RIMS Kôkyûroku Bessatsu B3 (22), Numerical verification methods for differential equations: Computer-assisted proofs based on infinite dimension

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(22), B3: 45-55 Issue Date 22-4 URL http://hdl.handle.

1 微分方程式の精度保証付き数値計算 : 逐次反復に基づく Title計算機援用証明 (Progress in Mathematics of Systems) Author(s) 渡部, 善隆 Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (22), B3: Issue Date 22-4 URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 RIMS Kôkyûroku Bessatsu B3 (22), Numerical verification methods for differential equations: Computer-assisted proofs based on infinite dimensional sequential iteration (Yoshitaka Watanabe) Research Institute for Information Technology, Kyushu University Newton Abstract: This paper proposes numerical verification methods of solutions for infinite dimensional functional equations based on residual form and sequential iteration. Some computer-assisted proofs for differential equations including nonlinear partial differential equations will be shown. Comparing with other verification procedures as typified by infinite dimensional Newton-type iteration, the proposed algorithm can be done at low computational cost although it needs that the formulated compact map is retractive in some neighborhood of the fixed-point to be verified. 2. ˆX,X,Y Hilbert : ˆX X Y ˆX X A ˆX Y f X Y Au = f(u) () u () A f Hilbert X, Y ( u, v ) X,(u, v ) Y u X = ( u, u ) X, u Y = ( u, u ) Y.2. Banach A f A. φ Y Aψ = φ ψ ˆX ˆX c 22 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

3 46 X A : Y X A2. A C p > A u X C p u Y, u Y, (2) A3. f : X Y X Y A Y ˆX A A ˆX X I ˆX X A A Y I ˆX ˆX X X : φ ψ ψ A Y X.2 () u h 2.2. u h X Au h Y () Aw = f(w + u h ) Au h (3) w = u u h (3) g(w) Aw = g(w) u h u (3).2.2 X u h Au h / Y X [5] X X h X h h X h X P h : X X h ( v P h v, v h ) X =, v h X h (4) h C(h) > ( C(h) h (I P h )v X C(h) Av Y, v ˆX (5) A : ˆX Y ( u, v ) X =(Au, v ) Y, u ˆX, v X (6) (6) () ( u, v ) X =(f(u),v) Y, v X (7) 2

4 47 u X (5), A Aubin-Nitsche [7, 8] (I P h )v Y C(h) (I P h )v X, v X (8) u h X h ( u h,v h ) X =(f(u h ),v h ) Y, v h X h (9) u h [, 4] f(u h ) Y A Aū = f(u h ) () ū X (6) (ū, v ) X =(f(u h ),v) Y, v X () () v X X h (9) (ū u h,v h ) X =, v h X h (2) (2) P h ū u h ū u h v := ū u h (3) (5), (8) v X C(h) f(u h ) Y, v Y C(h) 2 f(u h ) Y (4) w = u ū () Aw = f(w + u h + v ) f(u h ) (5) (5) g(w) Aw = g(w) (5) w X () u u = u h + v + w Au h Y A X h v a posteriori [8, 6, 7]..2.3 X A (6) (2) C p X Y : u Y C p u X, u X (6) Schwarz A u 2 X =(A u, A u ) X =(u, A u ) Y u Y A u Y C p u Y A u X 3

5 48 2 : IS-Res IS-Res * 2. Aw = g(w) (7) w X g : X Y f A3 A g F F := A g : X X (8) A () u = F (u) (9) u X A3 g A : Y X F X X F g A X Y X F : u g(u) A g(u) 2.2 F 2. W X α> W := {w X w X α} (2) C p sup g(w) Y α (2) w W F W : W X (2), (2) sup F (w) X C p sup g(w) Y α w W w W F (W ) W Schauder [9] 2.3 W * Infinite, Sequential, Residual 4

