Mixed finite element methods for the ferrofluid model with magnetization paralleled to the magnetic field

TL;DR

This paper develops mixed finite element methods for a ferrofluid model coupling Maxwell and Navier-Stokes equations, introducing variable transformations to decouple the system and providing theoretical analysis and numerical validation.

Contribution

It introduces a novel decoupling approach for the ferrofluid model and establishes existence, uniqueness, and error estimates for the finite element solutions.

Findings

The methods achieve optimal error estimates.

Numerical experiments confirm theoretical predictions.

Decoupling simplifies the computational process.

Abstract

Mixed finite element methods are considered for a ferrofluid flow model with magnetization paralleled to the magnetic field. The ferrofluid model is a coupled system of the Maxwell equations and the incompressible Navier-Stokes equations. By skillfully introducing some new variables, the model is rewritten as several decoupled subsystems that can be solved independently. Mixed finite element formulations are given to discretize the decoupled systems with proper finite element spaces. Existence and uniqueness of the mixed finite element solutions are shown, and optimal order error estimates are obtained under some reasonable assumptions. Numerical experiments confirm the theoretical results.