No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Algorithm 949: MATLAB tools for HDG in three dimensions
Abstract
In this article, we provide some MATLAB tools for efficient vectorized implementation of the Hybridizable Discontinuous Galerkin for linear variable coefficient reaction-diffusion problems in polyhedral domains. The resulting tools are modular and include enhanced structures to deal with convection-diffusion problems, plus several projection operators and the postprocessing implementation that is necessary to realize the superconvergence property of the method. Loops over the elements are exclusively local and, as such, have been parallelized.
... Scientific computing and applications were an integral part of Javier's research activities. His publications cover, for instance, Euler-MacLaurin expansions [38], [4] (with Ricardo Celorrio), error estimates for convolution quadrature [20] (with Hasan Eruslu), and MATLAB implementations of the Argyris element [15] (with Victor Domínguez) and the HDG method in three dimensions [21] (with Zhixing Fu and Luis F. Gatica). Together with his students-Team Pancho-he made their MATLAB codes for the boundary element and HDG methods available to the community [46,47]. ...
This is the preface of a special issue dedicated to the memory of Francisco Javier Sayas who passed away on April 2, 2019. The articles reflect Sayas’ main research interests in the numerical analysis of partial differential equations, containing contributions on the scattering and propagation of acoustic and electromagnetic waves, and the analysis of discontinuous Galerkin schemes, boundary element methods, and coupled schemes. We discuss the main contributions of Sayas and give an overview of the results covered by this special issue.
... It is this inaccessibility, which is often (falsely) attributed to OOP, that drives researchers to develop their own small, highly specialized FEM codes, that are often written purely within more easily accessible scripting languages like Matlab or Python; see, e.g., [FPW11;Che09;FGS15]. They are written with a specific problem setting in mind and often do not provide general purpose routines, but comprise a small collection of tightly interacting, imperative scripts. ...
We present an easily accessible, object oriented code (written exclusively in Matlab) for finite element simulations in 2D. The object oriented programming paradigm allows for fast implementation of higher-order FEM on triangular meshes for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.
... It is, more or less, around this time that Javier took an interest in HDG methods for wave propagation. Javier developed MATLAB software for the HDG method in 3D, see the 2015 paper [43], which was used, for example, in his 2018 paper on viscoelastic waves [1]. During that period, he also became interested in the application of HDG methods to wave propagation phenomena. ...
Franciso Javier Sayas, man of grit and determination, left his hometown of Zaragoza in 2007 in pursuit of a dream, to become a scholar in the USA. I hosted him in Minneapolis, where he spent three long years of an arduous transition before obtaining a permanent position at the University of Delaware. There, he enthusiastically worked on the unfolding of his dream until his life was tragically cut short by cancer, at only 50. In this paper, I try to bring to light the part of his academic life he shared with me. As we both worked on hybridizable discontinuous Galerkin methods, and he wrote a book on the subject, I will tell Javier’s life as it developed around this topic. First, I will show how the ideas of static condensation and hybridization , proposed back in the mid 60s, lead to the introduction of those methods. This background material will allow me to tell the story of the evolution of the hybridizable discontinuous Galerkin methods and describe Javier’s participation in it. Javier faced death with open eyes and poised dignity. I will end with a poem he liked.
... • HDG3D [9,143] • FESTUNG [5,166] ...
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.
... • HDG3D [9,143] • FESTUNG [5,166] ...
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.
... The calculation amount of security check changes dynamically according to the application requirements, and dynamic allocation is adopted. After receiving the calculation request received by the service port, the security verification service terminal will select the calculation method according to the specific calculation content, and evaluate the accuracy of the calculation results [13]. As shown in Figure 1, the scanning speed modulation architecture of the power prediction model. ...
The scale of China’s power grid is continuously expanding, the amount of collected data is increasing rapidly, and data processing and analysis services are developing towards clustering. Therefore, at present, task management in a stand-alone mode faces severe challenges in ensuring the high efficiency and reliability of distributed tasks. Information security risks in cyberspace can cause fatal threats to smart grid entities through the destruction of grid dispatching control systems and communication networks. Security check is one of the application functions of intelligent power dispatching control system. It is an important security defense line to ensure the stable operation of power grid and provides security check service for power grid dispatching plan and power grid operation. The existing operation and maintenance management mode of the power grid dispatching control system is distributed, and the data of each system and unit is not effectively integrated and connected, and lacks overall consideration. Based on the development history of power grid dispatching automation, this paper comprehensively analyzes the current status of the overall structure of the power grid dispatching system, and summarizes the key technology innovations and application effects.
... Because the time series corresponding to adjacent points have a strong correlation, this process is simple but ignores a lot of information. Regarding production planning and dispatching applications, the relevant equipment resources should be reasonably allocated in conjunction with the main power distribution process [10]. Grey system theory is a theory that studies the analysis, modeling, prediction, decision and control of grey systems. ...
