U sup u U\{0 U }, q P \{0 P } b (u, q) u U q P 83 / 113
Hilbert 5.6.1 ( Hilbert U div ) b : U P R 1 U div = {v U b (v, q) = 0 for all q P } U Hilbert 84 / 113
5.6.2 ( ) a : U U R b : U P R l U r P (v, q) U P a (u, v) + b (v, p) = l, v, b (u, q) = r, q (u, p) U P 85 / 113
5.6.1 5.6.2 ( [4] p.61 Corollary 4.1, [1] p.42 Theorem 1.1, [6] p.135 7.3, [9] p.116 4.3) 86 / 113
5.6.3 ( ) a : U U R U div ( v U div α > 0 a (v, v) α v 2 U ) b : U P R β > 0 inf sup q P \{0 P } v U\{0 U } b (v, q) v U q P β (5.6.1) 87 / 113
5.6.3 ( ) 5.6.2 (u, p) U P α, β, a b c > 0 u U + p P c ( l U + r P ) (5.6.1) Ladysenskaja-Babuška-Brezzi Babuška-Brezzi-Kikuchi 88 / 113
5.6.3 Stokes 5.6.4 (Stokes ) 5.5.2 ũ = u u D U div ũ = 0 R d (5.5.14) (5.5.15) (ũ, p) U P 5.6.3 U P Hilbert 5.5.2 (v, q) U P a (ũ, v) + b (v, p) = ˆl (v), b (ũ, q) = ˆr (q), (ũ, p) U P ˆl (v) = l (v) a (ud, v), ˆr (q) = b (u D, q) a U div 5.4.4 a U b 89 / 113
U div U div div U div τ τ ( divv / v U < ) (v 1, v 2 U div τv 1 = τv 2 v 1 = v 2) v U div τv = divv = 0 v U div v U div U div = {0 U } τ U div P ( [4] ) inf sup q P \{0 P } v U\{0 U } inf q P \{0 P } v U\{0 U } b (v, q) v U q P = inf sup 1 > 0 τ 1 L(P ;U div) sup q P \{0 P } v U\{0 U } ( τv, q) P τ 1 ( q) U q P inf q P \{0 P } ( divv, q) L 2 (;R) v U q P (q, q) P τ 1 ( q) U q P ˆl U 5.4.4 (5.5.1) ˆr (q) = 0 P 5.6.3 5.5.2 (u u D, p) U P 90 / 113
5.6.2 ( 5.6.2) a : U U R 5.6.5 ( ) a : U U R b : U P R (l, r) U P L (v, q) = 1 a (v, v) + b (v, q) l, v r, q 2 (v, q) U P L (u, q) L (u, p) L (v, p) (u, p) U P 91 / 113
5.6.5 ( [4] p.62 Theorem 4.2, [9] p.118 4.4) 5.6.6 ( ) a (a (u, v) = a (v, u)) ( v U a (v, v) 0 ) 5.6.2 5.6.5 ( 5.6.5) q P Lagrange 5.6.5 f (v) = 1 a (v, v) l, v 2 min {f (v) b (v, q) r, q = 0} (5.6.2) (v,q) U P 92 / 113
(v, q) 5.6.5 L (v, q) Lagrange q P Lagrange (5.6.2) 5.6.5 5.6.6 ( 2.9.2) 93 / 113
5 5.7 5 5 1 (Poisson ) Lax-Milgram (5.1, 5.2 ) 2 (5.3 ) 3 Lax-Milgram (5.4 ) 4 Stokes Stokes (5.5 ) 94 / 113
5 5.8 5 5.1 b : R, u D : R u + u = b in, u = u D on u : R u b u D 5.2 5.8.1 x A x B 95 / 113
5 5.8 5 (cnt.) x A x B Γ D u D=(0, 0) T Γ p p N=(0, 1) T b=(0, 0) T 5.8.1: 96 / 113
5 5.8 5 (cnt.) 5.3 b : (0, t T ) R d, p N : Γ N (0, t T ) R d, u D : (0, t T ) R d, u D0 : R d, u DT : R d ρ > 0 ρü T S (u) = b T in (0, t T ), S (u) ν = p N on Γ N (0, t T ), u = u D on Γ D (0, t T ), u = u D0 in {0}, u = u DT in {t T } u : (0, t T ) R d t (0, t T ) u = u/ t 97 / 113
5 5.8 5 (cnt.) 5.4 5.3 b = 0 R d, p N = 0 R d u D = 0 R d (x, t) (0, t T ) ( ) u (x, t) = ϕ (x) e λt ϕ : R d λ R 5.5 b : (0, t T ) R d, u D : (0, t T ) R d, µ > 0 ρ > 0 ρ u + ρ (u ) u µ u + p = b in (0, t T ), u = 0 in (0, t T ), u = u D on { (0, t T )} { {0}} (u, p) : (0, t T ) R d R Navier-Stokes 1 Navier-Stokes 2 98 / 113
5 5.8 5 (cnt.) 5.6 e Y µ L Poisson ν P e Y = 2µ L (1 + ν P ) 99 / 113
5 5.1 Dirichlet U = H0 1 (; R) v U ( u + u) v dx = ( u + u) v dx = v u ν dγ + ( u v + uv) dx = ( u v + uv) dx = bv dx v U a (u, v) = l (v) 100 / 113
