Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics Special Session 36 Reaction Diffusion Systems 8th AIMS Conference on Dynamical Systems, Differential Equations & Applications Dresden University of Technology, May 25 28, 2010 Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems Annegret Glitzky Mohrenstrasse Berlin Germany Tel th AIMS Conference

2 Contents Model equations Energy estimates for the non-discretized problem Discretization scheme Results for finite-dimensional problems Remarks and generalizations Formulation in space of piecewise constant functions Uniform exponential decay of the free energy for classes of discretizations References Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 2 (20)

3 Model equations X i species, i = 1,...,m ū i reference densities v i chemical potentials u i = ū i e v i densities reversible reactions α 1 X α m X m β 1 X β m X m, (α,β) R Z m + Z m + reaction rates k αβ (e v α e v β ), v = (v 1,...,v m ) generation rate of species X i R i := k αβ (e v α e v β )(β i α i ) (α,β) R flux density j i = u i µ i v i, i = 1,...,m, continuity equations u i t + j i = R i in R + Ω, ν j i = 0 on R + Γ, u i (0) = U i in Ω, i = 1,...,m. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 3 (20)

4 Weak formulation v = (v 1,...,v m ) V = H 1 (Ω;R m ), u = (u 1,...,u m ) V Operators A, E : V L (Ω,R m ) V { m Av, v V := Ω ū i e v i µ i v i v i + i=1 m Ev, v V := ū i e vi v i dx, v V Ω i=1 (α, β) R k αβ (e α v e β v) } (α β) v dx Problem (P) u (t) + Av(t) = 0, u(t) = Ev(t) f.a.a. t R +, u(0) = U, u H 1 loc (R +;V ), v L 2 loc (R +;V ) L loc (R +;L (Ω) m ). } (P) Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 4 (20)

5 Energy functionals Dissipation rate m D(v) = ū i e v i µ i v i v i dx Ω i=1 + k αβ (e v α e v β )(α β) vdx 0 v V L (Ω,R m ) Ω (α,β) R Free energy F(u) = Ω m i=1 ( ui (ln u i ū i 1) + ū i ) dx Stoichiometric subspace S = span { α β : (α,β) R } Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 5 (20)

6 Results for the continuous problem (G./Hünlich 97, G./Gärtner 09) Invariants: (u, v) solution to (P) = { } u(t) U U := u V : ( u 1,1,..., u m,1 ) S t > 0. Thermodynamic equilibrium: There exists a unique solution (u,v ) to Av = 0, u = Ev, u U U. (S) Moreover, v = 0 and v S. Monotonous decay of the free energy: Let (u,v) be a solution to (P). Then F(u(t 2 )) F(u(t 1 )) F(U) for t 2 t 1 0. Exponential decay of the free energy: Let (u,v) be a solution to (P) and let (u,v ) be the thermodynamic equilibrium. Then there exists a constant λ > 0 such that F(u(t)) F(u ) e λt (F(U) F(u )) t 0. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 6 (20)

7 Voronoi finite volume discretization Ω R N, N 3, open, bounded, polyhedral domain M = (P,T,E ) set P of grid points in Ω, P is assumed to be boundary conforming Delaunay family T of control volumes, family E of faces of the Voronoi boxes E int E, E K E, M = #P xk K mσ L σ = K L K L x L dσ xl x K For x K P the Voronoi box K is given by K = {x Ω : x x K < x x L x L P, x L x K }, K T. m σ = (n 1)-dimensional measure of Voronoi face σ, d σ = x K x L for σ = K L E int Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 7 (20)

8 Discretization scheme u K i Mass of species X i in the box K u = (u K i ) K T,i=1,...,m R Mm v ik chemical potential of the species X i in x K v = (v ik ) K T,i=1,...,m R Mm Voronoi finite volume - implicit Euler - discretization u K i (t n) u K i (t n 1) t n t n 1 Yi σ σ E K E int Z σ i (t n)(v il (t n ) v ik (t n )) m σ d σ u K i (t n) = ū ik e v ik(t n ) mes(k), n 1, u K i (0) = U i dx, K T, i = 1,...,m, K = R K i (t n), (P M ) Zi σ = 2 1 (ev ik + e v il ), σ = K L, Yi σ suitable average of ū i µ i corresponding to face σ ū ik, k αβk average of ū i, k αβ on the box K, R K i (t n) = k αβk (e α v K(t n ) e β v K(t n ) )(β i α i )mes(k) (α,β) R Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 8 (20)

