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The phenomenon of phase separation has been observed in lipid membranes. This process is remarkable, since both in-membrane and solvent-mediated hydrodynamic effects affect separation dynamics. The Cahn–Hilliard model for phase separation is here considered, coupled with the overdamped (Stokes) fluid equations. The convection term of the Cahn–Hilliard equations, which is due to hydrodynamic effects, is here treated by a Lagrangian method, in which fluid particles move along the velocity field carrying the concentration field. The method is combined with a projection onto a fixed regular mesh, where the rest of the equations are solved in Fourier space. In this space, spatial derivatives are evaluated quite easily. Moreover, the effect of the underlying fluid is straightforward in Fourier space, through the modification of the Oseen tensor. This hybrid treatment is the main contribution of this work. Results are in good agreement with experimental findings. Some agreement is found with previous simulations, but some striking differences are present.READ MORE