Numerical Approximation of Phase Field Models

Transcription

1 Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School 2017 R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

2 Outline Lecture 2 1 Allen Cahn Equation Introduction Finite Element Approximations for a Smooth Potential 2 Cahn Hilliard Equation Introduction Gradient Flow Finite Element Approximations for a Smooth Potential R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

3 Outline 1 Allen Cahn Equation Introduction Finite Element Approximations for a Smooth Potential 2 Cahn Hilliard Equation Introduction Gradient Flow Finite Element Approximations for a Smooth Potential R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

4 Allen Cahn Equation Allen Cahn equation ε u t = ε u 1 ε Ψ (u) in Ω T, u(, 0) = u 0 in Ω, u ν Ω = 0 on Ω (0, T ), is a weighted L 2 gradient flow for the energy ˆ 1 E(u) = 2 ε u ε Ψ(u) dld. Ω Here Ψ : R R, with Ψ C 2 (R), is a smooth double-well potential satisfying Ψ(s) 0 for all s R, Ψ(s) = 0 if and only if s { 1, 1}, Ψ( s) = Ψ(s). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

5 Allen Cahn Equation For example, Ψ(s) = 1 4 (s2 1) R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

6 Allen Cahn Equation ε u t = ε u 1 ε Ψ (u) in Ω T, u(, 0) = u 0 in Ω, u ν Ω = 0 on Ω (0, T ), For some initial function ϕ 0 : Ω R, the flow for the ODE ε ϕ t = 1 ε Ψ (ϕ) drives positive values of ϕ 0 to +1 and negative values to 1. However, the Laplacian term has a smoothing effect, which will diffuse large gradients. Hence, after a short time solutions to the Allen Cahn equation will develop a structure consisting of bulk regions in which u takes the values ±1, and separating these regions there will be transition layers across which u changes rapidly from one bulk value to the other. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

7 Allen Cahn Equation It can be shown that the motion of these interfacial layers approximates mean curvature flow (MCF), see lectures by Patrick Dondl. Moreover, the profile of u across the interfacial layers can be described as follows. Let φ be the solution of ε φ (s) + 1 ε Ψ (φ(s)) = 0, s R, lim φ(s) = ±1, φ(0) = 0, s ± φ (s) > 0, s R. If Ψ = 1 4 (s2 1) 2, i.e. Ψ (s) = s 3 s, then the solution to this ODE is ( ) s φ(s) = tanh ε, 2 on recalling that tanh (s) = (1 tanh 2 (s)) = 2 tanh(s) (1 tanh 2 (s)). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

8 Allen Cahn Equation ( ) s φ(s) = tanh ε, 2 In particular, the interface width scales with ε. Ω 1 ϕ 1 2ε ε ε ϕ 1 1 R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

9 Mean Curvature Flow Let (Γ(t)) t 0 be a smoothly evolving family of hypersurfaces in R d. Let ν(t) denote the unit normal to Γ(t), and let V be the normal velocity of Γ(t). Then mean curvature flow (MCF) is defined by V = κ on Γ(t), where the mean curvature κ can be defined as the first variation of surface area, so that ˆ d dt Hd 1 (Γ(t)) = κ V dh d 1. Γ(t) Hence MCF can be interpreted as the L 2 gradient flow of surface area, and for solutions of MCF it holds that ˆ d dt Hd 1 (Γ(t)) + V 2 dh d 1 = 0. Γ(t) R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

10 Mean Curvature Flow Curvature for a curve in 2d For a curve in the plane, let y : I R 2 be a smooth parameterization by arc-length, so that y (s) = 1 for all s I. Then τ = y (s) is the unit tangent of the curve and y (s) describes how fast the tangent changes locally. For a curve Γ R 2 with unit normal ν, for example such that ( ν, τ) is positively oriented, we obtain that 0 = ( y (s) 2) = ( y (s). y (s) ) = 2 y (s). y (s) = 2 y (s). τ, and so y (s) is a multiple of ν. We define as the curvature of the curve Γ. κ = y (s). ν R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

11 Mean Curvature Flow Mean curvature for a surface in 3d Now consider a smooth surface Γ R 3 with unit normal ν. Here we can define a normal curvature in each tangential direction τ. For a fixed tangential vector τ, consider the hyperplane G spanned by ν and τ. The intersection G Γ yields a curve γ, with normal ν. For this curve in the plane G we can define the planar curvature as in 2d. This curvature we call normal curvature of Γ R 3 with respect to the chosen τ. γ Γ ν τ R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

12 Mean Curvature Flow Mean curvature for a surface in 3d γ Γ ν τ Now let κ 1 and κ 2 be the minimal and maximal values of the normal curvature over all possible tangential directions. The values κ 1 and κ 2 are called the principal curvatures and the mean curvature is defined as κ = κ 1 + κ 2. The classical definition is 1 2 (κ 1 + κ 2 ), but most mathematicians now prefer the sum of principal curvatures, as this leads to simpler expressions in most situations. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

