Jacobus J.W. van der Vegt

A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations

We introduce a space-time discontinuous Galerkin (DG) finite element method for the incompressible Navier-Stokes equations. Our formulation can be made arbitrarily high-order accurate in both space and time and can be directly applied to deforming domains. Different stabilizing approaches are discussed which ensure stability of the method. A numerical study is performed to compare the effect of the stabilizing approaches, to show the methods robustness on deforming domains and to investigate the behavior of the convergence rates of the solution. Recently we introduced a space-time hybridizable DG (HDG) method for incompressible flows [S. Rhebergen, B. Cockburn, A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012) 4185-4204]. We will compare numerical results of the space-time DG and space-time HDG methods. This constitutes the first comparison between DG and HDG methods.

A Local Discontinuous Galerkin Method for the Propagation of Phase Transition in Solids and Fluids

A local discontinuous Galerkin (LDG) finite element method for the solution of a hyperbolic–elliptic system modeling the propagation of phase transition in solids and fluids is presented. Viscosity and capillarity terms are added to select the physically relevant solution. The

Three-dimensional crystal of cavities in a 3D photonic band gap crystal

L^2-

A space-time discontinuous Galerkin finite element method for two-fluid problems

L2-stability of the LDG method is proven for basis functions of arbitrary polynomial order. In addition, using a priori error analysis, we provide an error estimate for the LDG discretization of the phase transition model when the stress–strain relation is linear, assuming that the solution is sufficiently smooth and the system is hyperbolic. Also, results of a linear stability analysis to determine the time step are presented. To obtain a reference exact solution we solved a Riemann problem for a trilinear strain–stress relation using a kinetic relation to select the unique admissible solution. This exact solution contains both shocks and phase transitions. The LDG method is demonstrated by computing several model problems representing phase transition in solids and in fluids with a Van der Waals equation of state. The results show the convergence properties of the LDG method.

HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis

We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass” and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.

A (Dis)continuous finite element model for generalized 2D vorticity dynamics

We investigate theoretically and numerically the transport of light in a three-dimensional (3D) crystal of cavities inside a 3D inverse woodpile photonic crystal. This class of crystals consists of two perpendicular arrays of pores and has a very broad 3D photonic band gap. An individual point defect is formed by reducing the radius of two intersecting pores. An earlier study revealed that an isolated point defect supports up to five cavity resonances within the band gap of the infinite crystal [1]. We explore light transport in a three-dimensional crystal of such cavities. Our question is what physics we can expect and whether there is an analogy to condensed matter physics. Indeed, inside the band gap light hops from cavity to cavity by evanescent mode coupling. Our freedom in choosing the cavity locations gives us a high level of control over light transport. This makes a crystal of cavities a promising framework for studying Anderson localization [2]. Two-dimensional arrays of coupled cavities [3] are used as lasers [4] and have an application in discrete solitons [5,6]. Hopping transport can be described by an expansion in eigenstates. In this work we have made a quantitative study of the eigenfunctions and resonance frequencies in a finite crystal of cavities.