On Multigrid for Phase Field

On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004

Synopsis adaptivity in time and space fast, robust solution of discrete spatial problems vector-valued Allen Cahn equations (Kh. & Krause) obstacle potential: monotone multigrid (MMG) by successive 1-D minimization logarithmic potential: robustness by constrained Newton linearization numerical experiments diffusion induced grain boundary motion (Kh., Krause & Styles) phase field models multigrid algorithms for the phase field and the solute Cahn Hilliard equation with obstacle potential (Gräser & Kh.) minimization, saddle point formulation and Uzawa-type iteration numerical experiments

Ginzburg Landau Approach Ginzburg Landau free energy: E(u) = phase field: u [ 1, +1] Ω ε u 2 + 1 ψ(u) dx ε ψ T 500 400 logarithmic free energy: 300 200 100 0 100 200 300 ψ T (u) = 1 2T((1 u) ln(1 u 2 ) + (1 + u) ln( 1+u 2 )) + 1 2 T c(1 u 2 ) temperature T, critical temperature T c 400 500 1.5 1 0.5 0 0.5 1 1.5 u deep quench limit: T 0 ψ T ψ 0 robustness of numerical solvers for T 0: no classical Newton linearization

Phase Field Models isothermal case: T = const. phase transition: Allen-Cahn equation (non-conserving) εu t = d du E(u) = ε u 1 ε ψ (u) (Cahn 60, Allen & Cahn 77) phase separation: Cahn-Hilliard equations (conserving) εu t = w, w = d du E(u) = ε u + 1 ε ψ (u) (Cahn & Hilliard 58) Lyapunov functional: d dt E(u(t)) 0 deep quench limit: complementarity problems, variational inequalities

Phase Transition with N Phases concentrations: u1,...,un vector-valued phase field: u = (u1,...,un) R N Gibbs simplex: u(x, t) G = {v R N 0 vi, N i=1 vi = 1} (closed, convex) examples: N=2: u 1 u 2 G N=3: u 1 u 3 u 2 G

Vector-valued Allen-Cahn Equation: Obstacle Potential Ginzburg-Landau free energy: E(u) = Ω ε N 2 i=1 u i 2 + 1 εψ(u) dx, ε > 0 obstacle potential: ψ = ψ 0 = T c N 2 N i=1 u i(1 u i ) u G parabolic variational inequality (Bronsard & Reitich 93, Garcke et al. 98, 99a, 99b): u(t) G : Gibbs constraints: G = ε(u t, v u) + ε( u, (v u)) T cn ε (u,v u) 0 v {v ( H 1 (Ω) ) N v(x) G a.e. in Ω } G special case N = 2: scalar free energy for u := u 1 u 2

Discretization semi-implicit Euler discretization (concave part explicit): stepsize τ (unconditionally stable) linear finite elements S N j : triangulation T j, meshsize h j = O(2 j ), nodes N j, nodal basis λ (j) p discrete spatial problems: u k j G j : u k j, v u k j + τ( u k j, (v u k j)) }{{} a(u k j,v uk j ) τ 1 + T c N ε 2uk 1 j, v u k j }{{} l(v u k j ) k uj G j discrete Gibbs constraints: G j = { v S N j v(p) G p N j }

Equivalent Formulation: Convex Minimization u j G j : J (u j ) J (v) v G j quadratic energy: J (v) = 1 2a(v, w) l(v) s.p.d. bilinear form: a(v, w) = v, w + τ( v, w) linear functional: l(v) = 1 + T c N τ u k 1 ε 2 j,v closed convex set: G j = { v S N j v(p) G p N j } Sj iterative solution by descent methods

Polygonal Gauß-Seidel Relaxation (Kh. & Krause 02) descent directions: λ (j) p E m, p N j, m = 1,...,M edge vectors of G: E 1,...,E M R N, M := N(N 1) 2 = O(N 2 ) E 2 E 3 G hyperplane: H j = span{λ (j) p E m p N j,m = 1,...,M} E 1 polygonal splitting: H j = p N j M m=1 V (j) p,m, V (j) p,m = span{λ (j) p E m } complexity: O(N 2 n j ) successive minimization of J = J + χ Gj on 1-D subspaces V (j) p,m for p N j and m = 1,...,M

convergence of subsequence: u ν k j Convergence u = u pλ (j) p G j u is fixed point: a(u,λ (j) p η i ) l(λ (j) p η i ) η i span{e i }, u p + η i G Lemma (Kh., Krause & Ziegler) u p, v p G there is a decomposition N 1 v p u p = η i, η i span{e mi }, u p + η i G i=1 v p u p u = u j is the solution: a(u,λ (j) p (v p u p)) l(λ (j) p (v p u p)) v p G a(u,v u ) l(v u ) v G j