13 Sharp Interface Limits Let ϕ : Ω R be a phase field approximation of Γ Ω, so that {ϕ = 0} Γ. Then it can be shown for ε small that 1 E(ϕ) = 1 ˆ 1 σ Ψ σ 2 ε ϕ Ψ ε Ψ(ϕ) dld H d 1 (Γ), where Ω σ Ψ = ˆ 1 1 (2 Ψ) 1 2 dl 1. Moreover, again for ε small, we have that 2 grad σ L 2 E(ϕ) = 2 [ ε ϕ + 1 ] Ψ σ Ψ ε Ψ (ϕ) κ on {ϕ = 0}. Together with 2 σ Ψ ε ϕ t V one can prove that V = κ is the sharp interface limit of the Allen Cahn equation. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

14 Allen Cahn Equation Let u 0 H 1 (Ω). Then a weak solution of the Allen Cahn equation is given by a function u L 2 (0, T ; H 1 (Ω)) with u t L 2 (0, T ; L 2 (Ω)) and such that for all t (0, T ) ε (u t (t), v) + ε ( u(t), v) + 1 ε (Ψ (u(t)), v) = 0 v H 1 (Ω), u(, 0) = u 0. Theorem It holds that 1 2 ε u(t) ˆ t ε (Ψ(u(t)), 1) + ε 0 u t (s) 2 0 dl 1 (s) = E(u 0 ), for all t (0, T ). In particular, u L (0, T ; H 1 (Ω)), if e.g. Ψ(s) = 1 4 (s2 1) 2. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

15 Allen Cahn Equation Proof. This directly follows from the gradient flow structure. In particular, choosing v = u t (t) yields d dt E(u(t)) = ε u t(t) 2 0. Integration over (0, t) gives the desired result. Finally, the bound implies that ess Ψ(s) = 1 4 (s2 1) 2 = 1 4 s4 1 2 s s4 1 4, sup t (0,T ) and so u L (0, T ; H 1 (Ω)). { 1 2 ε u(t) ε 1 u 4 L 4 (Ω) } E(u 0) + 4 ε L d (Ω), R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

16 Outline 1 Allen Cahn Equation Introduction Finite Element Approximations for a Smooth Potential 2 Cahn Hilliard Equation Introduction Gradient Flow Finite Element Approximations for a Smooth Potential R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

17 Allen Cahn Equation Finite element approximations ε u t = ε u 1 ε Ψ (u) in Ω T, u(, 0) = u 0 in Ω. u ν Ω = 0 on Ω (0, T ), Fully explicit scheme Given U 0 S h (Ω), for n 0 find U n+1 S h (Ω) such that ( U n+1 U n ) h ε, χ + ε ( U n, χ) + 1 t ε (Ψ (U n ), χ) h = 0 χ S h (Ω), where we recall the mass lumped L 2 inner product (, ) h. Stability only if t C h 2, similarly to the linear heat equation. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

18 Allen Cahn Equation Finite element approximations ε u t = ε u 1 ε Ψ (u) in Ω T, u(, 0) = u 0 in Ω. u ν Ω = 0 on Ω (0, T ), Fully implicit scheme Given U 0 S h (Ω), for n 0 find U n+1 S h (Ω) such that ( U n+1 U n ) h ε, χ + ε ( U n+1, χ) + 1 t ε (Ψ (U n+1 ), χ) h = 0 χ S h (Ω). Let E h (U) = ε 2 U ε (Ψ(U), 1)h. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

19 Allen Cahn Equation Assume from now on that there exists a constant C Ψ 0 such that C Ψ Ψ (s) s R. For example, when Ψ(s) = 1 4 (s2 1) 2, then Ψ (s) = 1 3 s 2 1 = C Ψ. Theorem For a solution U n+1 S h (Ω) of the fully implicit scheme it holds that ( ε E h (U n+1 ) + t C ) Ψ U n+1 U n 2 h 2 ε Eh (U n ), where we recall that χ h = [(χ, χ) h ] 1 2. In addition, if t < ε2 C Ψ, then there exists a unique solution U n+1 S h (Ω). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

20 Allen Cahn Equation Proof. Choosing χ = U n+1 U n yields that ε t Un+1 U n 2 h + ε ( Un+1, (U n+1 U n )) + 1 ε (Ψ (U n+1 ), U n+1 U n ) h = 0. Moreover, noting the Taylor expansion yields that Ψ(s) = Ψ(r) + Ψ (r) (s r) Ψ (ξ) (s r) 2 Ψ(r) Ψ(s) Ψ (r) (r s) = 1 2 Ψ (ξ) (r s) C Ψ (r s) 2. Combining the above, and recalling r (r s) 1 2 r s2, gives R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

21 Allen Cahn Equation Proof (Cont.) ε 2 Un ε 2 Un ε (Ψ(Un+1 ) Ψ(U n ), 1) h + ε t Un+1 U n 2 h and hence ε ( U n+1, (U n+1 U n )) + 1 ε (Ψ (U n+1 ), U n+1 U n ) h + C Ψ 2 ε Un+1 U n 2 h + ε t Un+1 U n 2 h = C Ψ 2 ε Un+1 U n 2 h, ( ε E h (U n+1 ) + t C ) Ψ U n+1 U n 2 h 2 ε Eh (U n ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

