Primal-dual active set methods for Allen-Cahn variational inequalities with nonlocal constraints. (English) Zbl 1272.65060
In this outstanding work the authors study the Allen-Cahn model which describes interface motion with applications including materials science, image processing, biology and geology. Here, an interface in which a phase field (or order parameter) rapidly changes its value is modeled to have a thickness of order \(\varepsilon\) where \(\varepsilon > 0\) is a small parameter.
The main goal of this paper is to introduce and analyze a primal-dual active set method for finite element discretization of a (semi-) implicit Euler discretization and respectively, the given variational inequality. The local convergence of the proposed primal-dual active set method is proved. To demonstrate specific features of introduced approach, the authors also discuss elliptic obstacle problems. The method to numerically solve an implicit time discretization of the Allen-Cahn variational inequality with integral constraint with the help of a primal-dual active set method is presented. The authors show efficiency and accuracy for a problem where the explicit solution of the corresponding sharp interface problem is known. Finally, two numerical simulations for the Allen-Cahn variational inequality with integral constraint are presented.
The main goal of this paper is to introduce and analyze a primal-dual active set method for finite element discretization of a (semi-) implicit Euler discretization and respectively, the given variational inequality. The local convergence of the proposed primal-dual active set method is proved. To demonstrate specific features of introduced approach, the authors also discuss elliptic obstacle problems. The method to numerically solve an implicit time discretization of the Allen-Cahn variational inequality with integral constraint with the help of a primal-dual active set method is presented. The authors show efficiency and accuracy for a problem where the explicit solution of the corresponding sharp interface problem is known. Finally, two numerical simulations for the Allen-Cahn variational inequality with integral constraint are presented.
Reviewer: Jan Lovíšek (Bratislava)
MSC:
| 65K15 | Numerical methods for variational inequalities and related problems |
| 49J40 | Variational inequalities |
| 49M37 | Numerical methods based on nonlinear programming |
Keywords:
Allen-Cahn variational inequality; finite element approximation; nonlocal constraints; primal-dual active set methods; semismooth Newton methods; Neumann boundary condition; convergence; numerical examples; implicit Euler discretizationReferences:
| [1] | J. F.Blowey and C. M.Elliott, Curvature dependent phase boundary motion and parabolic double obstacle problems, Degenerate diffusions (Minneapolis, MN, 1991), 19-60, IMA Vol. Math. Appl., Vol. 47, Wei‐Ming Ni (ed.), L.A.Peletier (ed.), and J.L.Vazquez (ed.), editors, Springer, New York, 1993. · Zbl 0794.35092 |
| [2] | J.Taylor and J. W.Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J Stat Phys77 ( 1994), 183-197. · Zbl 0844.35044 |
| [3] | H.Garcke, B.Nestler, B.Stinner, and F.Wendler, Allen‐Cahn systems with volume constraints, M3AS: Math Models Methods Appl Sci18 ( 2008), 1347-1381. · Zbl 1147.49036 |
| [4] | F.Wendler, J. K.Becker, B.Nestler, and P.Bons, Phase‐field simulations of partial melts in geological materials, Comput Geosci35 ( 2009), 1907-1916. |
| [5] | T.Blesgen and U.Weikard, Multi‐component Allen-Cahn equation for elastically stressed solids, Electron J Differential Equations ( 2005), 1-17. · Zbl 1090.35085 |
| [6] | M.Benes, V.Chalupecky, and K.Mikula, Geometrical image segmentation by the Allen‐Cahn equation, Appl Numer Math51 ( 2004), 187-205. · Zbl 1055.94502 |
| [7] | H.Garcke, R.Nürnberg, and V.Styles, Stress‐ and diffusion‐induced interface motion: modelling and numerical simulations, Eur J Appl Math18 ( 2007), 631-657. · Zbl 1128.74004 |
| [8] | D.Italo Capuzzo, S.Finzi Vita, and R.March, Area‐preserving curve‐shortening flows: from phase separation to image processing, Interfaces Free Bound4 ( 2002), 325-343. · Zbl 1021.35129 |
| [9] | L.Bronsard and B.Stoth, Volume‐preserving mean curvature flow as a limit of a nonlocal Ginzburg‐Landau equation, SIAM J Math Anal28 ( 1997), 769-807. · Zbl 0874.35009 |
| [10] | J.Rubinstein and P.Sternberg, Nonlocal reaction diffusion equations and nucleation, IMA J Appl Math48 ( 1992), 249-264. · Zbl 0763.35051 |
| [11] | M.Hintermüller, K.Ito, and K.Kunisch, The primal‐dual active set strategy as a semismooth Newton method, SIAM J Optim13 ( 2003), 865-888. · Zbl 1080.90074 |
| [12] | J. F.Bonnans, R.Bessi Fourati, and H.Smaoui, The obstacle problem for water tanks, J Math Pures Appl82 ( 2003), 1527-1553. · Zbl 1034.49037 |
| [13] | T. B.Benjamin and A. D.Cocker, Liquid drops suspended by soap films. II. Simplified model, Proc R Soc Lond A394 ( 1984), 33-45. · Zbl 0573.76091 |
| [14] | A. D.Cocker, A.Friedman, and B.McLeod, A variational inequality associated with liquid on a soap film, Arch Rational Mech Anal93 ( 1986), 15-44. |
| [15] | J. W.Barrett and C. M.Elliott, Finite element approximation of a free boundary problem arising in the theory of liquid drops and plasma physics, M2AN Math Model Numer Anal25 ( 1991), 213-252. · Zbl 0709.76086 |
