Unconditionally maximum principle preserving finite element schemes for the surface Allen-Cahn type equations

In this paper, we present two types of unconditionally maximum principle preserving finite element schemes to the standard and conservative surface Allen-Cahn equations. The surface finite element method is applied to the spatial discretization. For the temporal discretization of the standard Allen-Cahn equation, the stabilized semi-implicit and the convex splitting schemes are modified as lumped mass forms which enable schemes to preserve the discrete maximum principle. Based on the above schemes, an operator splitting approach is utilized to solve the conservative Allen-Cahn equation. The proofs of the unconditionally discrete maximum principle preservations of the proposed schemes are provided both for semi- (in time) and fully discrete cases. Numerical examples including simulations of the phase separations and mean curvature flows on various surfaces are presented to illustrate the validity of the proposed schemes.