An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations

We present a time-implicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the system of three-dimensional (3D) time-domain Maxwell equations. This method can be seen as a fully implicit variant of classical so-called DGTD (Discontinuous Galerkin Time-Domain) methods that have been extensively studied during the last 10 years for the simulation of time-domain electromagnetic wave propagation. The proposed method has been implemented for dealing with general 3D problems discretized using unstructured tetrahedral meshes. We provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand, and its performance when applied to more realistic problems. We also include some performance comparisons with a centered flux time-implicit DGTD method

An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations

International audienceWe present a time-implicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the system of three-dimensional (3D) time-domain Maxwell equations. This method can be seen as a fully implicit variant of classical so-called DGTD (Discontinuous Galerkin Time-Domain) methods that have been extensively studied during the last 10 years for the simulation of time-domain electromagnetic wave propagation. The proposed method has been implemented for dealing with general 3D problems discretized using unstructured tetrahedral meshes. We provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand, and its performance when applied to more realistic problems. We also include some performance comparisons with a centered flux time-implicit DGTD method

A hybridizable discontinuous Galerkin method for electromagnetics with a view on subsurface applications

Two Hybridizable Discontinuous Galerkin (HDG) schemes for the solution of Maxwell's equations in the time domain are presented. The first method is based on an electromagnetic diffusion equation, while the second is based on Faraday's and Maxwell--Amp\`ere's laws. Both formulations include the diffusive term depending on the conductivity of the medium. The three-dimensional formulation of the electromagnetic diffusion equation in the framework of HDG methods, the introduction of the conduction current term and the choice of the electric field as hybrid variable in a mixed formulation are the key points of the current study. Numerical results are provided for validation purposes and convergence studies of spatial and temporal discretizations are carried out. The test cases include both simulation in dielectric and conductive media

Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

We introduce a \textit{non-modal} analysis technique that characterizes the diffusion properties of spectral element methods for linear convection-diffusion systems. While strictly speaking only valid for linear problems, the analysis is devised so that it can give critical insights on two questions: (i) Why do spectral element methods suffer from stability issues in under-resolved computations of nonlinear problems? And, (ii) why do they successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these two questions can in turn provide crucial guidelines to construct more robust and accurate schemes for complex under-resolved flows, commonly found in industrial applications. For illustration purposes, this analysis technique is applied to the hybridized discontinuous Galerkin methods as representatives of spectral element methods. The effect of the polynomial order, the upwinding parameter and the P\'eclet number on the so-called \textit{short-term diffusion} of the scheme are investigated. From a purely non-modal analysis point of view, polynomial orders between $2$ and $4$ with standard upwinding are well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected. The non-modal analysis results are then tested against under-resolved turbulence simulations of the Burgers, Euler and Navier-Stokes equations. While devised in the linear setting, our non-modal analysis succeeds to predict the behavior of the scheme in the nonlinear problems considered

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

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

An explicit hybridizable discontinuous Galerkin method for the 3D time-domain Maxwell equations

International audienceWe present an explicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the system of three-dimensional (3D) time-domain Maxwell equations. The method is fully explicit similarly to classical so-called DGTD (Dis-continuous Galerkin Time-Domain) methods, is also high-order accurate in both space and time and can be seen as a generalization of the classical DGTD scheme based on upwind fluxes. We provide numerical results aiming at assessing its numerical convergence properties by considering a model problem and we present preliminary results of the superconvergence property on the H curl norm

An implicit hybridized discontinuous Galerkin method for time-domain Maxwell's equations

Discontinuous Galerkin (DG) methods have been the subject of numerous research activities in the last 15 years and have been successfully developed for various physical contexts modeled by elliptic, mixed hyperbolic-parabolic and hyperbolic systems of PDEs. One major drawback of high order DG methods is their intrinsic cost due to the very large number of globally coupled degrees of freedom as compared to classical high order conforming finite element methods. Different attempts have been made in the recent past to improve this situation and one promising strategy has been recently proposed by Cockburn (Cockburn et al., 2009) in the form of so-called hybridizable DG formulations. The distinctive feature of these methods is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements of the discretization mesh. The present work is concerned with the study of such a hybridizable DG method for the solution of the system of Maxwell equations. In this preliminary investigation, a hybridizable DG method is proposed for the two-dimensional time-domain Maxwell equations time integrated by an implicit scheme