Energy-stable mixed finite element methods for the Rosensweig ferrofluid flow model

TL;DR

This paper investigates energy-stable mixed finite element discretizations for the Rosensweig ferrofluid flow model, addressing stability, regularity, and convergence of numerical schemes. The authors derive regularity results for the model under basic assumptions. They prove that energy stability of the weak solutions is exactly preserved by both semi-discrete and fully discrete finite element schemes. They establish existence and uniqueness of discrete solutions and obtain optimal error estimates for the discretizations. Numerical experiments verify the theory and demonstrate the practical reliability of the proposed methods.

Abstract

In this paper, we consider mixed finite element semi-/full discretizations of the Rosensweig ferrofluid flow model. We first establish some regularity results for the model under several basic assumptions. Then we show that the energy stability of the weak solutions is preserved exactly for both the semi-discrete and fully discrete finite element solutions. Moreover, we prove the existence and uniqueness of the discrete solutions. We also derive optimal error estimates for the discrete schemes. Finally, we provide numerical experiments to verify the theoretical results.