On multigrid methods for the CahnHilliard equation with obstacle potential

Transkript

1 On multigrid methods for the CahnHilliard equation with obstacle potential ubomír Ba as Department of Mathematics Imperial College London Joint work with Robert Nürnberg

2 Overview 1 1. Introduction 2. Continuous model 3. Numerical Method 4. Solvers for the discrete system 5. Numerical experiments ubomír Ba as Harrachov /39

3 Introduction 2 Evolution of surfaces applications in material science (microstructure prediction, material proterties, void electromigration in semiconductors), image processing, etc. Overview Deckekelnick, Dzuik, Elliott (2005) ubomír Ba as Harrachov /39

4 Introduction 3 Surface diusion sharp interface model V = s κ on Γ(t) Γ(t) void surface s surface Laplacian V velocity of Γ(t) κ curvature ubomír Ba as Harrachov /39

5 Phase-eld model Diuse interface with interface width γπ 4 Alternatives to phase-eld approach Direct methods for approximation of the surface diusion model, topological changes problems with Level set methods can handle topological changes ubomír Ba as Harrachov /39

6 Phase-eld model 5 γ > 0 interfacial parameter u γ (, t) K := [ 1, 1], t [0, T ] conserved order parameter; u γ (, t) = 1 void, u γ (, t) = 1 conductor w γ (, t) chemical potential φ γ (, t) electric potential Phase eld approximation of surface diusion (diuse interface) γ u γ t.( b(u γ) w γ ) = 0 in Ω T := Ω (0, T ], w γ = γ u γ + γ 1 Ψ (u γ ) in Ω T, where u γ < 1, + I.C. + B.C. ubomír Ba as Harrachov /39

7 Degenerate coecients b(s) := 1 s 2, s K Obstacle-free energy Phase-eld model 6 { 1 ( ) Ψ(s) := 2 1 s 2 if s K, if s K, restricts u γ (, ) K. Approximation of the sharp interface model γ 0 then {x; u γ (x, t) = 0} Γ(t), Γ(t) is the solution of the sharp interface problem Advantages of phase-eld approach no explicit tracking of the interface needed can handle topological changes ubomír Ba as Harrachov /39

8 Numerical approximation 7 U n W n S h u γ K h w γ Φ n S h φ γ Double obstacle formulation ( regularisation parameter) γ ( U n U n 1 τ n ) h, χ + (Ξ (U n 1 ) W n, χ) = 0 χ S h, γ ( U n, [χ U n ]) (W n + γ 1 U n 1, χ U n ) h χ K h, discrete inner product (mass lumping) (η 1, η 2 ) h := Ω πh (η 1 (x) η 2 (x)) dx Ξ ( ) b( ) Convergence (Existence) 2D: Barrett, Nürnberg, Styles (2004), 3D: Ba as, Nürnberg (2006) h 0, 0, τ = O(h 2 ) ubomír Ba as Harrachov /39

9 Matrix formulation 8 The discrete system Find {U n, W n } K J R J such that γ (V U n ) T B U n (V U n ) T M W n (V U n ) T s V K J, γ M U n + τ n A n 1 W n = r M ij := (χ i, χ j ) h, B ij := ( χ i, χ j ), A n 1 ij := (Ξ (U n 1 ) χ i, χ j ) r := γ M U n 1 α τ n A n 1 Φ n R J, s := γ 1 M U n 1 R J. ubomír Ba as Harrachov /39

10 Block Gauss-Seidel algorithm with projection 9 Projected block Gauss-Seidel (V U n,k ) T (γ (B D B L ) U n,k γ M U n,k 2 2 system for every vertex; explicit solution [ [ M W n,k ) (V U n,k ) T (s + γ BL T U n,k 1 + τ n (A D A L ) W n,k = r + τ n A T L W n,k 1 U n,k ] W n,k j ] j = [ Mjj r j + τ n A n 1 jj γ [M jj ] 2 + τ n γ A n 1 jj = r j γ M jj [U n,k τ n A n 1 jj ] j ŝ j B jj ] K ) ubomír Ba as Harrachov /39

11 Uzawa algorithm 10 Uzawa-Multigrid algorithm Gräser, Kornhuber (2005), derived from the formulation of Blowey, Elliott (1991, 1992), Outer Uzawa-type iterations constrained minimisation, two sub-steps γ (V U n,k ) T B U n,k W n,k (V U n,k ( = W n,k 1 + S 1 γ M U n,k ) T s + (V U n,k ) T M W n,k 1 τ n A n 1 W n,k 1 ) + r V K J S 1 - preconditioner ubomír Ba as Harrachov /39

12 Uzawa algorithm Preconditioner If we know the exact coincidence/contact set Ĵ(U n ) = { j J : } [U n ] j = 1, 11 the problem becomes linear ( γ n B(U ) M(U n ) γ M τ n A n 1 ) ( U n W n ) = ( ŝ(u n ) r ). with and B ij = M ij = ŝ i = { δij i Ĵ B ij else { 0 i Ĵ M ij else { γ [U n ] i i Ĵ s i,, j J, else. ubomír Ba as Harrachov /39

