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In this paper, we construct first- and second-order fully discrete schemes for the Cahn–Hilliard–Hele–Shaw system based on the Fourier-spectral method for spatial discretization. For temporal discretization, we combine two efficient approaches, including the scalar auxiliary variable (SAV) method for linearizing nonlinear potentials and the zero-energy-contribution method (ZEC) for decoupling nonlinear couplings. These schemes are linear, fully decoupled, and unconditionally energy stable, requiring only the solution of a sequence of elliptic equations with constant coefficients at each time step. The rigorous proof of the error analysis for the first-order scheme is shown. In addition, several numerical examples are presented to demonstrate the stability, accuracy, and efficiency of the proposed scheme.

In this paper, a fully discrete scheme is proposed to solve the nonlinear Schrödinger-Possion equations. The scheme is developed by the scalar auxiliary variable (SAV) approach, the Crank-Nicolson temproal discretization and the Galerkin-Legendre spectral spatial discretization. The fully discrete scheme is proved to be mass- and energy- conserved. Moreover, unconditional energy stability and convergence of the scheme are obtained by the use of the Gagliardo-Nirenberg and some Sobolev inequalities. Numerical results are presented to confirm our theoretical findings.