6 (2) W X C g > g(w ) g(w 2 ) Y C g w w 2 X, w,w 2 W (22) 2. (2) C p C g < (23) F W : W w,w 2 W (2), (22) F (w ) F (w 2 ) X = A (g(w ) g(w 2 )) X C p C g w w 2 X. (23) Banach [9] F g W 2. (2) W X g W Fréchet (22) C g > sup g [ŵ]w Y C g w X, w X (24) ŵ W : W [7, 4.4] g(w ) g(w 2 ) Y sup g [ŵ](w w 2 ) Y, w,w 2 W. (25) ŵ W w = w w 2 X (24) (22) g [ŵ] :X Y ŵ W (24) : sup g [ŵ](w w 2 ) Y C g w w 2 X, w,w 2 W (26) ŵ W 2.4 () A u = A f(u) u h (2) W U = u h + W A f(u) u h W Schauder (2) u U A f(u) u h = A (f(u) Au h ) Au h Y Au h / Y 5

7 5 2.5 IS-Res 2., 2.2 IS-Res epsiloninflation [2] k = α () >. k. ε> IS-Res ˆα (k) := ( + ε)α (k ). 2. k W (k). W (k) := {w X w X ˆα (k) }. 3. k α (k) := C p sup g(w) Y. w W (k) 4. α (k) ˆα (k). W (k) X W (k) C g C p C g < W (k) 2.2 Ŵ 5. k := k +.. k α (k). 2.6 IS-Res C p W (k) w g(w) Y C g Newton (C p ) [9] [4] IS-Res [8, 7., 7.2 ] 2.7 (23) C g 6

8 5 2.2 (2) W X F w ˆα α> Ŵ := {w X w X ˆα} (27) Ĉg > g(w ) g(w 2 ) Y Ĉg w w 2 X, w,w 2 Ŵ (28) C p Ĉ g < F Ŵ : ŵ w F ŵ Ŵ (2), (28) w ŵ X = F (w) F (ŵ) X = A (g(w) g(ŵ)) X C p Ĉ g w ŵ X w ŵ X C p Ĉ g C p Ĉ g < w 2.2 W W Ŵ 3 3 IS-Res MATLAB R2a INTLAB [3] version 6 Ω m L 2 (Ω) H m (Ω) Ω Ω H (Ω) := {u H (Ω) u = on Ω} (29) Ω [ u u =,..., u ] T x x n ( u, v ) L 2 H (Ω) ( u, v ) L 2 L K> u Ku = ( π K ) sin πx, π u() = u() =. <x<, (3) (3) u(x) = sin πx π Au = u Ω =(, ) ˆX = H 2 (Ω) H (Ω), X = H (Ω), Y = L 2 (Ω) 7

9 52 ( u, v ) X =(u,v ) L 2, ( u, v ) Y =(u, v ) L 2 IS-Res (, ) N h =/N X h (5) C(h) =h/π [7, 4.4] (2) (6) C p =/π [5, Theorem.2] u h X h Au h / Y X (9) u h X h (2) K <π 2 v L 2 (Ω) π K ( Kπ 2 ) α α> K π 2 IS-Res Newton [4] N K (2) W IS-Res 2 N K = π K = π 2 /2 K =99/π Ω=(, ) (, ) { Δu = λ( + u + u 2 au 3 ) in Ω, u = on Ω. (3) λ, a > (3) a λ [3] Ω u λ λ λ u X h x y N h =/N (5) C(h) =.493h [2] (2) (6) C p =/(π 2) (9) u h X h Krawczyk 2 8

10 53 2 W H (Ω) Ŵ H (Ω) λ 6 2 IS-Res N = 3, a =. λ u h v H (Ω) W H (Ω) Ŵ H (Ω) λ = λ =3 λ = u h =.78 u h =.275 u h = Newton [9] Ω=(,π/a) (,π) { P Δ 2 u = PRξ u z Δu x + u x Δu z, in Ω u =Δu =, on Ω. (32) P, R, a > ξ u (32) 2 Navier-Stokes 4 [] 2 Fourier X k := a mn sin(amx)sin(ny) a mn R, ((am) 2k + n 2k )a 2 mn < m= n= m= n= A = P Δ 2, ˆX = X 4, X = X 3, Y = X 9

