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

Abstract

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 ? l 4 ) 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 ? l 2 ) 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 $$\ell ^2$$ l 2 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 ≤ C h . The key estimates for the error analyses take full advantage of the unconditional $$\ell ^\infty (0, T; \ell ^4)$$ l ? ( 0 , T ? l 4 ) stability of the numerical solution and an interpolation estimate of the form $$\left\| \phi \right\| _4 \le C \left\| \phi \right\| _2^\alpha \left\| \nabla _h\phi \right\| _2^{1-\alpha },\alpha = \frac{4-D}{4},D=1,2,3$$ ? 4 ≤ C ? 2 ? ? h ? 2 1 - ? , ? = 4 - D 4 , D = 1 , 2 , 3 , which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.