11th GAMM-Workshop on

Multigrid and Hierarchic Solution Techniques


  A. Almendral  
  M. Bader  
  R. Bank  
  M. Bebendorf  
  S. Beuchler  
  D. Braess  
  C. Douglas  
  L. Grasedyck  
  B. Khoromskij  
  R. Kornhuber  
  B. Krukier  
  U. Langer  
  C. Oosterlee  
  G. Pöplau  
  A. Reusken  
  J. Schöberl  
  M.A. Schweitzer  
  S. Serra Capizzano  
  B. Seynaeve  
  D. Smits  
  O. Steinbach  
  R. Stevenson  
  M. Wabro  
  R. Wienands  
  Ralf Kornhuber : Monotone multigrid for Allen-Cahn equations

Solidification of liquids or phase separation, e.g.,in binary alloys is frequently described mathematically by phase field models. The micromolecular phenomena at the interface are represented by an order parameter. We aim at the construction of robust multigrid solvers for large algebraic systems as resulting from implicit time discretization of Allen-Cahn equations (non-conserved order parameter) and Cahn-Hilliard equations (conserved order parameter). Robustness means that convergence behavior should be insensitive not only with respect to discretization parameters such as mesh size or time step but also with respect to relevant parameters of the continuous problem such as the amount of interfacial energy or temperature.

In this lecture we concentrate on multigrid algorithms for scalar and vector-valued Allen-Cahn equations with obstacle potential. The construction is based on successive minimization of energy with respect to a sequence of d-dimensional subspaces representing a scale of frequencies. Suitable selection of these subspaces together with monotone reduction of energy provides global convergence. Robust extensions to logarithmic free energy is obtained by so-called constrained Newton linearization. Possible variants for Cahn-Hilliard equations are also discussed.

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