22 Allen Cahn Equation Proof (Cont.) To prove existence, consider the variational problem: Find U S h (Ω) such that J h (U) J h (χ) χ S h (Ω), where J h (χ) = E h (χ) + ε 2 t U Un 2 h. Then J h : S h (Ω) R 0 is continuous, bounded from below, and such that J h (U) as U U n h. It follows that there exists a solution U to the above variational problem. Hence, for every χ S h (Ω), the function λ J h (U + λ χ) must have vanishing derivative at λ = 0, i.e. [ ] δ δu J h (U) (χ) = d dλ J h (U + λ χ) λ=0 = 0 χ S h (Ω). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

23 Allen Cahn Equation Proof (Cont.) Hence, for all χ S h (Ω), we have that [ ] δ δu J h (U) (χ) = ε t (U Un, χ) h + ε ( U, χ) + 1 ε (Ψ (U), χ) h = 0, which proves existence. In order to show uniqueness, observe that Ψ(r) = Ψ(s) + Ψ (s) (r s) Ψ (ξ 1 ) (r s) 2 Ψ(s) = Ψ(r) Ψ (r) (r s) Ψ (ξ 2 ) (r s) 2 0 = (Ψ (s) Ψ (r)) (r s) (Ψ (ξ 1 ) + Ψ (ξ 2 )) (r s) 2, and so (Ψ (r) Ψ (s)) (r s) C Ψ (r s) 2. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

24 Allen Cahn Equation Proof (Cont.) Now, if U 1 and U 2 are two solutions, their difference satisfies ε t (U 1 U 2, χ) h + ε ( (U 1 U 2 ), χ) + 1 ε (Ψ (U 1 ) Ψ (U 2 ), χ) h = 0, which for χ = U 1 U 2 yields that If 0 = ε t U 1 U 2 2 h + ε U 1 U ε (Ψ (U 1 ) Ψ (U 2 ), U 1 U 2 ) h ε t U 1 U 2 2 h + ε U 1 U C Ψ ε U 1 U 2 2 h ( ε t C ) Ψ U 1 U 2 2 h ε + ε U 1 U ε t C Ψ ε ε2 > 0, i.e. if t < C Ψ, then we obtain U 1 = U 2. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

25 Allen Cahn Equation Finite element approximations Both the fully explicit and the fully implicit scheme require a time step constraint for stability. They are conditionally stable. An unconditionally stable scheme is stable for any choice of time step t. If Ψ were convex, then Ψ (s) 0, and so formally we could choose C Ψ = 0 in the proof of the fully implicit scheme, giving the desired unconditionally stable scheme. Of course, the double well potential is non-convex, but we can try to find a splitting Ψ(s) = Ψ + (s) + Ψ (s), where Ψ + C 2 (R) is convex, and where Ψ C 2 (R) is concave. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

26 Allen Cahn Equation Finite element approximations Note that such a splitting always exists. For example, one can choose Ψ + (s) = Ψ(s) C Ψ s 2 and Ψ (s) = 1 2 C Ψ s 2. Then Ψ +(s) = Ψ (s) + C Ψ 0, and Ψ (s) = C Ψ 0. Given such a splitting, we would treat the term corresponding to 1 ε Ψ +(u) implicitly, and the term corresponding to 1 ε Ψ (u) explicitly. In practice it is often preferable to have a splitting, where the resultant system of equations is easy to solve. For Ψ(s) = 1 4 (s2 1) 2, such a splitting is Ψ + (s) = 1 4 s4 and Ψ (s) = 1 2 s The idea of utilizing a convex/concave splitting goes back to Elliott & Stuart (1993), in the context of general discrete semilinear parabolic equations. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

27 Allen Cahn Equation Finite element approximations Semi-implicit scheme Given U 0 S h (Ω), for n 0 find U n+1 S h (Ω) such that ( U n+1 U n h ε, χ) + ε ( U n+1, χ) t + 1 ε (Ψ +(U n+1 ), χ) h + 1 ε (Ψ (U n ), χ) h = 0 χ S h (Ω). Theorem There exists a unique solution U n+1 S h (Ω) to the semi-implicit scheme, and it holds that E h (U n+1 ) + ε t Un+1 U n 2 h Eh (U n ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

28 Allen Cahn Equation Proof. Existence, and uniqueness, of a solution can be shown via a variational argument as before, on considering J h U n(u) = 1 2 ε U ε (Ψ +(U), 1) h + 1 ε (Ψ (U n ), U) h + ε 2 t U Un 2 h. Alternatively, note that the coefficients of U n+1 in terms of the finite element basis {χ k } K k=1 satisfy F (U n+1 ) = ( M + t A) U n+1 + ϕ(u n+1 ) = b, where ( M + t A) is symmetric positive definite, and where ϕ : R K R K is diagonal and monotone, since [ϕ(u)] k = t Mkk Ψ ε +(U 2 k ), k = 1 K, and since Ψ +(s) 0. Hence F : R K R K is a homeomorphism. See e.g. Ortega & Rheinboldt (1970). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

