Preconditioning nonlocal multi-phase flow

Highlights: • We propose an efficient, mesh independent solver for discretisations of multi-phase Allen–Cahn variational inequalities. • The solver has only a very mild dependence on the number of phase field variables. • For two phase scalar Allen–Cahn variational inequalities we prove convergence of the approximation in 3 GMRES iterations. We propose an efficient solver for saddle point problems arising from finite element approximations of nonlocal multi-phase Allen–Cahn variational inequalities. The solver is seen to behave mesh independently and to have only a very mild dependence on the number of phase field variables. In addition we prove convergence, in three GMRES iterations, of the approximation of the two phase problem, regardless of mesh size or interfacial width. Numerical results are presented that illustrate the competitiveness of this approach.