9 Results for each finite-dimensional problem (P M ) (Glitzky 08) Invariants: ( u, v) (R Mm ) 2 solution to the discretized problem (P M ) = u(t n ) U { u R Mm : ( u K 1,..., u K ) } m S K T K T n N Stationary solutions: There is exactly one stationary solution ( u, v ) to (P M ) fulfilling the above invariance property. ( u, v ) is a discrete thermodynamic equilibrium. The scheme is dissipative and the discrete free energy m ˆF( u) = i=1 K T ( ) u K i v ik u K i + ū ik mes(k) decays monotonously and exponentially to its equilibrium value. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 9 (20)

10 Remarks Results are also valid for (Gärtner/G. 09, G. 08) anisotropic diffusion S i positive definite, symmetric flux densities j i = u i S i v i Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 10 (20)

11 Remarks Results are also valid for (Gärtner/G. 09, G. 08) anisotropic diffusion S i positive definite, symmetric flux densities j i = u i S i v i more general statistical relations u i = ū i g i (v i v i ) flux densities j i = ū i g i (v i v i )S i v i Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 10 (20)

12 Remarks Results are also valid for (Gärtner/G. 09, G. 08) anisotropic diffusion S i positive definite, symmetric flux densities j i = u i S i v i more general statistical relations u i = ū i g i (v i v i ) flux densities j i = ū i g i (v i v i )S i v i charged species X i with charge numbers q i in addition Poisson equation for the electrostatic potential v 0 m (S 0 v 0 ) = f + q i u i on R + Ω + BCs i=1 flux densities j i = ū i g i (v i v i )S i (v i + q i v 0 ) Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 10 (20)

13 Remarks concerning anisotropies Use for each species X i, i = 1,...,m, their own anisotropic Voronoi boxes K (i) = {x Ω : (x x K ) T S 1 i (x x K ) (x x L ) T S 1 i (x x L ) x L P} fluxes through faces σ (i) additionally contain the factor S i ν σ (i) reaction terms have to be evaluated on all different intersection sets K (1) k 1 K (2) k 2 K (m) k m, k i {1,...,M} Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 11 (20)

14 Formulation of (P M ) by piecewise constant functions Aim: Uniform exponential decay of the free energy for a class of Voronoi finite volume discretizations M of Ω with the properties diam(σ) κ 1 d σ σ E int, r K,out κ 2 r K,in K T (Q) X(M ) = set of functions from Ω to R, being constant on each K T. v K = value of v X(M ) on K. Discrete H 1 -seminorm for v X(M ) v 2 1,M = D σ v 2 m σ, d σ E int σ where D σ v = v K v L for σ = K L. use functions v i u i ū i k αβ a i = e v i /2 X(M ) v ik u ik ū ik k αβk e v ik/2 on K where u ik := uk i mes(k) Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 12 (20)

15 Formulation of (P M ) by piecewise constant functions E M : X(M ) X(M ), E M (v) = (ū i e v i )i=1,...,m, u = E M (v) Free energy: F M : X(M ) R, m F M (u) = (u i v i u i + ū i )dx F(u) Ω i=1 Dissipation rate: m D M (v) = Yi σ i=1 σ E int m c i=1 Zi σ D σ v i 2 m σ + d σ a i 2 1,M + c Ω (α,β) R [ m i=1 [ k αβ e α v e β v] (α β) v dx Ω (α,β) R a α i i m i=1 a β ] i 2 i dx 0, ai = e v i /2. Invariants: u(t n ) U U n N For the thermodynamic equilibrium (u,v ) to (P M ) (resulting from ( u, v )) we get u i = ūi ū i u i, i = 1,...,m, v = v S. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 13 (20)

16 Formulation of (P M ) by piecewise constant functions (G. Prep. 1443) Theorem A. [Uniform estimate of the free energy by the dissipation rate] Let a class of Voronoi finite volume discretizations fulfill (Q), moreover let (u,v ) be the thermodynamic equilibrium to (P M ). Then, for all ρ > 0 there exists a constant c ρ > 0 such that F M (E M v) F M (u ) c ρ D M (v) { } for all v N ρm := v X(M ) : F M (E M v) F M (u ) ρ, E M v U U uniformly for all Voronoi finite volume discretizations M. Indirect proof: Suppose there is {M l } with size(m l ) 0 and corresponding v l N ρml such that construct a contradiction. F Ml (E Ml v l ) F Ml (u l ) = C ld Ml (v l ) with C l, Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 14 (20)

