Error analysis of a fully discrete scheme for the Cahn-Hilliard-Magneto-hydrodynamics problem

TL;DR

This paper presents an analysis of a fully discrete finite element scheme for the coupled Cahn-Hilliard and Magneto-hydrodynamics equations, proving unconditional energy stability and deriving optimal error estimates.

Contribution

It introduces a novel fully discrete scheme combining finite elements and semi-implicit Euler discretization with convex splitting, providing rigorous stability and error analysis.

Findings

Unconditionally energy stable scheme

Optimal error estimates for multiple fields

Numerical validation of convergence rates

Abstract

In this paper we analyze a fully discrete scheme for a general Cahn-Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stability and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence.