Mahdi Gharehbaygloo, a doctoral candidate in mathematics, will defend their dissertation titled “An Energy-Stable Mixed CG-DG Scheme for a Full Shliomis Phase Field Model of Two-Phase Ferrofluid Flows.” Their advisor, Dr. Xiaoming He, is a professor in mathematics and statistics. The dissertation abstract is provided below.

Ferrofluids are magnetic nanoparticle suspensions whose motion couples surface tension, flow field, magnetostatics, and magnetization dynamics. This dissertation develops and analyzes an energy-stable finite element method for a two-phase ferrofluid model that couples the Cahn-Hilliard equations with the full Shliomis model of single-phase ferrofluids, retaining its Shliomis relaxation, rotational coupling, and antisymmetric magnetic stress. The spatial discretization is a mixed continuous Galerkin and discontinuous Galerkin formulation. It uses continuous L2 elements for the phase field, chemical potential, ve-locity, and magnetostatic potential, discontinuous L2 elements for the magnetization, and discontinuous M1 elements for the pressure. The temporal discretization is semi-implicit and semi-explicit with stabilization. The coupled nonlinear system is solved by a block-Gauss-Seidel Picard iteration. The target model is proved to satisfy a continuous energy law. The fully discrete scheme is proved to satisfy the corresponding discrete free-energy law. Stability is showed to be unconditional in term of the mesh size and time step size. It is also proved that adiscrete solution exist at each time step, by a Leray-Schauder degree argument. The method is implemented in both sequential and parallel adaptive C++ codes using deal.II, p4est, and Trilinos. Verification uses manufactured-solution tests and discrete diagnostics. Numerical studies include diamond-droplet relaxation, field-driven elongation, parameter sensitivity, and Lotus-type spike formation under a non-uniform magnetic field.