A sharper error estimate of verified computations for nonlinear heat equations.

Transcription

1 A sharper error estimate of verified computations for nonlinear heat equations. Makoto Mizuguchi (Waseda University) Akitoshi Takayasu Takayuki Kubo Shin ichi Oishi Waseda University (University of Tsukuba) Waseda University/ CREST JST 16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics 214 Sep /3

2 Contents 1. Introduction and remark 2. Sharper estimation (Main theme) 3. Numerical example 2/3

3 1. Introduction and remark 3/3

4 Problem Let Ω be a bounded and convex polygonal domain. We consider existence and uniqueness of the solution in the following Dirichlet type heat equations: t u u = f(u) in (, ) Ω, u Ω = on (, ) Ω, u(, x) = u (x) in Ω. (1) is represented Laplace operator. u H 1 (Ω) H 2 (Ω). 4/3

5 Assumption f : H 1 (Ω) L 2 (Ω) is a twice Fréchet differentiable nonlinear mapping There exists a non decreasing function L ρ > such that f(u) f(v) L 2 (Ω) L ρ ρ u, v U ρ, where for given ρ >, let U ρ := {w H 1 (Ω) : w H 1 ρ}. 5/3

6 Remark My abstruct is written in Scan214, Book of Abstructs, p.119. We will talk shaper estimation of contents of the abstruct. 6/3

7 Essential estimate to obtain absolute error According to the previous talk, to get rigorous estimate absolute error L (T 1 ;H 1 (Ω)), we need to obtain sharp estimates: δ := t t e (t s)a ( s ω(s) + Aω(s) f(ω))ds L (T 1 ;H 1(Ω) ), ε := u(t ) û H 1. We talk about a shaper estimations of δ by changing operator A and the semigroup e ta generated by A. 7/3

8 Difference form the previous talk Differential operator A (in this talk, A is represented ) A : H(Ω) 1 H 1 (Ω) (Weak solution) A : H(Ω) 1 H 2 (Ω) L 2 (Ω) (Strong solution) Semigroup e ta : H 1 (Ω) H 1 (Ω) e ta : L 2 (Ω) L 2 (Ω) 8/3

9 In the case of A (Weak solutions) For fixed time t, from t e ta = Ae ta holds, it sees that t e (t s)a ( s ω(s) + Aω(s) f(ω(s)))ds H 1 t = Ae (t s)a ( s ω(s) + Aω(s) f(ω(s)))ds H 1 t = s e (t s)a ( s ω(s) + Aω(s) f(ω(s)))ds H 1(2) (2) becomes a rough bound of δ. 9/3

10 In the case of A (Strong solutions) For fixed time t, it sees that t e (t s)a ( s ω(s) + Aω(s) f(ω(s))) H 1 ds t t A 1/2 e (t s)a ( s ω(s) + Aω(s) f(ω(s))) L 2ds e 1/2 (t s) 1/2 ( s ω(s) + Aω(s) f(ω(s))) L 2ds 2e 1/2 t 1/2 ( s ω + Aω f(ω)) L (T 1 ;L 2 )ds O( τ). (3) 1/3

11 2. Sharper estimation (Main theme) 11/3

12 Setting of approximate solutions Let V h be a finite dimensional subspace of D(A) and fix θ =, 1/2 or 1 1 and û,θ V h. We employ the following full discretization schemes to obtain u 1,θ V h such that for all v h V h, ( u h 1,θ û h,θ τ, v h )L 2 + ((1 θ)aû h,θ + θau h 1, v h ) L 2 = ((1 θ)f(û h ) + θf(u h 1), v h ) L 2. (4) We create ω,θ L ((t, t 1 ]; D(A)) by ω,θ (t) := û,θ φ (t) + û 1,θ φ 1 (t), t (t, t 1 ], where let û 1,θ V h be an approximation of u h 1. 1 Forward Euler (θ = ), Crank Nicolson (θ = 1/2), Backward Euler (θ = 1). 12/3

13 Main theme in this talk For ω,θ (θ =, 1 or 1/2), we give the following estimation δ defined by t e (t s)a ( s ω,θ (s) + Aω,θ (s) f(ω,θ (s)))ds t (5) By estimating δ, we consider that which scheme gives shaper estimation of absolute error u ω,θ L (T 1 ;H 1) in the case θ =, 1 and 1/2. L (T 1 ;H 1 (Ω) ). 13/3

14 Estimation of δ For given θ 1, C θ L 2 (Ω) is defined by C θ := û1,θ û,θ +(1 θ)aû,θ +θaû 1,θ (1 θ)f(û,θ ) θf(û 1,θ ). τ Let g(t) := f(û 1,θ )φ 1 (t) + f(û,θ )φ (t). We decompose t e (t s)a ( s ω,θ (s) + Aω,θ (s) f(ω,θ (s)))ds t L (T 1 ;H 1 (Ω) ) into t + t A 1 t 2 e (t s)a (f(ω,θ (s)) g(s))ds t A 1 2 e (t s)a (C 1 φ 1 (s) + C φ (s))ds L (J;L 2 (Ω)) L (T 1 ;L 2 (Ω)) (6).(7) 14/3

15 Estimation of δ Since û,θ and û 1,θ V h L (Ω), an upper bound of a classical linear interpolation is given for fixed x Ω: f(ω,θ )(t) g(t) τ 2 8 max d 2 f(ω(t)) t T 1 dt 2 = τ 2 ( ) 2 8 max t T 1 f dω,θ [ω(t)] dt = 1 8 max t T 1 f [ω(t)] (û1,θ û,θ ) 2, where we denote f [ω(t)] be the twice Fréchet derivative of f at ω(t) for fixed t T 1 15/3

