Eulerian Geometric Discretizations of Manifolds and Dynamics

This thesis explores new methods for geometric, structure-preserving Eulerian discretizations of dynamics, including Lie advection and incompressible fluids, and the manifolds in which these dynamics occur. The result is a novel method for discrete Lie advection of differential forms, a new family of structure-preserving fluid integrators, and a new set of energies for optimizing meshes appropriate for some discrete geometric operators. First, high-resolution finite volume methods are leveraged to introduce a new method for discretizing the Lie advection of discrete differential forms, along with the related contraction operator, on regular grids. Through its geometric approach, the method exactly preserves properties such as the closedness of Lie advected closed forms. This results in an extension of finite volume techniques applicable to forms of arbitrary degree. After this, attention is turned to simplicial meshes, where new meshing techniques are developed to give formal error bounds on the discrete diagonal Hodge star, an important operator for geometric computations. Utilizing weighted Delaunay triangulations, both the primal mesh and its dual are optimized simultaneously over the entire space of orthogonal primal/dual pairs. Improved accuracy of the solution of Poisson equations is demonstrated as a practical application, as well as an increase in percentage of wellcentered elements. Finally, a new structure-preserving method for the incompressible Navier-Stokes equations on simplicial meshes is developed, offering in the inviscid case the exact conservation of either the discrete energy or symplectic form. This leads to capturing the correct energy decay when viscosity is added, resulting in dissipation independent of grid and time resolution.