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

Enrico Zampa - PhD in Mathematics, University of Trento

Abstract: 
We present key contributions to the structure-preserving discretization of partial differential equations, developing and analyzing innovative numerical schemes with applications ranging from compressible flows to magnetohydrodynamics in tokamak geometries for nuclear fusion. In particular, we devise and analyze a rotational formulation of Stokes problem with Navier's slip boundary conditions and we introduce an asymptotic preserving and mass conservative method for weakly compressible flows. When the Mach number approaches zero, our method converges to an exactly divergence-free method for the Navier Stokes equations. Additionally, we investigate the Lie advection-diffusion problem in both stationary and time-dependent regimes, constructing structure-preserving stabilizations. These 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. Finally, we address 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-Falk-Winther conforming method, which satisfies the multisymplectic property only in a weak sense.