| [16] | S.Hildebrandt and M.Meier, On variational problems with obstacles and integral constraints for vector‐valued functions, Manuscripta Math28 ( 1979), 185-206. · Zbl 0432.35002 |
| [17] | G.Eisen, On the obstacle problem with a volume constraint, Manuscripta Math43 ( 1983), 73-83. · Zbl 0524.49005 |
| [18] | K.Deckelnick, G.Dziuk, and C. M.Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer14 ( 2005), 139-232. · Zbl 1113.65097 |
| [19] | R.Kornhuber and R.Krause, On multigrid methods for vector‐valued Allen‐Cahn equations with obstacle potential, Domain decomposition methods in science and engineering, 307314 (electronic), National Autonomous University of Mexico, Mexico, 2003. |
| [20] | L.Baňas and R.Nürnberg, A multigrid method for the Cahn‐Hilliard equation with obstacle potential, Appl Math Comput213 ( 2009), 290-303. · Zbl 1168.65386 |
| [21] | J. W.Barrett and J. F.Blowey, An error bound for the finite element approximation of a model for phase separation of a multi���component alloy, IMA J Numer Anal16 ( 1996), 257-287. · Zbl 0849.65069 |
| [22] | J. F.Blowey and C. M.Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I, Mathematical analysis. Eur J Appl Math2 ( 1991), 233-280. · Zbl 0797.35172 |
| [23] | L. C.Evans, Partial differential equations, Graduate studies in mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002 |
| [24] | D.Gilbarg and N. S.Trudinger, Elliptic partial differential equations of second order, Springer‐Verlag, Berlin, 2001. · Zbl 1042.35002 |
| [25] | K.Ito and K.Kunisch, Semi‐smooth Newton methods for variational inequalities of the first kind, M2AN Math Model Numer Anal37 ( 2003), 41-62. · Zbl 1027.49007 |
| [26] | X.Chen, Z.Nashed, and L.Qi, Smoothing methods and semismooth methods for non‐differentiable operator equations, SIAM J Numer Anal2000 ( 38), 1200-1216. · Zbl 0979.65046 |
| [27] | F.Kikuchi, K.Nakazato, and T.Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria, Jpn J Appl Math1 ( 1984), 369-403. · Zbl 0634.76117 |
| [28] | H.Garcke, K.Ito, and Y.Kohsaka, Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions, SIAM J Math Anal36 ( 2005), 1031-1056. · Zbl 1097.35030 |
| [29] | C. M.Elliott and J. R.Ockendon, Weak and variational methods for moving boundary problems, Research notes in mathematics, Vol. 59, Pitman, Boston, MA, 1982. · Zbl 0476.35080 |
| [30] | M.Bergounioux and K.Kunisch, Primal‐dual active set strategy for state constrained optimal control problems, Comput Optim Appl22 ( 2002), 193-224. · Zbl 1015.49026 |
| [31] | C.Meyer, A.Rösch, and F.Tröltzsch, Optimal control problems of PDEs with regularized pointwise state constraints, Comput Optim Appl33 ( 2006), 209-228. · Zbl 1103.90072 |
| [32] | I.Neitzel and F.Tröltzsch, On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints, ESAIM: Control Optim Calc Var15 ( 2009), 426-453. · Zbl 1171.49017 |
| [33] | C. M.Elliott, Approximation of curvature dependent interface motion, State of the art in Numerical Analysis, IMA Conference Proceedings, Vol. 63, Clarendon Press, Oxford, 1997, pp. 407-440. · Zbl 0881.65131 |
| [34] | C. M.Elliott and V.Styles, Computations of bi‐directional grain boundary dynamics in thin films, J Comput Phys187 ( 2003), 524-543. · Zbl 1020.82006 |
| [35] | A.Schmidt and K. G.Siebert, Design of adaptive finite element software. The finite element toolbox ALBERTA, Lecture notes in computational science and engineering, Vol. 42, Springer‐Verlag, Berlin, 2005, pp. xii+315. · Zbl 1068.65138 |
| [36] | J. W. Barrett, R.Nürnberg, and V.Styles, Finite element approximation of a phase field model for void electromigration, SIAM J Numer Anal46 ( 2004), 738-772. · Zbl 1076.78012 |
| [37] | L.Blank, M.Butz, and H.Garcke, Solving the Cahn-Hilliard variational inequality with a semi‐smooth Newton method, EASIM: Control Optim Calc Var17 ( 2011), 931-954. · Zbl 1233.35132 |
| [38] | D.Kinderlehrer and G.Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, Vol. 88, Academic Press, Inc., New York/London, 1980, pp. xx+313. · Zbl 0988.49003 |
| [39] | R. H.Nochetto, M.Paolini, and C.Verdi, A dynamic mesh algorithm for curvature dependent evolving interfaces, J Comput Phys123 ( 1996), 296-310. · Zbl 0851.65067 |
| [40] | B.Bourdin and A.Chambolle, Design‐dependent loads in topology optimization, ESAIM Control Optim Calc Var9 ( 2003), 19-48, (electronic). · Zbl 1066.49029 |
| [41] | M.Burger and R.Stainko, Phase‐field relaxation of topology optimization with local stress constraints, SIAM J Control Optim45 ( 2006), 1447-1466, (electronic). · Zbl 1116.74053 |
| [42] | S.Zhou and M. Y.Wang, Multimaterial structural topology optimization with a generalized Cahn‐Hilliard model of multiphase transition, Struct Multidisc Optim33 ( 2007), 89-111. · Zbl 1245.74077 |
| [43] | M. E.Gurtin, The linear theory of elasticity, Handbuch der Physik, Band VIa/2: Festkörpermechanik II, C.Truesdell (ed.), editor, Springer‐Verlag, Berlin, 1972. · Zbl 0277.73001 |
| [44] | J. L.Lions, Optimal control of systems goverend by partial differential equations, Springer‐Verlag, Berlin, 1971. · Zbl 0203.09001 |
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