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 ∞(0, T ; 4) stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the ∞(0, T ; 2) and ∞(0, T ; ∞) 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 2 convergence proof requires no refinement path constraint, while the one involving the ∞ norm requires only a mild linear refinement constraint, s ≤ Ch. The key estimates for the error analyses take full advantage of the unconditional ∞(0, T ; 4) stability of the numerical solution and an interpolation estimate of the form ‖φ‖4 ≤ C ‖φ‖2 ‖∇hφ‖1−α 2 , α = 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. Mathematics Subject Classification 65M12 · 65R20 Z. Guan Mathematics Department, University of California, Irvine, CA 92697, USA e-mail: [email protected] C. Wang Mathematics Department, University ofMassachusetts Dartmouth, North Dartmouth,MA 02747, USA e-mail: [email protected] C. Wang School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, People’s Republic of China S. M. Wise (B) Mathematics Department, University of Tennessee, Knoxville, TN 37996, USA e-mail: [email protected]