A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation

In this paper we devise a first-order-in-time, second-order-in-space, convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. The unconditional unique solvability, energy stability and $$\ell ^\infty (0, T; \ell ^4)$$ℓ∞(0,T;ℓ4) stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the $$\ell ^\infty (0, T; \ell ^2)$$ℓ∞(0,T;ℓ2) and $$\ell ^\infty (0, T; \ell ^\infty )$$ℓ∞(0,T;ℓ∞) norms. The proofs of these estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The $$\ell ^2$$ℓ2 convergence proof requires no refinement path constraint, while the one involving the $$\ell ^\infty $$ℓ∞ norm requires only a mild linear refinement constraint, $$s \le C h$$s≤Ch. The key estimates for the error analyses take full advantage of the unconditional $$\ell ^\infty (0, T; \ell ^4)$$ℓ∞(0,T;ℓ4) stability of the numerical solution and an interpolation estimate of the form $$\left\| \phi \right\| _4 \le C \left\| \phi \right\| _2^\alpha \left\| \nabla _h\phi \right\| _2^{1-\alpha },\alpha = \frac{4-D}{4},D=1,2,3$$ϕ4≤Cϕ2α∇hϕ21-α,α=4-D4,D=1,2,3, which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.