Preconditioning nonlocal multi-phase flow

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