A posteriori analysis of a discontinuous Galerkin scheme for a diffuse interface model

Transcription

1 A posteriori analysis of a discontinuous Galerkin scheme for a diffuse interface model Jan Giesselmann joint work with Ch. Makridakis (Univ. of Sussex), T. Pryer (Univ. of Reading) 9th DFG-CNRS WORKSHOP Micro-Macro Modeling and Simulation of Liquid-Vapor Flows February / 22

2 Outline. Van-der-Waals fluid model 2. Relative entropy stability framework convex energy non-convex energy 3. Semi-discrete dg scheme 4. Reconstruction and error estimate 2 / 22

3 Van-der-Waals fluid model One space dimension; describing fluid flows undergoing liquid-vapor phase transition: u t v x = 0 v t W (u) x = µv xx γu xxx (vdw) u specific volume, v velocity W non-convex energy density (vdw) is hyperbolic-elliptic W C 3 (R, [0, )) γ > 0 capillarity parameter, µ 0 viscosity. We consider the problem on the flat circle, denoted S. 3 / 22

4 Van-der-Waals fluid model One space dimension; describing fluid flows undergoing liquid-vapor phase transition: u t v x = 0 v t W (u) x = µv xx γu xxx (vdw) u specific volume, v velocity W non-convex energy density (vdw) is hyperbolic-elliptic W C 3 (R, [0, )) γ > 0 capillarity parameter, µ 0 viscosity. We consider the problem on the flat circle, denoted S. Associated energy balance ( W (u)+ γ 2 (u x) v2) t ( vw (u) γvu xx +γv x u x +µvv x ) x +µ(v x) 2 = 0. 3 / 22

5 Recall: Standard relative entropy (W strictly convex) Consider the non-regularized, hyperbolic problem u t v x = 0 v t W (u) x = 0 with W strictly convex. Solutions (u, v), (ũ, ṽ) can be compared by their relative entropy S W (ũ) W (u) W (u)(ũ u) + 2 (ṽ v)2 d x, which is equivalent to ũ u 2 L 2 (S ) + ṽ v 2 L 2 (S ). 4 / 22

6 Recall: Standard relative entropy (W strictly convex) The relative entropy satisfies d W (ũ) W (u) W (u)(ũ u) + d t S 2 (ṽ v)2 d x ( ) ṽw (ũ) vw (u) W (u)(ṽ v) v(w (ũ) W (u)) d x S x ( v x W (ũ) W (u) W (u)(ũ u) ) d x. S 5 / 22

7 Standard relative entropy argument (W strictly convex) For W bounded and v Lipschitz this implies d W (ũ) W (u) W (u)(ũ u) + d t S 2 (ṽ v)2 d x C ( ũ u 2 L 2 (S ) + ṽ ) v 2 L 2 (S ) C W (ũ) W (u) W (u)(ũ u) + S 2 (ṽ v)2 d x. Therefore, by Gronwall s Lemma, the relative entropy grows at most exponentially and, therefore, for t > s ũ(t, ) u(t, ) L 2 + ṽ(t, ) v(t, ) L 2 e C(t s)( ũ(s, ) u(s, ) L 2 + ṽ(s, ) v(s, ) L 2 ). 6 / 22

8 Back to the multi-phase case, i.e., W not convex Regularized model: u t v x = 0 v t W (u) x = µv xx γu xxx. For different (u, v) (ũ, ṽ) their relative entropy is given by S W (ũ) W (u) W (u)(ũ u) + 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x, which is not convex, as W is not convex and γ is small. On S there are no boundary terms, and we obtain d W (ũ) W (u) W (u)(ũ u) + d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x v x (W (ũ) W (u) W (u)(ũ u)) µ(v x ṽ x ) 2 d x. S 7 / 22

9 Estimating the partial relative entropy rate Idea: Remove the W terms from the relative entropy. Indeed ) t (W (ũ) W (u) W (u)(ũ u) =W (ũ)ũ t W (u)u t W (u)u t (ũ u) W (u)ũ t + W (u)u t =ṽ x W (ũ) ṽ x W (u) v x W (u)(ũ u). Thus, we can shift the W terms to the right hand side and find d d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x (v ṽ) x (W (ũ) W (u)) µ(v x ṽ x ) 2 d x. S 8 / 22

