2016/01/13 by Pawan Kumar, Kumar, Pawan
Computer Science · Engineering · Materials Science · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Aluminum Alloy Microstructure Properties #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Quasicrystal Structures and Properties #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1601.03230
openalex publication_date 2016/01/13 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28
The phase separation processes are typically modeled by Cahn-Hilliard\nequations. This equation was originally introduced to model phase separation in\nbinary alloys, where phase stands for concentration of different components in\nalloy. When the binary alloy under preparation is subjected to a rapid\nreduction in temperature below a critical temperature, it has been\nexperimentally observed that the concentration changes from a mixed state to a\nvisibly distinct spatially separated two phase for binary alloy. This rapid\nreduction in the temperature, the so-called "deep quench limit", is modeled\neffectively by obstacle potential. The discretization of Cahn-Hilliard equation\nwith obstacle potential leads to a block 2 \× 2 em non-linear system,\nwhere the (1,1) block has a non-linear and non-smooth term. Recently a\nglobally convergent Newton Schur method was proposed for the non-linear Schur\ncomplement corresponding to this non-linear system. The proposed method is\nsimilar to an inexact active set method in the sense that the active sets are\nfirst approximately identified by solving a quadratic obstacle problem\ncorresponding to the (1,1) block of the block 2 \× 2 system, and later\nsolving a reduced linear system by annihilating the rows and columns\ncorresponding to identified active sets. For solving the quadratic obstacle\nproblem, various optimal multigrid like methods have been proposed. In this\npaper, we study a non-standard norm that is equivalent to applying a block\ndiagonal preconditioner to the reduced linear systems. Numerical experiments\nconfirm the optimality of the solver and convergence independent of problem\nparameters on sufficiently fine mesh.\n