29 Allen Cahn Equation Proof (Cont.) Choosing χ = U n+1 U n yields that ε t Un+1 U n 2 h + ε ( Un+1, (U n+1 U n )) + 1 ε (Ψ +(U n+1 ), U n+1 U n ) h + 1 ε (Ψ (U n ), U n+1 U n ) h = 0. The two Taylor expansions imply that Ψ + (r) Ψ + (s) Ψ +(r) (r s) = 1 2 Ψ +(ξ) (r s) 2 0, Ψ (r) Ψ (s) Ψ (s) (r s) = 1 2 Ψ (ξ) (r s) 2 0, Ψ(r) Ψ(s) Ψ +(r) (r s) + Ψ (s) (r s). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

30 Allen Cahn Equation Proof (Cont.) Combining the above, and recalling r (r s) 1 2 r s2, yields that ε t Un+1 U n 2 h + ε 2 Un ε 2 Un ε (Ψ(Un+1 ) Ψ(U n ), 1) h ε t Un+1 U n 2 h + ε ( Un+1, (U n+1 U n )) = 0, + 1 ε (Ψ +(U n+1 ), U n+1 U n ) h + 1 ε (Ψ (U n ), U n+1 U n ) h and hence E h (U n+1 ) + ε t Un+1 U n 2 h Eh (U n ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

31 Allen Cahn Equation Recall that the zero level sets of u(, t), the solution to ε u t = ε u 1 ε Ψ (u) in Ω T, u(, 0) = u 0 in Ω, u ν Ω = 0 on Ω (0, T ), converge as ε 0 to hypersurfaces {Γ(t)} t 0 that move by mean curvature flow (MCF). Hence a natural question for discretizations of the Allen Cahn equation is, whether the zero level sets of the discrete approximations also converge to {Γ(t)} t 0, as t, h, ε 0. This has been an active research area over the last view decades. For example, Feng & Prohl (2003) showed a rigorous O(ε 2 ) error bound under the assumption that t C 1 h 2 C 2 ε 7. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

32 Allen Cahn Equation Solution methods Fully explicit scheme The solution is trivial, as the coefficient vector satisfies U n+1 = (Id t M 1 A) U n + t ε 2 Ψ (U n ). Fully and semi-implicit schemes The solution boils down to finding U n+1 R K such that ( M + t A) U n+1 + t ε 2 M Ψ (+) (Un+1 ) = b. This system of nonlinear algebraic equations can be solved with: Newton method, nonlinear SOR method, nonlinear multigrid method. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

33 Outline 1 Allen Cahn Equation Introduction Finite Element Approximations for a Smooth Potential 2 Cahn Hilliard Equation Introduction Gradient Flow Finite Element Approximations for a Smooth Potential R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

34 Cahn Hilliard Equation Cahn Hilliard equation u t = (ε u ε 1 Ψ (u)) in Ω T, u(, 0) = u 0 in Ω, u ν Ω = (ε u ε 1 Ψ (u)) ν Ω = 0 on Ω (0, T ). The Cahn Hilliard equation was originally introduced by Cahn & Hilliard (1958) to model phase separation in binary alloys. Here u is defined to be the difference of the local concentrations of the two components of an alloy. Hence physically meaningful values of u lie in the interval [ 1, 1], with the pure phases corresponding to the values ±1. In addition, and in contrast to the Allen Cahn equation, the total concentration is conserved, i.e. (u(t), 1) = (u(0), 1) t (0, T ). For more details, see the lectures by Dirk Blömker. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

35 Cahn Hilliard Equation Recall that for the Ginzburg Landau energy ˆ 1 E(u) = 2 ε u Ψ(u) dld ε Ω we had shown that [ ] δ δu E(u) (v) = ( ε u + 1 ε Ψ (u), v) v H 1 (Ω), if u is sufficiently smooth with u ν Ω = 0 on Ω. Hence the Allen Cahn equation is the L 2 gradient flow of E(u). Choosing a different inner product, we would now like to interpret the Cahn Hilliard equation as an alternative gradient flow of E(u). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

36 Outline 1 Allen Cahn Equation Introduction Finite Element Approximations for a Smooth Potential 2 Cahn Hilliard Equation Introduction Gradient Flow Finite Element Approximations for a Smooth Potential R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

37 Cahn Hilliard Equation To this end, for m ( 1, 1), consider the sets where denotes the mean value of η over Ω. Ĥ 1 m(ω) = {η H 1 (Ω) : η = m}, (η, 1) η = (1, 1) Clearly, Ĥ1 m(ω) = m + Ĥ1 0 (Ω) are affine subspaces of H1 (Ω), and for u Ĥ1 m(ω) it holds that u + v Ĥ1 m(ω) v Ĥ1 0 (Ω). Let H 1 (Ω) = (Ĥ1 0 (Ω)). In the following, we extend l H 1 (Ω) to H 1 (Ω) by setting l(1) = 0. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

38 Cahn Hilliard Equation Remark Let f L 2 0 (Ω) := {η L2 (Ω) : η = 0}. Then l : v (f, v) defines an element l H 1 (Ω). In particular, Ĥ1 0 (Ω) is canonically embedded into H 1 (Ω). We now introduce the H 1 inner product on H 1 (Ω). Definition For l H 1 (Ω), let w = ( ) 1 l be the weak solution to w = l in Ω, w ν Ω = 0 on Ω, i.e. w Ĥ1 0 (Ω) is such that ( w, v) = l(v) v Ĥ1 0 (Ω). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