A Variant: Block Gauß-Seidel nodal splitting: H j = p N j V p, V p = {λ (j) p E E span{e 1,...,E M }} successive minimization of J + χ Gj on V p : convergence proof similar (but simpler) solution of (N 1) D subproblems complicated for large N typical drawback of any Gauß-Seidel approach: convergence speed might deteriorate exponentially (i.e. ρ j = 1 O(2 j ))

Multilevel Version of Polygonal Gauß-Seidel adaptive refinement: T 0 T 1 T j nodes: N 0 N 1 N j nested finite element spaces: S 0 S 1 S j, λ (j 2) q S k = span{λ (k) p p N k } λ (j) p q p polygonal multilevel splitting: H j = j k=0 p N k M m=1 V (k) p,m V (k) p,m = span{λ (k) p E m }

Polygonal Monotone Multigrid (Kh. & Krause 02) sucessive minimization of J = J + χ Gj on subspaces V (k) p,m for p N k, m = 1,...,M and k = j,j 1,...,0 k = j polygonal Gauß-Seidel relaxation on S N j : fine grid smoother: M j (convergent descent method) k < j coarse grid correction: C j (monotone: J (C j w)) J (C j w)) technical modifications (monotone restriction, truncation) monotone multigrid: u ν+1 j = C j ū ν j ū ν j = M ju ν j properties: implementation as multigrid V -cycle: complexity O(N 2 n j ) globally convergent: u ν j u j ν asymptotic multigrid convergence rates for N = 2

logarithmic potential: Logarithmic Potential ψ T (u) = Tφ(u) + φ 0 (u) convex part: φ(u) = N i=1 u i ln(u i ) concave part: φ 0 (u) = T c N 2 N i=1 u i(1 u i ) ψ T has N distinct minima for 0 T< T c spatial problems: u j G j : a(u j,v) + τ ε 2 T φ (u j ),v = l(v) v H j minimization of convex energy J T (v) = 1 2 a(v, v) + τ ε 2 T φ(v), 1 l(v): u j G j : J T (u j ) J T (v) v G j family of problems: T > 0 logarithmic potential, T = 0 obstacle potential robustness: family of convergent algorithms T 0

What can we do? nonlinear polygonal Gauß-Seidel: robust but slow multilevel version: too expensive (no representation of φ on S k, k < j) Newton s method: not robust (not applicable for T = 0) 500 500 400 400 constrained Newton linearization: 300 200 100 300 200 100 0 0 100 100 200 200 300 300 400 400 500 1.5 1 0.5 0 0.5 1 1.5 small Newton corrections 500 1.5 1 0.5 0 0.5 1 1.5 implicit local damping

Robust Multigrid for the Logarithmic Potential (Kh. & Krause 04) given iterate u ν j fine grid smoothing: M j (globally convergent descent method) nonlinear polygonal Gauß-Seidel iteration: smoothed iterate ū ν j = M ju ν j coarse grid correction: C j (monotone) Newton linearization of φ at ū ν j constrain corrections to smooth regime G u ν j of φ 1 step of polygonal MMG with local damping G Gūν j (p) ū ν j (p) new iterate u ν+1 j = C j ū ν j family of multigrid algorithms T 0 : V -cycle with complexity O(N 2 n j ), (asymptotic) multigrid convergence

Numerical Experiments: The Scalar Case N = 2 time discretization: backward Euler parameter: ε = 0.005, T = 1 10 T c, T c = 1, τ = 1 200 ε2, j = 7 (h j = 1/256 ε/2) initial condition: u(x, 0) 1, random evolution: t = 100τ t = 500τ t = 2000τ t = 3000τ

Robust Convergence parameter: ε = 0.005, T c = 1, τ = 1 2 ε2 initial condition: u(x, 0) 1, random algorithm: multigrid V (1, 1)-cycle with nested iteration, uniform refinement various time steps (T = 0.001, h j = 1/256) meshsize: h j 0 (T = 0.001, 1. time step) temperature T 0 (1. time step, h j = 1/256) 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 100 200 300 400 time steps k 0 2 4 6 8 10 refinement level j 10 0 10 1 10 2 10 3 inverse temperature 1/T

