NKS Method for the Implicit Solution of a Coupled Allen-Cahn/Cahn-Hilliard System ∗ Chao Yang 1 , Xiao-Chuan Cai 2 , David E. Keyes 3 , and Michael Pernice 4 1 Coupled Allen-Cahn/Cahn-Hilliard system Coupled Allen-Cahn/Cahn-Hilliard (AC/CH) systems, often found in phase-field simulations, are prototype systems that admit simultaneous ordering and phase sep- aration. Numerical methods to solve coupled AC/CH systems are studied in e.g., [2, 6, 8, 9, 10, 11]. However, except for [9] and [10], the above works are based on explicit methods that require very small time step size to advance the solution and need many time steps for long time integrations. Fully implicit methods enjoy an advantage that the stability limit on the time step size is greatly relaxed. The purpose of this paper is to study efficient and scalable algorithms based on domain decomposition methods for the fully implicit solution of a coupled AC/CH system. There are several different ways to couple the AC and the CH equations. Among them we restrict our study to the original form introduced in [3], which is        ∂ u ∂ t = ∇ · c(u, v)∇ δ E (u, v) δ u , ∂ v ∂ t = − c(u, v) ρ δ E (u, v) δ v . (1) where u and v are functions of x ∈ Ω ⊂ R 2 and t ∈ [0, +∞). Both u and v are bounded with restrictions: u ∈ [0, 1], v ∈ [−1/2, 1/2] and (u ± v) ∈ [0, 1]. Here the first equation in (1) is the Cahn-Hilliard equation in which u represents a conserved concentration field for the phase separation; the second equation in (1) is the Allen- Cahn equation where v denotes a non-conserved order parameter for the anti-phase coarsening. In (1), the mobility c(u, v)= u(1 − u)(1/4 − v 2 ) is degenerate at pure phases and the density ρ is a positive constant. The free energy functional E (u, v) reads 1 Chao Yang, Institute of Software, Chinese Academy of Sciences, Beijing 100190, and State Key Laboratory of High Performance Computing, Changsha 410073, China, e-mail: yangchao@ iscas.ac.cn · 2 Xiao-Chuan Cai, Department of Computer Science, University of Colorado Boulder, Boulder, CO 80309, USA, e-mail: cai@cs.colorado.edu · 3 David E. Keyes, CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia, e-mail: david.keyes@kaust.edu.sa · 4 Michael Pernice, Idaho National Labora- tory, Idaho Falls, ID 83415, USA e-mail: michael.pernice@inl.gov ∗ This work was supported in part by DE-FC02-06ER25784. The first author also received sup- ports from NSFC under 61170075, 91130023 and 61120106005, and from 973 Program of China under 2011CB309701. 1