13 Uzawa algorithm 12 Optimal choice Schur complement S(U n ) = M B(U n ) 1 M(U n ) + τ n A n 1 Approximation U n,k U n S = S(U n,k ) = M B(U n,k ) 1 n,k M(U ) + τ n A n 1 Uzawa with the preconditioner S(U k ) γ (V U n,k ) T B U n,k (V U n,k ( W n,k = S(U n,k ) 1 M ) T s + (V U n,k B(U n,k ) T M W n,k 1 ) V K J, ) + r. ) 1 ŝ(u n,k ubomír Ba as Harrachov /39

14 Uzawa algorithm 13 Solution of the subproblems rst step, elliptic variational inequality with double obstacle, we can use standard methods: projected Gauss-Seidel or Monotone multigrid; iterations can be stopped when we obtain convergence in the coincidence step - only few iterations. input W n,k 1, output U n,k second step is equivalent to the solution of linear symmetric saddle point problem ( γ γ 2 B γ n,k M(U n,k M(U ) τ n A n 1 ) ) ( Ũ k W n,k ) = ( γ s r ) standard W -cycle multigrid method for saddle point problems (Stokes equations, mixed FEM), canonical restriction and prolongation, block Gauss-Seidel smoother (1 smoothing step), alternative (Vanka type (1986)) smoother Schröberl, Zulehner (2003). input Ĵ k n,k = Ĵ(U ), output W n,k ubomír Ba as Harrachov /39

15 Uzawa algorithm 14 Numerical implementation (more natural, but no proof of convergence) Ξ (U n 1 ) π h [b(u n 1 )], π[b(u n 1 )] 0 on J deg A n 1 - has zero rows due to the degeneracy of b( ) Degenerate set j J deg := { j J : π [ h b(u n 1 ) ] 0 on supp(χ j ) } Solution We use the fact (J deg Ĵ k ) U n,k j = U n 1 j saddle point problem with regular matrix A 1 deg ( γ 2 B γ M γ M τ n A n 1 deg for all j J deg We obtain an equivalent ) ( Ũ k W k ) = ( γ s r deg ), where W j = W n 1 j for j J deg. ubomír Ba as Harrachov /39

16 Complete algorithm Uzawa algorithm Initialization: Start with inital guess U n,0 = U n 1, set Ĵ 0 = W n,0 by solving the linear saddle point problem with coincidence set Ĵ Uzawa iterations: for k = 1,... do n,0 Ĵ(U ) and compute Ĵ(U n,k 1st sub-step Compute the approximate coincidence set Ĵ k = is obtained from the elliptic variational inequality by PGS or MMG. If Ĵ k = Ĵ k 1 go to step 3. ), where U n,k 2nd sub-step Solve a linear symmetric saddle point problem by the multigrid method with block GaussSeidel smoother to obtain W n,k [ If max j J to step 3. W n,k ] j [ W n,k 1 ] j. < tol, with tol being the prescribed tolerance, go 3. Uzawa iterations have converged: Compute U n,k+1 up to the desired accuracy from the elliptic variational inequality from the 1st sub-step using W n,k 4. Set U n = U n,k+1, W n = W n,k.. ubomír Ba as Harrachov /39

17 Numerical experiments 16 FEM code Alberta; adaptive meshes: if U n < 1 (i.e. in the interfacial region) set h = h min 1 N f h = h max = 1 N. c Comparison of Uzawa and Gauss-Seidel methods γ = 1 12π else ( U n = 1) set N f τ GS Uzawa-MG ratio 128 1e m 3445m e-6 252m 146m e-5 9m40s 11m20s 0.85 Table 1: Computation times for dierent values of h γ N f τ GS Uzawa-MG ratio 1/12π 128 1e m 3445m /6π 64 4e-6 853m 259m /3π 32 1.e-5 93m 30m 3.1 Table 2: Computational times for dierent values of γ ubomír Ba as Harrachov /39

18 Multigrid algorithm - notation 17 Problem matrix on ne mesh ( B M A = M A Saddle point problem with variational inequality ( ) ( ) U r A W s ) U K J Intergrid transfer operators (canonical restriction and prolongation) I c f, If c Coarse matrix A c A c = ( I c f B I f c I c f M If c ) I c f M If c I c f A If c ubomír Ba as Harrachov /39

19 Multigrid algorithm 18 Two-grid scheme for the solution of ( U A W ) ( r s ) U K J pre smoothing m iteration of projected Gauss-Seidel (U 0, W 0 ) (U m, W m ) coarse grid correction solve the coarse problem exactly 1. compute residual (Q u, Q w ) = (r, s) A(U m, W m ) 2. ( ) ( V u I c A c V w f Q u ) If c V u K J c Qw 3. update solution (U m+1, W m+1 ) = (U m + I f c V u, W m + I f c V w ) post smoothing m iteration of projected Gauss-Seidel (U m+1, W m+1 ) (U 2m+1, W 2m+1 ) ubomír Ba as Harrachov /39

