Optimal L2 error estimates for a fully discrete method of the Cahn-Hilliard-MHD model

Abstract

This paper carries out the optimal L² error estimation for a fully discrete method, which was proposed in [J. Sci. Comput., 95(1):16, 2023] for the Cahn-Hilliard-MHD model with constant coefficients. Compared with the original scheme, we adopt weaker regularity requirement for the finite element space of the discrete phase field. To obtain the optimal convergence order, $H^{-1}$-superconvergence error estimates of Ritz projection and Ritz quasi-projection are employed. The H¹ semi-norm and L² norm terms of the phase-field system, arising from the $H^{-1}-H^1$ norm pair, are estimated by two sets of test functions. Furthermore, the first-order differential operator induces a loss of convergence rate in the projection error terms. Hence Ritz quasi-projection and Stokes quasi-projection, incorporating the above terms, are adopted from [SIAM J. Numer. Anal.,61(3):1218-1245, 2023]. Additionally, operator transfer based on Green’s formula provides an alternative strategy. We also carry out numerical examples to validate the numerical scheme.