16 Estimation of δ It follows f(ω,θ ) g(t) L 2 M 2 8 f [ω,θ ] L (T 1 ;L (Ω)) û 1,θ û,θ 2 H, 1 where M 2 is a computable constant such that u 2 L 2 M 2 u 2 H 1, for u H 1 (Ω). Let λ min be the minimum eigenvalue of and erf(x) := 2 x π e x2 dx. From the spectrum mapping theorem implies that t A 1 2 e (t s)a (f(ω,θ (s)) p(s)) L 2ds t is bounded by ( ) 2π λ min e erf λmin t f(ω,θ ) g 2 L (T 1 ;L 2 (Ω)). 16/3

17 Estimation of δ Therefore, we have t A 1 2 e (t s)a (f(ω,θ (s)) g(s))ds t ( ) 2π C p α 2 λmin τ erf, eλ min 2 where L (J;L 2 (Ω)) C p := M 2 8 f [ω,θ ] L (T 1 ;L (Ω)) and α := û 1,θ û,θ H 1. 17/3

18 Estimation of δ We consider the estimation of t 2 e (t s)a (C 1 φ 1 (s) + C φ (s))ds t A 1 L (T 1 ;L 2 (Ω)) For s T 1, since φ (s) + φ 1 (s) = 1, it sees that C 1 φ 1 (s) + C φ (s) = (C 1 C θ )φ 1 (s) + (C C θ )φ (s) + C θ = C θ + τ 1 (1 θ)(c 1 C ) (s t ) + τ 1 θ(c 1 C ) (t 1 s).. 18/3

19 Estimation of δ Therefore, an upper bound of t 2 e (t s)a (C 1 φ 1 (s) + C φ (s))ds t A 1 is given as follows: t 1 2 e (t s) λ min L (T 1 ;L 2 (Ω)) e 1 2 Cθ L 2 (t s) 2 ds t t +d (t s) 1 2 e (t s) λ min 2 ((1 θ)φ 1 (s) + θφ (s))ds, t where let d = e 1 2 C 1 C L 2. 19/3

20 Estimation of δ Let r = t s. Then, we have e 1 2 Cθ L 2 t t + 1 θ τ e C 1 C L 2 + θ τ e C 1 C L 2 = e 1 2 Cθ L 2 t t r 1 2 e r λ min 2 dr t t t t r 1 2 e r λ min 2 dr + (1 θ)φ 1(t) + θφ (t) τ C 1 C L 2 e + 2θ 1 τ e C 1 C L 2 t t r 1 2 e r λ min 2 (t t r)dr r 1 2 e r λ min 2 (t 1 t + r)dr t t r 1 2 e r λ min 2 dr. r 1 2 e r λ min 2 dr 2/3

21 Estimation of δ From t t r 1 2 e rλ min /2 dr = 2 t t λ 1 min e λ min(t t )/2 t t +λ 1 min r 1/2 e rλmin/2 dr, the upper bound is given ( ) 2θ 1 d 1 C θ L 2 + d 1 + ((1 θ)φ 1 (t) + θφ (t)) C 1 C L 2 τλ min 2(1 2θ) + τ C 1 C L 2 t t e λ min(t t )/2, eλ min ( ) 2π λ where d 1 = eλ min erf min (t t ). 2 21/3

22 Summation of estimation of δ Let I(t) : T 1 R I(t) : = d 1 C θ L 2 ( ) 2θ 1 +d 1 + ((1 θ)φ 1 (t) + θφ (t)) C 1 C L 2 τλ min 2(1 2θ) + τ C 1 C L 2 t t e λ min(t t )/2, eλ min ( ) 2π λ where d 1 = eλ min erf min (t t ). Let 2 C p := M 2 f [ω 8,θ ] L (T 1 ;L (Ω)) and α := û 1,θ û,θ H 1. δ is bounded as follows: ( ) 2π δ C p α 2 λmin τ erf eλ min 2 + max I(t). t T 1 22/3

23 3. Numerical example 23/3

24 Numerical example Let Ω be an unit square domain. We consider existence and uniqueness of the solution in the following heat equations: t u u = u 2 in (, ) Ω, u Ω = on (, ) Ω, (8) u(, x) = sin(πx) sin(πy) in Ω. to consider which scheme gives a shaper absolute error estimate (θ =, 1 and 1/2). 24/3

25 Setting for numerical computation Windows 7 Professional, Intel(R) Core(TM) i7 CPU GHz 16 GB Memory MATLAB 212a (toolbox INTLAB ver7 [1]) For N N, let a finite dimensional space V N D(A) by { } N V N = u h = a k,l sin(kπx) sin(lπy) : a k,l R. k,l=1 We set N = 7 and τ = 2 1 equally. [1] S.M. Rump. INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages Kluwer Academic Publishers, Dordrecht, /3

26 Verification of δ θ θ θ 26/3

27 Verification of u ω L (T k ;H 1 ) θ θ θ 27/3

28 Essential estimate to obtain absolute error To get rigorous estimate absolute error L (T 1 ;H 1 (Ω)), we need to obtain sharp estimates: δ := t t e (t s)a ( s ω (s) + Aω (s) f(ω (s)))ds L (T 1 ;H 1(Ω)), ε := u(t ) û H 1. 28/3

29 Verification of u(t k ) û k H 1 θ θ θ 29/3

30 Conclusion The scheme with θ = 1/2 gives a sharper estimate of δ. (Crank-Nicolson scheme ) The scheme with θ = 1 gives a sharper estimate of the absolute error: u ω L (T 1 ;H 1 (Ω)) (Backward Euler scheme) The reason is that the value of the absolute error is dependent not only δ but also ε The scheme with θ = 1 has an effective estimate in this numerical example. (Backward Euler scheme) 3/3