39 Cahn Hilliard Equation Remark For l H 1 (Ω), it holds that [( ) 1 l] = l. w ν Ω Similarly, for w H 1 (Ω) with = 0 on Ω, it holds that ( w) H 1 (Ω) and ( ) 1 ( w) = (I ) w. Proof. Let w H 1 (Ω) with Clearly, w ν Ω = 0 on Ω. Then l = w H 1 (Ω) with l(v) = ( w, v) v Ĥ1 0 (Ω). ( [(I ) w], v) = ( w, v) = l(v) v Ĥ1 0 (Ω), and so Ĥ1 0 (Ω) (I ) w = ( ) 1 l = ( ) 1 ( w). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

40 Cahn Hilliard Equation Definition For l 1, l 2 H 1 (Ω), let (l 1, l 2 ) H 1 = ( [( ) 1 l 1 ], [( ) 1 l 2 ]). Note that (, ) H 1 is indeed a scalar product on H 1 (Ω), since 0 = (l, l) H 1 = ( ) 1 l 2 1 ( ) 1 l = 0 Ĥ1 0 (Ω) [( ) 1 l] = l = 0 H 1 (Ω). For l 1, l 2 H 1 (Ω), it holds that (l 1, l 2 ) H 1 = l 1 (( ) 1 l 2 ) = l 2 (( ) 1 l 1 ). In particular, for v Ĥ1 0 (Ω) we obtain that (l, v) H 1 = (( ) 1 l, v) L 2. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

41 Cahn Hilliard Equation As a consequence, we have for any w H 1 (Ω) with (w, v) L 2 = ((I ) w, v) L 2 = (( ) 1 [ w], v) L 2 w ν Ω = ( w, v) H 1 v Ĥ1 0 (Ω). = 0 on Ω, that Now the H 1 gradient of E(u) is defined as grad H 1 E(u) H 1 (Ω) such that [ ] δ (grad H 1 E(u), v) H 1 = δu E(u) (v) v Ĥ1 0 (Ω). Hence, it holds that (grad H 1 E(u), v) H 1 = (grad L 2 E(u), v) L 2 if w = grad L 2 E(u) H 1 (Ω) with = ( grad L 2 E(u), v) H 1 v Ĥ1 0 (Ω), w ν Ω = 0 on Ω, i.e. that grad H 1 E(u) = grad L 2 E(u). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

42 Cahn Hilliard Equation But we already know that if u is sufficiently smooth with Hence grad L 2 E(u) = ε u + 1 ε Ψ (u), u ν Ω = 0 on Ω. [ grad H 1 E(u) = ε u + 1 ] ε Ψ (u), if u is sufficiently smooth with u ν Ω = 0 and (ε u 1 ε Ψ (u)) ν Ω Finally, the H 1 gradient flow of E(u) is given by = 0 on Ω. u t = (ε u ε 1 Ψ (u)) in Ω T, u ν Ω = (ε u ε 1 Ψ (u)) ν Ω = 0 on Ω (0, T ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

43 Cahn Hilliard Equation Cahn Hilliard equation u t = (ε u ε 1 Ψ (u)) in Ω T, u(, 0) = u 0 in Ω, u = (ε u ε 1 Ψ (u)) = 0 on Ω (0, T ). ν Ω ν Ω The Cahn Hilliard equation is a fourth order parabolic PDE. A weak formulation could be based on (u t, v) + (ε u ε 1 Ψ (u), v) = 0 v H 2 (Ω). However, the presence of the second derivative terms means that standard finite element spaces, which are conformal only in H 1 (Ω), can no longer be used. A solution is to introduce the auxiliary variable w = ε u + ε 1 Ψ (u), and then consider a coupled system of second order differential equations. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

44 Cahn Hilliard Equation Cahn Hilliard equation u t = w, w = ε u + ε 1 Ψ (u) in Ω T, u ν Ω = w ν Ω = 0 on Ω (0, T ), u(, 0) = u 0 in Ω. In the physical context of the Cahn Hilliard equation, the function w is called chemical potential. A weak formulation is now given by (u t, η) + ( w, η) = 0 η H 1 (Ω), ε ( u, χ) + ε 1 (Ψ (u), χ) = (w, χ) χ H 1 (Ω), u(, 0) = u 0 H 1 (Ω). Choosing η = 1 yields that d dt (u(t), 1) = (u t(t), 1) = 0 (u(t) u 0, 1) = 0 t (0, T ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

45 Cahn Hilliard Equation Moreover, choosing η = w and χ = u t yields that ˆ d t dt E(u(t)) + w 2 1 = 0 E(u(t)) + w 2 1 dl 1 = E(u 0 ). 0 This allows to prove existence of a weak solution (u, w) with u L (0, T ; H 1 (Ω)) and w L 2 (0, T ; H 1 (Ω)), see e.g. Elliott (1989) for details. Note also that existence of a unique global-in-time classical solution, given sufficiently smooth initial data, was shown in Elliott & Zheng (1986). Before introducing numerical approximations of the Cahn Hilliard equation, we note that finite element approximations based on the presented weak formulation are also called mixed methods, because they employ two different field variables. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

46 Outline 1 Allen Cahn Equation Introduction Finite Element Approximations for a Smooth Potential 2 Cahn Hilliard Equation Introduction Gradient Flow Finite Element Approximations for a Smooth Potential R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