5 (cnt.) ũ = u u D U a (u, v) = ( u v + uv) dx, l (v) = bv dx Lax-Milgram U = H 1 0 (; R) Hilbert v H 1 0 (; R) a (v, v) = v 2 H 1 (;R) a ˆl U ˆl ˆl (v) bv dx + ( u D v + u D v ) dx 101 / 113
5 (cnt.) b L 2 (;R) v L 2 (;R) + u D L 2 (;R d ) v L 2 (;R d ) + u D L 2 (;R) v L 2 (;R) ( ) b L2 (;R) + u D H1 (;R) v H1 (;R) b L 2 (; R) u D H 1 (; R) 5.2 x A Dirichlet Neumann α = π/2 5.3.2 (2) x A B A u H 2 ( B A ; R 2) x A x B Neumann Neumann α π/2 5.3.2 (1) p N x B B B (0, 0) T (0, 1) T Γ p p N L ( B B ; R 2) u C 0,1 ( B B ; R 2) H 2 ( B B ; R 2) 102 / 113
5 (cnt.) 5.3 U = { u H 1 ( (0, t T ) ; H 1 ( ; R d)) u = 0R d on Γ D (0, t T ), u = 0 R d on (0, t T ) } u D0, u DT H 1 ( ; R d) u D H 1 ( (0, t T ) ; H 1 ( ; R d)) b L 2 ( (0, t T ) ; L 2 ( ; R d)), p N L 2 ( (0, t T ) ; L 2 ( Γ N ; R d)) 1 v U (0, t T ) v U tt 0 (b ( u, v) a (u, v) + l (v)) dt = 0 103 / 113
5 (cnt.) ũ = u u D U b (u, v) = ρu v dx, a (u, v) = S(u) E(v) dx, l (v) = b v dx + p N v dγ Γ N 104 / 113
5 (cnt.) 5.4 ϕ U = { ϕ H 1 ( ; R d) ϕ = 0R d on Γ D } ϕ U u (x, t) = ϕ (x) e λt ρü T T S (u) = 0 T R v U d u = u D on Γ D (0, t T ) v U λ 2 b (ϕ, v) + a (ϕ, v) = 0 (ϕ, λ) U R ( ) U ( ) a (, ) (0 ) b (, ) ( ) U 105 / 113
5 (cnt.) (ϕ i, λ i ) i N λ 2 i 0 λ i = ±iω i (i ) ϕ i (x) ( e iωit + e iωit) = ϕ i cos ω i t ω i ϕ i 5.5 u p U = { u H 1 ( (0, t T ) ; H 1 ( ; R d)) u = 0 R d on (0, t T ) {0} }, V = { u H 1 ( (0, t T ) ; H 1 ( ; R d)) u = 0 R d on (0, t T ) {t T } }, P = {p L 2 ( (0, t T ) ; L 2 (; R) ) } p dx = 0 106 / 113
5 (cnt.) v V Navier-Stokes (0, t T ) u = u D on (0, t T ) {0} Navier-Stokes q P (0, t T ) (v, q) U Q tt (b( u, v) + c(u)(u, v) + a(u, v) + d(v, p)) dt = 0 0 tt 0 d(u, q) dt = 0 (u u D, p) U Q a (u, v) = µ ( u T) ( v T) dx, tt l (v) dt, 107 / 113
5 (cnt.) b (u, v) = ρu v dx, c(u)(w, v) = ρ ((u ) w) v dx, d (v, q) = q v dx, l (v) = b v dx + p N v dγ Γ N 108 / 113
5 (cnt.) 5.6 P.5.1 (a) ε 11 = ε 22 = 1 + ν P e Y σ 0 (1) P.5.1 (b) π/4 ε 11 = γ 12/ 2 2 = γ 12 2 = ε 12 = σ 0 2µ L (2) (1) (2) e Y = 2µ L (1 + ν P ) 109 / 113
5 (cnt.) σ0 σ0 σ0 σ0 x 2 x 1 σ0 σ0 σ0 γ 12 /2 σ0 (a) 2 (b) π/4 P.5.1: 110 / 113
[1] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, New York; Tokyo, 1991. [2] P. G. Ciarlet. Finite Element Methods. Handbook of Numerical Analysis, P.G. Ciarlet, J.L. Lions, general editors. Elsevier, Amsterdam; Tokyo: North-Holl, 1991. [3] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, 1992. [4] V. Girault and P. A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin; Tokyo, 1986. 111 / 113
(cnt.) [5] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman Advanced Pub. Program, Boston, 1985. [6]. :.,, 1994. [7] R. S. Lehman. Developments at an analytic corner of solutions of elliptic partial differential equations. Journal of Applied Mathematics and Mechanics, Vol. 8, pp. 727 760, 1959. [8] G. Strang, G. J. Fix,,..,, 1976. 112 / 113
(cnt.) [9]..,, 2010. [10],,.. bit, Vol. 5, pp. 1035 1040, 1973. 113 / 113