17 Formulation of (P M ) by piecewise constant functions Essential tool in this proof: Theorem B. [Discrete Sobolev-Poincaré inequality] (G./Griepentrog 10) u m Ω (u) L q (Ω) c q u 1,M u X(M ) for q [1, ) if N = 2, q [1, N 2 2N ) if N > 2 uniformly for all M with the property (Q). D Ml (v l ) 0 = a Ml i 1,Ml 0 = a Ml i m Ω (a Ml i) 0 in L q (Ω), m Ω (a Ml i) e v i /2 guarantees the limit passage l in parts of the dissipation rate resulting from the reaction terms. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 15 (20)

18 Uniform exponential decay of the free energy for solutions to (P M ) Theorem C. [Uniform exponential decay of the free energy] We consider Voronoi finite volume discretizations M fulfilling the property (Q). Let (u,v) be a solution to (P M ) and let (u,v ) be the thermodynamic equilibrium to (P M ). Then F M (u(t n )) F M (u ) e λ t n ( FM (U) F M (u ) ) n 1 with a unified constant λ > 0. Idea of the proof: Discrete Brézis formula: For n 2 > n 1 0 and λ 0 we obtain ) ( ) e λt n 2 (F M (u(t n2 )) F M (u ) e λt n 1 F M (u(t n1 )) F M (u ) n 2 { e λt r 1 (t r t r 1 ) e λ h ( ) } λ F M (u(t r )) F M (u ) D M (v(t r )). r=n 1 +1 Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 16 (20)

19 Idea of the proof For n 2 > n 1 0 and λ 0 we obtain ) ( ) e λt n 2 (F M (u(t n2 )) F M (u ) e λt n 1 F M (u(t n1 )) F M (u ) n 2 { e λt r 1 (t r t r 1 ) e λ h ( ) } λ F M (u(t r )) F M (u ) D M (v(t r )). r=n 1 +1 λ = 0: F M (u(t n2 )) + n 2 (t r t r 1 )D M (v(t r )) F M (u(t n1 )) F M (U) r=n 1 +1 F M decays monotonously, F M (U) F M (u ) ρ for all M with (Q), = v(t r ) N ρm for r 1 E M v(t r ) U U apply Theorem A, choose λ > 0 such that λ e λ h c ρ < 1 = result follows uniformly for all M with (Q). Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 17 (20)

20 References R. Eymard, T. Gallouët, and R. Herbin, The finite volume method, Handbook of numerical Analysis VII (2000), A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity 21 (2008), A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction- diffusion systems, WIAS-Preprint 1443 (2009). A. Glitzky and K. Gärtner, Energy estimates for continuous and discretized electroreaction-diffusion systems, Nonlin. Anal. 70 (2009), A. Glitzky and J. A. Giepentrog, Discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations, SINUM 48 (2010), A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for reaction diffusion systems, Z. Angew. Math. Mech. 77 (1997), Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 18 (20)

21 Assumptions Ω R N, N = 2,3, open, bounded polyhedral domain; µ i, u i, U i L+(Ω), µ i, u i δ > 0, q i Z, i = 1,...,m; R Z m + Zm + finite subset, k αβ L+(Ω), k αβ δ > 0 for all (α,β) R. { } If N = 3 then max (α,β) R max m i=1 α i, m i=1 β i 3; (< 3 for uniform exponential decay of the free energy) m i=1 U i,λ i > 0 λ S + \ {0} (Slater condition) {b R m + : bα = b β (α,β) R, u U U, u i = ū i b i } R m + = /0; (no false equilibria in the sense of Prigogine) M Voronoi finite volume discretization, P boundary conforming Delaunay grid, h = sup n N (t n t n 1 ) <. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 19 (20)

22 Assumptions for uniform results Ω R N is star shaped with respect to some ball B(y 0,R). exp { } R 2 if y y ρ(y) = R 2 y y < R, 0 if y y 0 R. ρ M X(M ): ρk M (x) = min y K ρ(y) for x K. For all finite volume meshes M of Ω we suppose the following properties: Ω ρm (x)dx κ 0, E K E ext /0 implies x K Ω, The geometric weights fulfill property (Q). Ω = I I Ω I finite disjoint union of subdomains s.t. discontinuities of ū i, i = 1,...,m, coincide with subdomain boundaries. (N 1) dimensional measure of all internal subdomain boundaries is bounded. There exists some γ (0,1] such that ū i C 0,γ (Ω I ) := {w ΩI,w C 0,γ (R N )}, i = 1,...,m, I I. Uniform exponential decay of the free energy Dresden, May 26, 2010 Page 20 (20)