20 Coarse grid correction 19 We require U m+1 + I f c V u 1 New obstacle for V u 1 U m+1 I f c V u 1 U m+1 associated with the ne mesh, but to compute V u on the coarse grid we need a coarse obstacle. Solution Mandel (1984) look for V u K J c { where } K J c V R J c ; Q c f ( 1 U m+1 ) V Rf c ( 1 U m+1 ) = obstacle restriction operators dened as with upper/lower ] [Q c f v (p) = max { } v(q); q N f int supp χ p, χ p Vh c, ] [Rf c v (p) = min { } v(q); q N f int supp χ p, χ p Vh c, with p N k 1, v V f h. Kornhuber (1994) slightly better obstacle restriction (suitable linear combination instead of min/max). ubomír Ba as Harrachov /39

21 Coarse grid correction - mltiple grids 20 Restrictions for numerical convergence grid on the lowest has to be ne enough; number of grid levels depends on γ and h min, we need: small γ for good approximation of the sharp interface problem small h min (depending on γ) for good approximation of the continuous phase-eld model Feasible parameter combnations in 3D γ = 1 12π, N f = 1 128, 6 mesh points in the interface, 2-level method γ = 1 9π, N f = 1 128, 8 mesh points in the interface, 3-level method Coarse grid solver inexact solution with projected GS, 30 iteration are enough for the 3-level method. ubomír Ba as Harrachov /39

22 Numerical experiments 21 Meshes on dierent levels ubomír Ba as Harrachov /39

23 Comparison of the Multigrid and Uzawa methods 22 3D computation with about degrees of freedom, γ = 1 9π method total iteration CPU time MG m Uzawa-MG m Table 3: 3-level MG vs. Uzawa (8-level); W-cycle, 1 smoothing step Multigrid computations are about 1.5 times faster with 2 less iterations. ubomír Ba as Harrachov /39

24 Numerical experiments γ = 1 12π, T = 0.06, τ = 10 6, N f = 128, N c = 2 23 Figure 1: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, 0.001, ubomír Ba as Harrachov /39

25 Numerical experiments γ = 1 12π, T = 0.06, τ = 10 6, N f = 128, N c = 2 24 Figure 2: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0.01, 0.015, T = ubomír Ba as Harrachov /39

26 Numerical experiments γ = 1 12π, T = 0.08, τ = , N f = 48, N c = 2 25 Figure 3: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, 0.001, ubomír Ba as Harrachov /39

27 Numerical experiments γ = 1 12π, T = 0.08, τ = , N f = 48, N c = 2 26 Figure 4: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0.01, 0.015, T = ubomír Ba as Harrachov /39

28 Numerical experiments γ = 1 12π, T = 0.06, τ = , N f = 128, N c = 2 27 Figure 5: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, , ubomír Ba as Harrachov /39

29 Numerical experiments γ = 1 12π, T = 0.06, τ = , N f = 128, N c = 2 28 Figure 6: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = , , T = ubomír Ba as Harrachov /39

30 Numerical experiments γ = 1 12π, T = 0.001,τ = 10 5, N f = 64, N c = 2 29 Figure 7: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, , ubomír Ba as Harrachov /39

31 Numerical experiments γ = 1 12π, T = 0.001,τ = 10 5, N f = 64, N c = 2 30 Figure 8: (α = 0) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = , , ubomír Ba as Harrachov /39

32 Numerical experiments α = 114, γ = 1 12π, T = , τ = , N f = 128, N c = Figure 9: (α = 114π) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, , ubomír Ba as Harrachov /39

33 Numerical experiments α = 114, γ = 1 12π, T = , τ = , N f = 128, N c = Figure 10: (α = 114π) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = , , ubomír Ba as Harrachov /39

34 Numerical experiments α = 114, γ = 1 12π, T = , τ = 10 7, N f = 128, N c = Figure 11: (α = 114π) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, , ubomír Ba as Harrachov /39

35 Numerical experiments α = 114, γ = 1 12π, T = , τ = 10 7, N f = 128, N c = Figure 12: (α = 114π) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = , , ubomír Ba as Harrachov /39

36 Numerical experiments α = 300, γ = 1 12π, T = , τ = 10 7, N f = 128, N c = Figure 13: (α = 300π) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = 0, , ubomír Ba as Harrachov /39

37 Numerical experiments α = 300, γ = 1 12π, T = , τ = 10 7, N f = 128, N c = Figure 14: (α = 300π) Zero level sets for U (x, t), with cut through the mesh at x 3 = 0 at times t = , , T = ubomír Ba as Harrachov /39

38 Numerical experiments α = 120, γ = 1 12π, T = , τ = 10 7, N f = 128, N c = Figure 15: (α = 120π) Zero level sets for U (x, t) at times t = 0, , ubomír Ba as Harrachov /39

39 Numerical experiments α = 120, γ = 1 12π, T = , τ = 10 7, N f = 128, N c = Figure 16: (α = 120π) Zero level sets for U (x, t) at times t = , , T = ubomír Ba as Harrachov /39

40 Final remarks 39 fast coarse solver needed for eciency limited exibility with respect to γ robust except above remarks 2 less iterations than Uzawa theory? ubomír Ba as Harrachov /39