On multigrid methods for vector-valued Allen-Cahn equations

Abstract

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. Similar multigrid techniques have been applied in a different context. Numerical experiments illustrate the reliability and efficiency of our method