47 Cahn Hilliard Equation Finite element approximations Fully explicit scheme Given U 0 S h (Ω), for n 0 find U n+1 S h (Ω) such that ( U n+1 U n h, η) + ( W n, η) = 0 η S h (Ω), t ε ( U n, χ) + ε 1 (Ψ (U n ), χ) h = (W n, χ) h χ S h (Ω). In matrix notation, this may be expressed as U n+1 + t M 1 A W n = U n, ε M 1 A U n + ε 1 Ψ (U n ) = W n. i.e. U n+1 = (Id ε t ( M 1 A) 2 ) U n ε 1 t M 1 A Ψ (U n ). Stability only if ε t C h 4. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

48 Cahn Hilliard Equation Finite element approximations Fully implicit scheme Given U 0 S h (Ω), for n 0 find (U n+1, W n+1 ) S h (Ω) S h (Ω) such that ( U n+1 U n h, η) + ( W n+1, η) = 0 η S h (Ω), t ε ( U n+1, χ) + ε 1 (Ψ (U n+1 ), χ) h = (W n+1, χ) h χ S h (Ω). Recall and that E h (U) = ε 2 U ε (Ψ(U), 1)h, C Ψ Ψ (s) s R. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

49 Cahn Hilliard Equation Theorem A solution (U n+1, W n+1 ) to the fully implicit scheme satisfies In addition, if (U n+1, 1) = (U 0, 1), ( E h (U n+1 ) + 1 t C Ψ 2 ) 8 ε 3 t W n E h (U n ). t < 4 ε3 C 2 Ψ then there exists a unique solution (U n+1, W n+1 ) S h (Ω) S h (Ω). Proof. The conservation property follows immediately on choosing η = 1., R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

50 Cahn Hilliard Equation Proof (Cont.) Choosing η = W n+1 and χ = U n+1 U n yields that (U n+1 U n, W n+1 ) h + t W n = 0, ε ( U n+1, (U n+1 U n )) + ε 1 (Ψ (U n+1 ), U n+1 U n ) h = (W n+1, U n+1 U n ) h, and so ε ( U n+1, (U n+1 U n ))+ε 1 (Ψ (U n+1 ), U n+1 U n ) h + t W n = 0. On recalling that 2 r (r s) = (r 2 s 2 ) + (r s) 2 and Ψ(r) Ψ(s) Ψ (r) (r s) = 1 2 Ψ (ξ) (r s) C Ψ (r s) 2, we obtain that... R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

51 Cahn Hilliard Equation Proof (Cont.) E h (U n+1 ) E h (U n ) + t W n ε 2 Un+1 U n 2 1 = ε 1 (Ψ(U n+1 ) Ψ(U n ), 1) h ε 1 (Ψ (U n+1 ), U n+1 U n ) h 1 2 ε 1 C Ψ U n+1 U n 2 h. Choosing η = t C Ψ 2 ε (U n+1 U n ) yields that 1 2 ε 1 C Ψ U n+1 U n 2 h = t C Ψ ( W n+1, (U n+1 U n )) 2 ε ( t C Ψ) 2 8 ε 3 W n ε 2 Un+1 U n 2 1. Combining the two bounds shows the desired stability bound. The existence and uniqueness result will be shown in the next lemma. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

52 Cahn Hilliard Equation Before proving the existence and uniqueness result, it is convenient to introduce the discrete Green s operator approximating the inverse Laplacian with Neumann boundary conditions. Definition Let Ŝ 0 h(ω) := {η S h (Ω) : (χ, 1) h = 0} and let G h : Ŝ 0 h(ω) Ŝ 0 h (Ω) be the linear operator, where for each η Ŝ 0 h(ω) we define Gh η S0 h(ω) such that ( G h η, χ) = (η, χ) h χ Ŝ 0 h (Ω). In addition, we define the norm η h = G h η 1 on Ŝ h 0 (Ω). Observe that η 2 h = ( Gh η, G h η) = (η, G h η) h η Ŝ h 0 (Ω). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

53 Cahn Hilliard Equation ( U n+1 U n h, η) + ( W n+1, η) = 0 t η S h (Ω), ε ( U n+1, χ) + ε 1 (Ψ (U n+1 ), χ) h = (W n+1, χ) h χ S h (Ω). Lemma If t < 4 ε3 C 2 Ψ then there exists a unique solution (U n+1, W n+1 ). Proof. Let Ŝ U h := {η S h (Ω) : (η U n, 1) h = 0}, and consider the variational n problem: Find U Ŝ U h such that n J h (U) J h (χ) χ Ŝ h U n, where J h (χ) = E h (χ) t U Un 2 h. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

54 Cahn Hilliard Equation Proof (Cont.) Then J h : Ŝ U h R n 0 is continuous, bounded from below, and such that J h (U) as U 1. It follows that there exists a solution U to the above variational problem. Hence, for every χ Ŝ 0 h (Ω), the function λ J h (U + λ χ) must have vanishing derivative at λ = 0, i.e. [ ] δ 0 = δu J h (U) (χ) = d dλ J h (U + λ χ) λ=0 = 1 t (Gh (U U n ), χ) h + ε ( U, χ) + ε 1 (Ψ (U), χ) h χ Ŝ h 0 (Ω). On setting W = G h U Un t this implies that... + ε 1 Ψ (U), R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

