Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the well-posedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the \({\varvec{L}}^2 \text -{\varvec{H}}^1\) ${\mathit{L}}^2\text{-}{\mathit{H}}^1$ error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional \({\varvec{L}}^2 \text -{\varvec{H}}^1\) ${\mathit{L}}^2\text{-}{\mathit{H}}^1$ error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.