10 Continuous dependence on initial data Lemma (JG 3) For µ > 0 let u 0, ũ 0 H 3 (S ), v 0, ṽ 0 H 2 (S ) be given with S (u 0 ũ 0 ) dx = 0. Then, for any T > 0, it exists a constant C = C(u 0, v 0, γ, µ, T) such that v(t, ) ṽ(t, ) L 2 (S ) + u(t, ) ũ(t, ) H (S ) ( ) C v 0 ṽ 0 L 2 (S ) + u 0 ũ 0 H (S ) for all t T. Sketch of the proof: For any T > 0 there exist strong solutions u, ũ C ((0, T), L 2 (S )) C 0 ((0, T), H 3 (S )) v, ṽ C ((0, T), L 2 (S )) C 0 ((0, T), H 2 (S )). 9 / 22

11 Sketch of the proof Due to the energy inequality and the continuous embedding H (S ) C 0 (S ), we find that u L, ũ L <. The partial relative entropy calculation leads to d d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x = (ṽ v) x (W (u) W (ũ)) µ(v x ṽ x ) 2 d x. S Such that by Young s and Poincaré s inequality d d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x C Gronwall s Lemma concludes the proof. S γ 2 (ũ x u x ) 2 d x. 0 / 22

12 Continuous dependence on initial data for µ = 0 Lemma (JG 3) For µ = 0 let solutions u, ũ C ((0, T), L 2 (S )) C 0 ((0, T), H 3 (S )) v, ṽ C ((0, T), L 2 (S )) C 0 ((0, T), H (S )) corresponding to initial data (u 0, v 0 ) and (ũ 0, ṽ 0 ) be given with S (u 0 ũ 0 ) dx = 0. Let W be bounded. Then it exists a constant C = C(u 0, v 0, γ, T) such that v(t, ) ṽ(t, ) L 2 (S ) + u(t, ) ũ(t, ) H (S ) ( ) C v 0 ṽ 0 L 2 (S ) + u 0 ũ 0 H (S ) for all t T. / 22

13 Sketch of the proof The partial relative entropy calculation leads to d d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x = (ṽ v) x (W (u) W (ũ)) d x S (ṽ v) 2 + ((W (u) W (ũ)) x ) 2 d x S C using Young s and Poincaré s inequality. Applying Gronwall s Lemma concludes the proof. S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x 2 / 22

14 Semi-discrete dg scheme (from now on µ = 0) Decompose [0, ] into 0 = x 0 < x < < x N =. Identify 0 and. V q := space of (discontinuous) piece-wise polynomials of degree q, V c q := V q C 0 (S ). Find u h, v h C ((0, T), V q ), τ h C 0 ([0, T], V q ) such that 0 = t u h Φ G[u h ]Φ d x Φ V q S 0 = t v h Ψ G[τ h ]Ψ d x Ψ V q S 0 = τ h Z W (u h )Z d x + γa h (u h, Z) Z V q, S where a h is an interior penalty discretisation of the Laplacian and G denotes a discrete gradient operator. 3 / 22

15 Discretisations For g h V q we define G[g h ] V q by S G[g h ]Φ dx = N i=0 ( x i+ x i (g h ) x Φ dx g h xi { Φ } xi ) for all Φ V q. For f h, g h V q we define a h (f h, g h ) := N i=0 ( x i+ x i (f h ) x (g h ) x dx f h xi { (g h ) x } xi g h xi { (f h ) x } xi + σ h f h xi g h xi ), for some σ, such that a h : V q V q R is coercive. 4 / 22

16 Reconstruction ˆτ V q+ is defined by 0 = (ˆτ x G[τ h ])Ψ d x Ψ V q S It can be shown that ˆτ(x n ) = τ h(x n ) + τ h (x + n ) 2 and n {0,..., N }. ˆτ is continuous. ˆτ τ h 2 L 2 (S ) ( ) N i=0 (x i+ x i ) τ h 2xi + τ h 2xi+. 5 / 22

17 Reconstruction 2 ũ H 3 (S ) is defined by 0 = γ xx ũ P c q+(w (u h )) + ˆτ, where P c q+ : L 2 (S ) V c q+ is the L 2 -projection, and S (u ũ) dx = 0. Using elliptic regularity and a posteriori control of elliptic reconstructions we have ( ũ u h dg := ũ uh 2 H ((xi,xi+)) + ũ u h 2) 2 xi C P c q+(w (u h )) W (u h ) H (S ) + C ˆτ τ h H (S ) + H [u h, W (u h ), τ h ], where H [u h, W (u h ), τ h ] is an explicitly computable estimator, expected to be of order h q. 6 / 22

