A fully discrete semi-implicit numerical scheme and its optimal error estimates for Cahn-Hilliard-MHD model with variable density

Abstract

This paper proposes and analyzes a fully discrete semi-implicit unconditionally energy stable numerical scheme to solve the Cahn-Hilliard Magnetohydrodynamics (Cahn-Hilliard-MHD) model with variable density. The unconditional energy stability and optimal $L^2$ error estimates are established for the fully discrete scheme. Major challenges in error estimation arise from the variable density, the strong nonlinearities, and the multi-physics coupling of the model. Under the mathematical induction framework, the Ritz quasi-projection and the Stokes quasi-projection, proposed in [SIAM J. Numer. Anal., 61(3):1218-1245, 2023], are utilized to avoid the gradient terms of the projection errors. The $H^{-1}$ superconvergence error estimates of Ritz projection and Ritz quasi-projection reduce the regularity requirement for the finite element space. Two different sets of test functions are also selected to avoid extra estimations of the $L^2$ norm terms. The summation by parts transfers the backward difference quotient operator from the test function to the trial function, avoiding the need for extra estimation of the test function. A numerical experiment is provided to verify the theoretical results.