With the large-scale development of the power industry, new requirements are put forward for the stable operation of the power system. Power system load forecasting refers to the use of historical load data to predict the future load value, which is an important part of energy management system. The short-term load forecasting process is often combined with the basic mechanism of power grid dispatching to achieve the balance of power grid supply and demand, reflecting the highly nonlinear computing ability. Power load forecasting is the premise of power grid real-time control, operation planning and development planning. At present, the grey model model used for power system load forecasting generally has the problems of large calculation amount, no mature theoretical basis for selecting structures and parameters, etc. This paper discusses the application of grey model in short-term power load forecasting, and puts forward a principal component analysis method suitable for ordinary daily power load forecasting data, which improves the accuracy of short-term power load forecasting.
... The computational implementation of the algorithm described in this paper benefited greatly from the detailed explanations and code templates for HDG and adaptive refinement provided respectively by Fu, Gatica and Sayas [49], and Funken, Praetorius and Wissgott [50]. Finally, sampling of the confinement regions from the analytic expressions given in [41] was done using chebfun [51]. ...
We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modelling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries.
... The computational implementation of the algorithm described in this paper benefited greatly from the detailed explanations and code templates for HDG and adaptive refinement provided respectively by Fu, Gatica and Sayas [49], and Funken, Praetorius and Wissgott [50]. Finally, sampling of the confinement regions from the analytic expressions given in [41] was done using chebfun [51]. ...
We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modelling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries.
... The reader familiar with mixed methods may object that this family imposes very tough restrictions on the possible choices for the approximation spaces, but one can rest assured that yet another positive feature of HDG is providing with ample flexibility for the spaces used. We refer the reader interested in the programming aspects of HDG methods to [34], where a very detailed explanation and coding strategies and tools are given. Our own implementation is based on the tools provided in that reference. ...
In axisymmetric fusion reactors, the equilibrium magnetic configuration can be expressed in terms of the solution to a semi-linear elliptic equation known as the Grad-Shafranov equation, the solution of which determines the poloidal component of the magnetic field. When the geometry of the confinement region is known, the problem becomes an interior Dirichlet boundary value problem. We propose a high order solver based on the Hybridizable Discontinuous Galerkin method. The resulting algorithm (1) provides high order of convergence for the flux function and its gradient, (2) incorporates a novel method for handling piecewise smooth geometries by extension from polygonal meshes, (3) can handle geometries with non-smooth boundaries and x-points, (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration, and (5) is ideally suited for parallel implementations. The effectiveness of the algorithm is verified with computations for cases where analytic solutions are known on configurations similar to those of actual devices (ITER, NSTX, ASDEX upgrade, and Field Reversed Configurations).
... The code expands the three dimensional HDG Matlab package of [7]. The code is written for the second-order-in-frequency form given in (6.3), which allows for easy comparisons with the physical quantities of interest and for almost straightforward changes to have all the methods we have considered in this paper. ...
We derive and analyze a hybridizable discontinuous Galerkin (HDG) method for approximating weak solutions to the equations of time-harmonic linear elasticity on a bounded Lipschitz domain in three dimensions. The real symmetry of the stress tensor is strongly enforced and its coefficients as well as those of the displacement vector field are approximated simultaneously at optimal convergence with respect to the choice of approximating spaces, wavenumber, and mesh size. Sufficient conditions are given so that the system is indeed transferable onto a global hybrid variable that, for larger polynomial degrees, may be approximated via a smaller-dimensional space than the original variables. We construct several variants of this method and discuss their advantages and disadvantages, and give a systematic approach to the error analysis for these methods. We touch briefly on the application of this error analysis to the time-dependent problem, and finally, we examine two different implementations of the method over various polynomial degrees and numerically demonstrate the convergence properties proven herein.
... In the following simulations we consider tetrahedral meshes and τ t = h −1 . The implementation is based on the work developed by [11]. u(x, y, z) = sin(π y) sin(π z), sin(π x) sin(π z), sin(π x) sin(π y) T , ...
We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier p is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell's problem. We use polynomials of degree k+1, k, k to approximate \bfu,\nabla \times \bfu and p respectively. In contrast, we only use a non-trivial subspace of polynomials of degree k+1 to approximate the numerical tangential trace of the electric field and polynomials of degree k+1 to approximate the numerical trace of the Lagrange multiplier on the faces. On the simplicial meshes, a special choice of the stabilization parameters is applied, and the HDG system is shown to be well-posed. Moreover, we show that the convergence rates for and are independent of the Lagrange multiplier p. If we assume the dual operator of the Maxwell equation on the domain has adequate regularity, we show that the convergence rate for is . From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. Finally, we show that on general polyhedral elements, by a particular choice of the stabilization parameters again, the HDG system is also well-posed and the superconvergence of the HDG method is derived.
