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The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution.

Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness.

Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.

Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion.

Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD.

This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.