Optimal error estimates for a fully discrete, highly efficient decoupled scheme for the 2D/3D diffuse interface two-phase MHD flows

TL;DR

This paper presents a fully discrete, decoupled finite element method for 2D/3D diffuse interface MHD flows, achieving optimal error estimates and unconditional energy stability, validated through numerical experiments.

Contribution

It introduces a novel decoupled FEM scheme with optimal error bounds and unconditional energy stability for two-phase MHD flows.

Findings

Optimal L2- and H1-norm error estimates derived

Unconditional energy stability of the scheme proven

Numerical results confirm theoretical accuracy and stability

Abstract

In this paper, we derive optimal L2- and H1-norm error estimates for a fully discrete convex-splitting decoupled finite element method (FEM) for the two-phase diffuse interface magnetohydrodynamics (MHD) system. We use the semi-implicit backward Euler scheme in time and employ the standard inf-sup stable Taylor--Hood or Mini elements to discretize the velocity and pressure. Furthermore, we apply a pressure-correction scheme to decouple the velocity from the pressure. The optimal error estimates are obtained via novel Ritz and Stokes quasi-projection techniques. In addition, the unconditional energy stability of the proposed scheme is ensured. Numerical examples are presented to validate the theoretical analysis.