We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.
Music appreciation is one of the indispensable courses. It plays a key role in cultivating students’ aesthetic ability and perfecting students’ personality. The purpose of the Ecology Curriculum is to create an equal and relaxed classroom atmosphere between the teachers and students during the course of the course. Let students and teachers grow together in this harmonious and happy environment, aiming at the problems that the existing courses are not valued, the teaching content is not cautious, and the teaching materials are not standardized. This paper proposes that on the one hand, it is necessary to improve the music appreciation ecological course, on the other hand, it is necessary to improve the teaching method. Measures to improve the Ecology Course for Music Appreciation include a complete teaching, a clear teaching plan and appropriate textbooks. The diversification of teaching methods and the enrichment of educational models are also important methods for innovative teaching methods. At the same time, it is important that students have the right learning concept in the music appreciation course and the school’s support for the faculty and equipment.
We propose a standard hybridizable discontinuous Galerkin (HDG) method to solve a classic problem in fluids mechanics: Darcy’s law. This model describes the behavior of a fluid trough a porous medium and it is relevant to the flow patterns on the macro scale. Here we present the theoretical results of existence and uniqueness of the weak and discontinuous solution of the second order elliptic equation, as well as the predicted convergence order of the HDG method. We highlight the use and implementation of Dubiner polynomial basis functions that allow us to develop a general and efficient high order numerical approximation. We also show some numerical examples that validate the theoretical results.
We analyze a hybridizable discontinuous Galerkin (HDG) method for the time harmonic Maxwell’s equations arising from modeling photovoltaic solar cells. The problem is set in an inhomogeneous domain with a polyhedral connected boundary and the divergence-free condition is imposed using a Lagrange multiplier. We prove the HDG scheme is well-posed up to some frequencies and derive a stability estimate. In particular, we prove that the method is absolutely stable without any mesh constraint even for solutions with low regularity, when the stabilization parameters are chosen as purely imaginary. Moreover, when the solution has enough regularity, we show that the L2-norm of the error of the approximation in the electric and magnetic fields, are of order hk+1 and hk+1∕2, where k is the polynomial degree of the local approximation spaces. Numerical examples are shown to validate the theory.
We propose a residual-type a posteriori error estimator for a hybridizable discontinuous Galerkin method applied to the Oseen problem in a gradient-velocity-pressure formulation. We state reliability and local efficiency results for our estimator with respect to an error measured in the natural norms, with constants depending explicitly on the physical parameters. A weighting function technique to control the L^2-error of the velocity, and the approximation properties of the Oswald interpolation operator are the main ingredients in the analysis. Numerical experiments in three dimensions validate our theoretical results.
This is the first in a series of papers on implementing a discontinuous
Galerkin method as a MATLAB / GNU Octave toolbox. The main goal is the
development of techniques that deliver optimized computational performance
combined with a compact, user-friendly interface. Our implementation relies on
fully vectorized matrix / vector operations and is carefully documented; in
addition, a direct mapping between discretization terms and code routines is
maintained throughout. The present work focuses on a two-dimensional
time-dependent diffusion equation with space / time-varying coefficients. The
spatial discretization is based on the local discontinuous Galerkin formulation
and is locally mass conservative. Approximations of orders zero through four
based on orthogonal polynomials have been implemented; more spaces of arbitrary
type and order can be easily accommodated by the code structure. Time
discretization is performed using an implicit Euler method.
http://arxiv-web3.library.cornell.edu/abs/1408.3877
arXiv:1408.3877 [math.NA]
In this paper the first error analyses of hybridizable discontinuous Galerkin (HDG) methods for convection–diffusion equations for variable-degree approximations and nonconforming meshes are presented. The analysis technique is an extension of the projection-based approach recently used to analyse the HDG method for the purely diffusive case. In particular, for approximations of degree k on all elements and conforming meshes, we show that the order of convergence of the error in the diffusive flux is k + 1 and that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k > 0. When nonconforming meshes are used our estimates do not rule out a degradation of 1/2 in the order of convergence in the diffusive flux and a loss of 1 in the order of convergence of the projection of the error in the scalar variable. However, they do guarantee the optimal convergence of order k + 1 of the scalar variable.
This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. Information can be returned in floating-point (numerical) form, or in exact symbolic form. The latter is useful for computer-algebra aided FEM work that carries along symbolic variables. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. Some unpublished non-product rules for pyramid regions were found and included in the collection. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. The collection embodies most FEM-useful formulas of low and moderate order for the seven regions noted above. Some gaps as regard region geometries and omission of non-product rules are noted in the conclusions. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. It can also be used as generator for low-level floating-point code modules in Fortran or C. Floating point accuracy can be selected arbitrarily. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before.