N = 5 Phases in 2D parameter: logarithmic potential ψ T, 0 T T c = 1, ε = 0.08 semi-implicit Euler method: 1. time step, τ 0 = ε 2 /80, linear finite elements: uniform refinement j = 6 (h j ε/5) multigrid algorithm: truncated MMG V (3, 3)-cycle with nested iteration time step τ (T = 0.0, h j = 1/64) mesh size: h j 0 (T = 0.0, τ = τ 0 ) temperature T 0 (τ = τ 0 h j = 1/128) 1 0.8 0.6 0.4 0.2 0 10 4 10 2 10 0 time step size τ 1 0.8 0.6 0.4 0.2 0 0 2 4 6 refinement level j 1 0.8 0.6 0.4 0.2 0 10 0 10 1 10 2 10 3 inverse temperature 1/T

N = 3 Phases in 3D parameter: logarithmic potential ψ 0, T c = 1, ε = 0.16 semi-implicit Euler method: 1. time step, τ 0 = ε 2 /2, linear finite elements: uniform refinement j = 5 (h j ε/5) multigrid algorithm: truncated MMG V (3, 3)-cycle with nested iteration timesteps: τ k (T = 0.0, h j = 1/32) meshsize: h j 0 (T = 0.0, τ = τ 0 ) temperature T 0 (τ = τ 0, h j = 1/32) 1 0.8 0.6 0.4 0.2 0 10 4 10 2 10 0 time step size τ 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 refinement level j 1 0.8 0.6 0.4 0.2 0 10 0 10 1 10 2 10 3 inverse temperature 1/T

Conclusion fast, robust solver for arbitrary N and 2 or 3 space dimensions convergence slows down for T > 0 and increasing τ: local damping strategy too pessimistic Current Work anisotropic interfacial energy (cf. Garcke, Nestler & Stoth 99) adaptivity in time and space heuristic refinement indicators, a posteriori error estimates (cf. Kessler, Nochetto & Schmidt 03)

Diffusion Induced Grain Boundary Motion (Garcke & Styles 04) un-alloyed and alloyed grains: ϕ i and ϕ i+n, i = 1,...,N order parameter: ϕ = (ϕ 1,...,ϕ 2N ), free energy with obstacle potential: E(ϕ,u) = ( 1 Ω 2ε u2 + ε N 2 i=1 (ϕ i + ϕ i+n ) 2 + 1 2 ψ 0(ϕ) + g(u) ) N i=1 ϕ i dx alloying solute: u semi-linear parabolic equation: u t (D(ϕ) u) = q(u) grain 3 grain 2 degenerate diffusion: D(ϕ) = 2N i=1 (1 ϕ i) grain 1

Multigrid for the Phase Field variational inequality: ϕ(t) G : ε(ϕ t, η ϕ) + ε( ϕ, ( ϕ ṽ)) ( 1 εϕ + g(u)e, ϕ ṽ) v G ṽ = (v i + v i+n ) N i=1 RN, e = (1,...,1,0,...,0) R 2N semi-implicit Euler method, finite elements: ϕ k+1 j, v ϕ k+1 j + τ( ϕ k+1 j, (ṽ ϕ k+1 j )) }{{} a(ϕ k+1 j,v ϕ k+1 j ) ( τ + 1)ϕ k ε 2 j + τ ε g(ϕk j )e, v ϕk+1 j }{{} l(v ϕ k+1 j ) fast solution by monotone multigrid complexity: O((2N) 2 n j ), alternative model (Elliott & Styles 03): N phases

Multigrid for the Solute semi-implicit Euler method, finite elements: u k+1 j S j : ϕ k+1,v + τ(d(ϕ k j ) uk+1 j, v) = u k j, v + τ(q(uk j ), v) v S j j observation: u k+1 j (p) = u k j (p) p N j = {p N j D(ϕ k j ) suppλ p (j) 0} reduced solution space: S j = {v Ω v S j} Ω = Ω \ Ω, Ω = p N j suppλ (j) p reduced Neumann problem: u k+1 j S j : b(uk+1 j,v) = f(v) v S j no representation of Ω on coarse grids: truncated multigrid (Kh. & Yserentant 94)

Current work numerical results comparison of thin/thick film models with 3D realistic geometries improved models effects of continuum mechanics