Port-Hamiltonian Discontinuous Galerkin Finite Element Methods

This dissertation discusses the formulation and implementation of structure preserving port-Hamiltonian discontinuous Galerkin finite element methods for a general class of partial differential equations (PDEs), which includes for example, the wave equation, the shallow water equations, and Maxwell’s equations. This dissertation focuses on three main topics: (i) formulation of the port-Hamiltonian discontinuous Galerkin finite element method (pH-DGFEM), (ii) existence of a canonical/standard Dirac structure for linear dynamical systems on Sobolev spaces of differential forms, and (iii) the implementation of the pH-DGFEM on the isotropic acoustic wave equation.
In Chapter 2, we start by deriving an elementwise Dirac structure for a certain class of PDEs (e.g., wave equation) defined on broken Sobolev
spaces. Next, we define a power preserving interconnection Dirac structure. We then approximate all variables using piecewise polynomial differential form spaces and state the port-Hamiltonian discontinuous Galerkin finite element method with appropriate power preserving numerical fluxes. In the second part of this chapter we state an a priori error analysis for the pH-DGFEM and apply the pH-DGFEM on a model problem - the isotropic acoustic wave equation.
In Chapter 3, we state a proof of the existence of a canonical Dirac structure on a Sobolev space of differential forms.
In Chapter 4, we first state the pH-DGFEM formulation of the isotropic acoustic wave equation with associated numerical fluxes. We then discuss the choices of polynomial differential forms and their associated finite elements for the implementation of the pH-DGFEM on general simplicial, quadrilateral and cubic meshes. Next, the implementation of different types of boundary conditions, i.e., Dirichlet and Neumann boundary conditions is discussed. In the final section of this chapter we present results of numerical accuracy tests of the pH-DGFEM.