55 Cahn Hilliard Equation Proof (Cont.) 0 = (W, χ) h + ε ( U, χ) + ε 1 (Ψ (U), χ) h χ Ŝ h 0 (Ω) and 0 = (W, 1) h + ε 1 (Ψ (U), 1) h, 0 = (W, χ) h + ε ( U, χ) + ε 1 (Ψ (U), χ) h χ S h (Ω). Moreover, it holds by definition that ( U U n t h, η) ( G h U ) Un, η = 0 η t Ŝ 0 h (Ω), ( U U n h, η) + ( W, η) = 0 η S h (Ω). t This shows the existence of a solution (U n+1, W n+1 ) = (U, W ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

56 Cahn Hilliard Equation Proof (Cont.) Now, if (U 1, W 1 ) and (U 2, W 2 ) are two solutions, their difference (Ū, W ) = (U 1 U 2, W 1 W 2 ) satisfies Ū Ŝ 0 h (Ω) and (Ū, η) h + t ( W, η) = 0 η S h (Ω), ε ( Ū, χ) + ε 1 (Ψ (U 1 ) Ψ (U 2 ), χ) h = ( W, χ) h χ S h (Ω). Choosing η = W and χ = Ū yields that ε Ū t W 2 1 ε 1 C Ψ (Ū, Ū) h = t ε 1 C Ψ ( W, Ū), where we have recalled (Ψ (r) Ψ (s)) (r s) C Ψ (r s) 2. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

57 Cahn Hilliard Equation Proof (Cont.) Hence it holds that ε Ū t W 2 1 t ε 1 C Ψ ( W, Ū) t ε 1 C Ψ W 1 Ū 1 t W t C 2 Ψ 4 ε 2 Ū 2 1. If t C 2 Ψ 4 ε 2 < ε we obtain that Ū 2 1 = 0, and on recalling that Ū Ŝ h 0 (Ω) this yields that Ū = 0, and then also W = 0. Hence U n+1 S h (Ω) is unique, and so is W n+1 = G h Un+1 U n t + ε 1 Ψ (U n+1 ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

58 Cahn Hilliard Equation Finite element approximations To recap: Fully explicit scheme: stable if ε t C h 4. Fully implicit scheme: unique solution exists and is stable if t < 4 ε3. CΨ 2 Similarly to the Allen Cahn equation, we now introduce an unconditionally stable semi-implicit finite element approximation that is based on the convex/concave splitting Ψ(s) = Ψ + (s) + Ψ (s), where Ψ + C 2 (R) is convex, and where Ψ C 2 (R) is concave. In the context of finite element approximations of the Cahn Hilliard equation, such a splitting was first used in Barrett & Blowey (1997). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

59 Cahn Hilliard Equation Semi-implicit scheme Given U 0 S h (Ω), for n 0 find (U n+1, W n+1 ) S h (Ω) S h (Ω) such that ( U n+1 U n h, η) + ( W n+1, η) = 0 t ε ( U n+1, χ) + ε 1 (Ψ +(U n+1 ) + Ψ (U n ), χ) h = (W n+1, χ) h for all η, χ S h (Ω). Theorem There exists a unique solution (U n+1, W n+1 ) S h (Ω) S h (Ω) to the semi-implicit scheme, and it holds that (U n+1, 1) = (U 0, 1) as well as E h (U n+1 ) + t W n E h (U n ). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

60 Cahn Hilliard Equation Proof. Existence, and uniqueness, of a solution can be shown via a variational argument as before, on considering J h U n(u) = 1 2 ε U 2 1+ε 1 (Ψ + (U), 1) h +ε 1 (Ψ (U n ), U) h t U Un 2 h. In particular, on assuming without loss of generality that Ψ + (s) 0 for all s R, it holds for U Ŝ U h that n J h U n(u) 1 2 ε U ε 1 (Ψ (U n ), U) h 1 2 ε U 2 1 ε 1 Ψ (U n ) h U h 1 2 ε U 2 1 C ε 1 Ψ (U n ) h U ε U 2 1 C(U n ). Hence J h U n is bounded from below on Ŝ h U n, and there exists a minimizer U. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

61 Cahn Hilliard Equation Proof (Cont.) Setting W = G h U Un + ε 1 (Ψ t +(U) + Ψ (U n )) then shows the existence of a solution (U, W ). The difference of two solutions, (Ū, W ) = (U 1 U 2, W 1 W 2 ), now satisfies (Ū, η) h + t ( W, η) = 0 η S h (Ω), ε ( Ū, χ) + ε 1 (Ψ +(U 1 ) Ψ +(U 2 ), χ) h = ( W, χ) h χ S h (Ω). Choosing η = W and χ = Ū yields that ε Ū t W 2 1 0, where we have recalled that (Ψ +(r) Ψ +(s)) (r s) 0. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

