Structure-preserving finite element and finite volume methods for nonlinear and time-dependent PDEs

In this thesis, we present significant advancements in the structure-preserving discretization of partial differential equations, developing and analyzing novel numerical schemes with applications ranging from compressible fluid dynamics to magnetohydrodynamics (MHD) in tokamak geometries for nuclear fusion research. We begin by introducing a rotational formulation of the Stokes problem with Navier’s slip boundary conditions, followed by the development of an asymptotic-preserving and mass-conservative method for weakly compressible flows. As the Mach number approaches zero, this method converges to an exactly divergence-free scheme for the Navier-Stokes equations. Additionally, we incorporate an a posteriori limiter based on the discrete maximum principle to effectively resolve shocks and reduce spurious oscillations. Next, we investigate the Lie advection-diffusion problem in both stationary and time-dependent regimes, constructing structure-preserving stabilization techniques. For the time-dependent case, we propose a generalization of the interpolation-contraction method originally introduced by Hiptmair and Pagliantini. The aforementioned results are applied to develop two novel schemes for viscous and resistive incompressible magnetohydrodynamics that preserve the magnetic field’s divergence-free property to machine precision. One of these schemes is also well-balanced and compatible with mixed-element meshes, enabling long-time simulations of the Soloviev equilibrium in simplified 3D tokamak geometries. We conclude by addressing the question of whether finite element methods satisfy a discrete multisymplectic conservation law. We find that all analyzed methods satisfy a strong version of this property, except the Arnold-FalkWinther conforming method, which satisfies the multisymplectic property only in a weak sense. The theoretical results are validated through extensive numerical experiments. The proposed schemes are implemented using the finite element library NGSolve and in Fortran. Additionally, some of the codes and data are made available for reproducibility