HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs

Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite element methods. Multisymplectic variational integrators are a class of geometric numerical integrators for Lagrangian and Hamiltonian field theories, and they yield methods that preserve the multisymplectic structure and momentum-conservation properties of the continuous system. In this paper, we investigate the synthesis of these two approaches, by constructing discretization of the variational principle for Lagrangian field theories utilizing structure-preserving finite element projections. In our investigation, compatible discretization by cochain projections plays a pivotal role in the preservation of the variational structure at the discrete level, allowing the discrete variational structure to essentially be the restriction of the continuum variational structure to a finite-dimensional subspace. The preservation of the variational structure at the discrete level will allow us to construct a discrete Cartan form, which encodes the variational structure of the discrete theory, and subsequently, we utilize the discrete Cartan form to naturally state discrete analogues of Noether's theorem and multisymplecticity, which generalize those introduced in the discrete Lagrangian variational framework by Marsden et al. [29]. We will study both covariant spacetime discretization and canonical spatial semi-discretization, and subsequently relate the two in the case of spacetime tensor product finite element spaces.Comment: 44 pages, 1 figur

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

Finite element methods as geometric structure preserving algorithms

Here we investigate finite element techniques aimed at preserving the underlying geometric structures for various problems, and, in doing so, develop new geometric structure preserving methods. We initially focus on systems of Hamiltonian ODEs, examining the place of existing methods as geometric numerical integrators. We then develop a new geometrical finite element method for Hamiltonian ODEs with a view to generalise it to be the temporal discretisation of a space-time adaptive finite element method. We go on to investigate how well finite element methods can preserve the structure of Hamiltonian PDEs, which are a large class of physically relevant PDEs possessing a conserved physical invariant, the Hamiltonian functional, which often physically represents the energy of the problem. Examples of this kind of problem include, but are not limited to, oceanographical models of wave propagation such as KdV type equations and the nonlinear Schr¨odinger equations, and the semi-geostrophic equations for atmospheric modelling. We construct a general methodology for the design of finite element schemes for such problems and go on to develop multiple schemes in this framework for not only Hamiltonian PDEs but also systems of Hamiltonian PDEs. Within the study of finite element methods for Hamiltonian PDEs we prove both a priori and a posteriori error bounds, in addition to examining the role of spatial adaptivity for our schemes

Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand

Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed

Multisymplecticity of hybridizable discontinuous Galerkin methods

This talk discusses the application of hybridizable discontinuous Galerkin (HDG) methods to canonical Hamiltonian PDEs. We present necessary and sufficient conditions for an HDG method to satisfy a multisymplectic conservation law, when applied to such a system, and show that these conditions are satisfied by "hybridized" versions of several of the most commonly-used finite element methods. These finite element methods may therefore be used for high-order, structure-preserving discretization of Hamiltonian PDEs on unstructured meshes. (Joint work with Robert McLachlan.)Non UBCUnreviewedAuthor affiliation: Washington University in St. LouisFacult