Preprint No. A-02-04

Ralf Kornhuber, Rolf Krause

On multigrid methods for vector-valued Allen-Cahn equation with obstacle potential

Abstract: Phase field models provide a well-established framework for the mathematical description to free boundary problems for phase transitions.
In this paper, we consider multicomponent phase transitions as described by a vector-valued Allen-Cahn equation with obstacle potential. Semi-implicit discretization in time is unconditionally stable but, after finite element discretization in space, leads to large non-smooth algebraic systems. So far, fast solvers for such kind of problems were not available. As a consequence, explicit schemes are applied, in spite of severe stability restrictions on the time step.
We present a new class of multigrid methods based on successive minimization in the direction of well selected search directions and prove global convergence. Numerical experiments illustrate the reliability and efficiency of our method.

Keywords: multigrid methods, constrained minimization, phase fields models

Mathematics Subject Classification (MSC00): 65K10, 65M55, 65N55, 65N22

Language: ENG

Available: Pr-A-02-04.ps, Pr-A-02-04.ps.gz

Contact: Ralf Kornhuber, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (kornhube@math.fu-berlin.de)

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