18 Reconstruction 2 cont d Pq+(W c (u h )) W (u h ) H (S ) C (x i+ x i ) u h 2 i + C sup(x i+ x i ) q+ W (u h ) H ((x i,x i+)). i According to arguments provided by Nochetto and Makridakis 06 terms of the structure (x i+ x i ) h 2 are expected to be of optimal order. i 7 / 22

19 Reconstruction 3 ˆv V q+ is defined by 0 = (ˆv x G[v h ])Ψ d x Ψ V q and ˆv(x n ) = v h(xn ) + v h (x n + ) n. S 2 ṽ H 2 (S ) is defined by ˆv is continuous and 0 = xx ṽ xt ũ and S ṽ v h dx = 0. N ˆv v h 2 L 2 (S ) ( ) (x i+ x i ) v h 2xi + v h 2xi+, i=0 while ṽ ˆv L 2 (S ) t ũ t u h dg. 8 / 22

20 Perturbed equation By definition the reconstructions satisfy (point-wise a.e.) ũ t ṽ x = 0 (v h ) t ˆτ x = 0 which implies, by definition of ũ, (v h ) t (Pq+(W c (u h ))) x + γũ xxx = 0. Thus, ũ t ṽ x = 0 ṽ t W (ũ) x = γũ xxx + E, where the residual E is given by E := t (ṽ v h ) + x (Pq+(W c (u h )) W (ũ)). By the estimates above, E is bounded explicitly in terms of u h, v h, τ h. 9 / 22

21 Partial relative entropy Then, the partial relative entropy calculation implies d d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x = (v ṽ)(w (ũ) W (u)) x d x + S S (v ṽ)e d x. We infer d d t S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x C S 2 (ṽ v)2 + γ 2 (ũ x u x ) 2 d x + E 2 d x. S 20 / 22

22 Error estimate Applying Gronwall s Lemma and triangle inequality gives Theorem (JG, Makridakis, Pryer 4) Let (u h, v h ) denote the solution of the semi-discrete dg scheme. Let (u, v) be a weak solution of (vdw) with u C ((0, T), L 2 (S )) C 0 ([0, T], H 3 (S )) v C ((0, T), L 2 (S )) C 0 ([0, T], H (S )) Then, there is C>0 such that, v h (t, ) v(t, ) 2 L 2 (S ) + u h(t, ) u(t, ) 2 dg C ( E + E 2 + E 3 ), ( ) where E := ṽ(0, ) v 0 2 L 2 (S ) + ũ(0, ) u 0 2 H (S ) exp(ct) t E 2 := E 2 dx dt exp(ct) 0 S E 3 := ṽ(t, ) v h (t, ) 2 L 2 (S ) + ũ(t, ) u h(t, ) 2 dg. 2 / 22

23 Error estimate Applying Gronwall s Lemma and triangle inequality gives Theorem (JG, Makridakis, Pryer 4) Let (u h, v h ) denote the solution of the semi-discrete dg scheme. Let (u, v) be a weak solution of (vdw) with u C ((0, T), L 2 (S )) C 0 ([0, T], H 3 (S )) v C ((0, T), L 2 (S )) C 0 ([0, T], H (S )) Then, there is C>0 such that, v h (t, ) v(t, ) 2 L 2 (S ) + u h(t, ) u(t, ) 2 dg C ( E + E 2 + E 3 ), ( ) where E := ṽ(0, ) v 0 2 L 2 (S ) + ũ(0, ) u 0 2 H (S ) exp(ct) t E 2 := E 2 dx dt exp(ct) 0 S E 3 := ṽ(t, ) v h (t, ) 2 L 2 (S ) + ũ(t, ) u h(t, ) 2 dg. Note that E, E 2, E 3 can be explicitly bounded in terms of u h, v h, τ h. 2 / 22

24 Summary and outlook Summary New stability framework for regularized hyperbolic-elliptic problems. Derived an a posteriori error estimate. Estimate depends sensitively on γ, blows up for γ 0. Stability framework can also be used for model convergence. Outlook Numerical experiments. Extension to fully discrete scheme. Extension to several space dimensions. Including viscosity. 22 / 22

25 Summary and outlook Summary New stability framework for regularized hyperbolic-elliptic problems. Derived an a posteriori error estimate. Estimate depends sensitively on γ, blows up for γ 0. Stability framework can also be used for model convergence. Outlook Numerical experiments. Extension to fully discrete scheme. Extension to several space dimensions. Including viscosity. Thank you for your attention! 22 / 22