Unified analysis of HDG methods using scalar and vector hybrid variables

In this paper, hybridizable discontinuous Galerkin (HDG) methods using scalar and vector hybrid variables for steady-state diffusion problems are considered. We propose a unified framework to analyze the methods, where both the hybrid variables are treated as double-valued functions. If either of them is single valued, the well-posedness is ensured under some assumptions on approximation spaces. Moreover, we prove that all methods are superconvergent, based on the so-called $M$-decomposition theory. Numerical results are presented to validate our theoretical results.Comment: 16 page

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces

Convergence to weak solutions of a space-time hybridized discontinuous Galerkin method for the incompressible Navier--Stokes equations

We prove that a space-time hybridized discontinuous Galerkin method for the evolutionary Navier--Stokes equations converges to a weak solution as the time step and mesh size tend to zero. Moreover, we show that this weak solution satisfies the energy inequality. To perform our analysis, we make use of discrete functional analysis tools and a discrete version of the Aubin--Lions--Simon theorem

Méthodes HDG et HDG+ pour des problèmes d’ondes convectées en régime harmonique

In this report, we introduce three variants of the HDG method based on two weak formulations of the convected Helmholtz equation. Two of them are standard HDG methods with the same interpolation degree for all the unknowns and one of them uses a higher interpolation degree for the volumetric scalar un- known. For those three numerical methods, a detailed analysis including local and global well-posedness, as well as convergence estimates is carried out. We then pro- vide implementation details and numerical experiments to illustrate our theoretical results.Dans ce rapport, nous construisons trois variantes de la méthode HDG, basées sur deux formulations faibles de l’équation d’Helmholtz convectée. Deux de ces méthodes sont des méthodes HDG standard qui utilisent le même degré d’interpolation polynomiale pour toutes les inconnues. La troisième méthode, quant à elle, utilise un degré d’interpolation plus élevé pour l’inconnue scalaire volumique, à l’instar des méthodes HDG+. Pour toutes ces méthodes, une analyse détaillée a été effectuée, elle inclut des résultats d’existence et unicité locale et globale ainsi qu’une étude de convergence. Pour finir, nous présentons les détails de l’implémentation de ces méthodes et des expériences numériques qui illustrent nos résultats théoriques

Numerical analysis of space-time hybridized discontinuous Galerkin methods for incompressible flows

Many industrial problems require the solution of the incompressible Navier-Stokes equations on moving and deforming domains. Notable examples include the simulation of rotating wind turbines in strong air flow, wave impact on offshore structures, and arterial blood flow in the human body. A viable candidate for the numerical solution of the Navier-Stokes equations on time-dependent domains is the space-time discontinuous Galerkin (DG) method, which makes no distinction between spatial and temporal variables. Space-time DG is well suited to handle moving and deforming domains but at a significant increase in computational cost in comparison to traditional time-stepping methods. Attempts to rectify this situation have led to the pairing of space-time DG with the hybridized discontinuous Galerkin (HDG) method, which was developed to reduce the computational expense of DG. The combination of the two methods results in a scheme that retains the high-order spatial and temporal accuracy and geometric flexibility of space-time DG at a reduced cost. Moreover, the use of hybridization allows for the design of pressure-robust space-time methods on time-dependent domains, which is a class of mimetic methods that inherit at the discrete level a fundamental invariance property of the incompressible Navier-Stokes equations. The space-time HDG method has been successfully applied to incompressible flow problems on time-dependent domains; however, at present, no supporting theoretical analysis can be found in the literature. This thesis is a first step toward such an analysis. In particular, we perform a thorough theoretical convergence analysis of a space-time HDG method for the incompressible Navier-Stokes equations on fixed domains, and of a space-time HDG method for the linear advection-diffusion equation on time-dependent domains. The former contribution elucidates the difficulties involved in the theoretical analysis of space-time HDG methods for the Navier-Stokes equations, while the latter contribution introduces a framework for the convergence analysis of space-time HDG methods on time-dependent domains. We begin with an a priori error analysis of a pressure-robust HDG method for the stationary Navier-Stokes equations. Then, we provide an a priori error analysis of a pressure-robust space-time HDG method from which we conclude that the space-time HDG method converges to strong solutions of the Navier-Stokes equations. This leaves open the question of convergence to weak solutions, which we answer in the affirmative using compactness techniques. Finally, we provide an a priori error analysis of a space-time HDG method for the linear advection-diffusion equation on time-dependent domains

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree , then the best approximation error in is of order . In this survey, the available finite element error analysis for the velocity error in is reviewed, where is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order , , and for the velocity error in . All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open