We consider the least-squares spectral element method to solve the phase field model for two immiscible, incompressible and density-matched fluids. The coupled Cahn-Hilliard and Navier-Stokes system is selected as the numerical model, which was introduced by Hohenberg et al. [1]. The least-squares spectral element scheme is combined with a time-space formulation where both time and space domains are discretized by the same finite element approach to cope with time dependent multidimensional problems in an efficient way. C1 Hermite basis functions are applied for approximating the coupled system. An element-by-element conjugated gradient method is used to facilitate parallelization of the solver. The convergence analysis is conducted to verify our solver, and two numerical experiments are addressed to show applicability of the solver in general situations. Energy dissipation with conserved phase field at equilibrium state is confirmed through the bubble coalescence case, and the influence of the interface mobility is studied with the two-phase lid-driven cavity flow example.

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