11 渡部善隆 54 と定めるこのとき u X := Δu L2 (Ω), u Y := u L2(Ω) に対し式 (2) は Cp = /(P ( a2 + )) で成立し IS-Res の適用が可能となるここでは ξ を式 (32) と移流拡散方程式の連立系 [6] の近似解として定め Fourier 級数展開を x, y とも N で打ち切った空間から uh を定め直接的な引き戻し P Δ2 w w = J(w + uh, Δw) + J(w, Δuh ) P Δ2 uh + = Δw =, w = u uh, P R ξ + J(uh, Δuh ) in Ω on Ω, (33) J(u, v) := ux vz vx uz を適用した解の検証では Plum による最大値ノルム評価 [] u L (Ω) K u L2 (Ω) + K2 u L2 (Ω) + K3 Δu L2(Ω), u X, a2 + 9a4 π a3 3 を用いた図 3 は近似解の形状を示すその結果 R = 5, P =, a = / 2, N = 5 において W X の意 K = π a, 2 K2 = a2, 6a K3 = 図3 近似解の概形味での範囲に解を包み込むことに成功したなおより大きな R や移流拡散方程式との系となる Oberbeck-Boussinesq 方程式に適用するには IS-Res では限界があり Newton 法に基づく検証手法 [6] などが必要となる謝辞本研究は科学研究費補助金基盤研究 (S) 課題番号 2224 および基盤研究 (C) 課題番号の補助を受けている参考文献 [] P. Grisvard: Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 985. [2] F. Kikuchi, and L. Xuefeng: Determination of the Babuska-Aziz constant for the linear triangular ﬁnite element, Japan Journal of Industrial and Applied Mathematics, vol. 23, no., pp , February, 26. [3] P.L. Lions: On the existence of positive solutions of semilinear elliptic equations, SIAM Review, vol. 24, no. 4, pp , October, 982. [4] M.T. Nakao: A numerical approach to the proof of existence of solutions for elliptic problems II, Japan Journal of Applied Mathematics, vol. 7, no. 3, pp , October, 99.

12 55 [5] M.T. Nakao: Solving nonlinear elliptic problems with result verification using an H type residual iteration, in Validation Numerics: theory and applications, eds. R. Albrecht, G. Alefeld, and H. J. Stetter, pp. 6 73, Computing Supplementum, vol. 9, Springer-Verlag, Wien, 993. [6] M.T. Nakao, and N. Yamamoto: Numerical verification of solutions for nonlinear elliptic problems using L residual method, Journal of Mathematical Analysis and Applications, vol. 27, no., pp , January, 998. [7], :,, 998. [8], :, 2, SCG -85,, 2. [9] M.T. Nakao, and Y. Watanabe: Numerical verification methods for solutions of semilinear elliptic boundary value problems, Nonlinear Theory and Its Applications, IEICE, vol. 2, no., pp. 2 3, January, 2. [] :,, 2. [] M. Plum: Explicit H 2 -estimates and pointwise bounds for solutions of second-order elliptic boundary value problems, Journal of Mathematical Analysis and Applications, vol. 65, no., pp. 36 6, March, 992. [2] S.M. Rump: A note on epsilon-inflation, Reliable Computing, vol. 4, no. 4, pp ,November, 998. [3] S.M. Rump: INTLAB INTerval LABoratory, in Developments in Reliable Computing, ed. T. Csendes, pp. 77 4, Kluwer Academic Publishers, Dordrecht, [4] S.M. Rump: Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, vol. 9, pp , May, 2. [5] M.H. Schultz: Spline Analysis, Prentice-Hall, London, 973. [6] Y. Watanabe, N. Yamamoto, M.T. Nakao, and T. Nishida: A numerical verification of nontrivial solutions for the heat convection problem, Journal of Mathematical Fluid Mechanics, vol. 6, no., pp. 2, March, 24. [7] N. Yamamoto, and M.T. Nakao: Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains, Numerische Mathematik, vol. 65, no., pp , December, 993. [8] N. Yamamoto, and M.T. Nakao: Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element, Journal of Computational and Applied Mathematics, vol. 6, no. -2, pp , June, 995. [9] E. Zeidler(Translated by P.R. Wadsack): Nonlinear Functional Analysis and its Applications, Part I: Fixed-Point Theorems, Springer-Verlag, 986.

第5章偏微分方程式の境界値問題

第5章偏微分方程式の境界値問題 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