2017/03/08 by Anke Böttcher, Herbert Egger, Böttcher, Anke +1
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1703.02778
openalex publication_date 2017/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study the systematic numerical approximation of a class of Allen-Cahn type\nproblems modeling the motion of phase interfaces. The common feature of these\nmodels is an underlying gradient flow structure which gives rise to a decay of\nan associated energy functional along solution trajectories. We first study the\ndiscretization in space by a conforming Galerkin approximation of a variational\nprinciple which characterizes smooth solutions of the problem. Well-posedness\nof the resulting semi-discretization is established and the energy decay along\ndiscrete solution trajectories is proven. A problem adapted implicit\ntime-stepping scheme is then proposed and we establish its well-posed and decay\nof the free energy for the fully discrete scheme. Some details about the\nnumerical realization by finite elements are discussed, in particular the\niterative solution of the nonlinear problems arising in every time-step. The\ntheoretical results are illustrated by numerical tests which also provide\nfurther evidence for asymptotic expansions of the interface velocities derived\nby Alber et al.\n