62 Cahn Hilliard Equation Proof (Cont.) Choosing η = W n+1 and χ = U n+1 U n yields that ε ( U n+1, (U n+1 U n )) + ε 1 (Ψ +(U n+1 ) + Ψ (U n ), U n+1 U n ) h + t W n = 0. On recalling that r (r s) 1 2 r s2 and that Ψ(r) Ψ(s) Ψ +(r) (r s) + Ψ (s) (r s). we obtain that E h (U n+1 ) E h (U n ) + t W n R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

63 Cahn Hilliard Equation Concentration dependent mobility In the physical context of the Cahn Hilliard equation, it is often of interest to consider a concentration dependent mobility, b(u), where b : R R 0. u t =. (b(u) w), w = ε u + ε 1 Ψ (u) in Ω T, u ν Ω = w ν Ω = 0 on Ω (0, T ), u(, 0) = u 0 in Ω. If 0 b min b(s) b max s R, with b min > 0, then all the stated results still hold. The time step constraints for the fully explicit and fully implicit schemes need to be adapted to respectively. ε t C h4 and t < 4 ε3 b max CΨ 2 b, max R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

64 Cahn Hilliard Equation Degenerate mobility A mobility with b min = 0 is called degenerate mobility, i.e. there exist s R such that b(s) = 0. A typical example is b(s) = [1 s 2 ] + = max{0, 1 s 2 }, which suppresses diffusion in the pure phase {±1}. This poses no additional difficulty for the fully explicit scheme. As the weighted Green s operator Gb h : Ŝ 0 h(ω) Ŝ 0 h (Ω) is no longer bijective, for the implicit schemes a numerical precursor can be considered, e.g. by defining b δ (s) = b(s) + δ, 0 < δ 1. One of the main reasons for a degenerate mobility is the desired sharp interface limit, as ε 0, of surface diffusion. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

65 Cahn Hilliard Equation Sharp interface limit For many years, the degenerate Cahn Hilliard equation ε u t =. ([1 u 2 ] + w), w = ε u + ε 1 (u 3 u), has been used in the literature as a phase field model for surface diffusion: V = s κ on Γ(t), (SD) where s is the Laplace Beltrami operator or surface Laplacian on Γ(t), i.e. s κ = κ ss if d = 2. However, it was recently shown in Lee, Münch & Süli (2016), that (SD) is not the sharp interface limit of the Cahn Hilliard equation above. As a remedy, the authors suggest to consider mobilities with a higher degeneracy, e.g. b(s) = ([1 s 2 ] + ) 2, or to employ an obstacle potential (see later). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

66 Cahn Hilliard Equation Sharp interface limit The sharp interface limit of the standard Cahn Hilliard equation u t = w, w = ε u + ε 1 Ψ (u), (CH) is given by the following Hele Shaw or Mullins Sekerka problem: [ w ν Γ ] + 0 = w in Ω ± (t), = 2 V on Γ(t), (MS) w = 1 2 σ Ψ κ on Γ(t), where Ω ± (t) denote the exterior/interior of Γ(t) in Ω, [ ] + denotes the jump across Γ(t), and where σ Ψ = 1 1 (2 Ψ) 1 2 dl 1. A rigorous convergence result for the zero level sets of the discrete solutions of the fully implicit scheme for (CH) to hypersurface that move by (MS) has been given in Feng & Prohl (2005). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

67 Cahn Hilliard Equation Solution methods Fully explicit scheme The solution is trivial, as the coefficient vector for U m+1 satisfies U n+1 = (Id ε t ( M 1 A) 2 ) U n ε 1 t M 1 A Ψ (U n ). Fully and semi-implicit schemes The solution boils down to finding U n+1 R K such that ( M + ε t A M 1 A) U n+1 + ε 1 t A Ψ (+) (Un+1 ) = b. This system of nonlinear algebraic equations can be solved with: Newton method, nonlinear SOR method, nonlinear multigrid method. R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

68 Cahn Hilliard Equation In many situations, in which Allen Cahn and Cahn Hilliard equations are considered, the only meaningful values for u lie in the interval [ 1, 1]. For discrete approximations of the Allen Cahn and Cahn Hilliard equations with a smooth potential Ψ, however, there is no guarantee that the approximations U n+1 remain inside [ 1, 1]. In fact, in practice it can be observed that U n+1 > 1. If we want to enforce U n+1 to always remain in the interval [ 1, 1], then we have to consider an obstacle potential Ψ, that assigns the value Ψ(s) = if s [ 1, 1]. Another practical advantage of such an obstacle potential is, that the discrete interfacial region is now clearly defined via { U n+1 < 1}, which aids the usage of adaptive meshes, for example. Finally, we had seen that the degenerate Cahn Hilliard equation, with b(s) = [1 s 2 ] +, yields surface diffusion as the sharp interface limit only if an obstacle potential is employed. See also Cahn, Elliott & Novick-Cohen (1996). R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69

69 Cahn Hilliard Equation Finite element approximations Of course, a disadvantage is that Ψ is no longer smooth, and so non-smooth numerical methods have to be employed. In particular, we need to consider variational inequalities. The simplest possible choice for an obstacle potential is Ψ(s) = { 1 2 (1 s2 ) s [ 1, 1], s / [ 1, 1], 1 which was first proposed by Blowey & Elliott (1991). 1 1 R. Nürnberg (Imperial College London) Numerical Approximation of PF Models 2/4 TUM Summer School / 69