We provide a MATLAB package p1afem for an adaptive P1-finite element method (AFEM).
This includes functions for the assembly of the data, different error estimators, and
an indicator-based adaptive meshrefining algorithm. Throughout, the focus is on an efficient
realization by use of MATLAB built-in functions and vectorization. Numerical experiments
underline the efficiency of the code which is observed to be of almost linear complexity
with respect to the runtime. Although the scope of this paper is on AFEM, the general ideas
can be understood as a guideline for writing efficient MATLAB code.
We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain. Mathematics subject classification: 65D32, 65D30, 65M60.
The L2-orthogonal polynomials used in finite and spectral element methods on nonrectangular elements may be defined in terms of collapsed coordinates, wherein the shapes are mapped to a square or cube by means of a singular change of variables. The orthogonal basis is a product of specific Jacobi polynomials in these new coordinates. Implementations of these polynomials require special handling of the coordinate singularities. We derive new recurrence relations for these polynomials on triangles and tetrahedra that work directly in the original coordinates. These relations, also applicable to pyramids and prisms, do not require any special treatment of singular points. These recurrences are seen to speed up both symbolic and numerical computation of the orthogonal polynomials.
We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, nonconforming, and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework 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. Since the associated matrix is sparse, symmetric, and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain, which are then automatically coupled. Finally, the framework brings about a new point of view, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.
In this article, we derive computational bases for Raviart–Thomas (RTRT) and Brezzi–Douglas–Marini (BDMBDM) (vector) approximation spaces on a triangulation of a domain in R2R2. The basis functions, defined on the reference triangle, have a Lagrangian property. The continuity of the normal component of the approximation across the edges in the triangulation is satisfied by the use of the Piola transformation and the Lagrangian property of the basis functions. A numerical example is given demonstrating the approximation property of the bases.
We present hybridizable discontinuous Galerkin methods for solving steady and time-dependent partial differential equations (PDEs) in continuum mechanics. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the numerical flux to provide stability and consistency; and a global jump condition that enforces the continuity of the numerical flux to arrive at a global weak formulation in terms of the numerical trace. The HDG methods are fully implicit, high-order accurate and endowed with several unique features which distinguish themselves from other discontinuous Galerkin methods. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, they provide, for smooth viscous-dominated problems, approximations of all the variables which converge with the optimal order of k + 1 in the L2-norm. Third, they possess some superconvergence properties that allow us to define inexpensive element-by-element postprocessing procedures to compute a new approximate solution which may converge with higher order than the original solution. And fourth, they allow for a novel and systematic way for imposing boundary conditions for the total stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the methods. In addition, they possess other interesting properties for specific problems. Their approximate solution can be postprocessed to yield an exactly divergence-free and H(div)-conforming velocity field for incompressible flows. They do not exhibit volumetric locking for nearly incompressible solids. We provide extensive numerical results to illustrate their distinct characteristics and compare their performance with that of continuous Galerkin methods.
Summary We introduce two families of mixed finite element on conforming inH(div) and one conforming inH(curl). These finite elements can be used to approximate the Stokes' system.
We discuss a technique of implementing certain mixed finite elements based on the use of Lagrange multipliers to impose interelement continuity. The matrices arising from this implementation are positive definite. Considering some well-known mixed methods, namely the Raviart-Thomas methods for second order elliptic problems and the Hellan-Hermann-Johnson method for biharmonic problems, we show that the computed Lagrange multipliers may be exploited in a simple postprocess to produce better approximation of the original variables. We further extablish an equivalence between the mixed methods and certain modified versions of well-known nonconforming methods, notably the Morley method in the case of the biharmonic problem. The equivalence is exploited to provide error estimates for both the mixed and nonconforming methods.
Two families of mixed finite elements, one based on simplices and the other on cubes, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. These spaces are analogues of those introduced by Brezzi, Douglas, and Marini in two space variables. Error estimates inL
2 andH
–s are derived.
Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called
static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical
and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence
results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous
Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating
a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally
competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization
strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement
our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.
We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than
the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show
that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals
the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before
taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle
hp-adaptivity.
KeywordsHigh-order finite elements–Spectral/hp elements–Discontinuous Galerkin method–Hybridization–Domain decomposition
We identify discontinuous Galerkin methods for second-order el- liptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k ≥ 0 for both the po- tential as well as the flux. We show that the approximate flux converges in L2 with the optimal order of k + 1, and that the approximate potential and its numerical trace superconverge, in L2-like norms, to suitably chosen pro- jections of the potential, with order k + 2. We also apply element-by-element postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have nor- mal components continuous across inter-element boundaries, to converge in L2 with order k + 1, and to have a divergence converging in L2 also with order k + 1. The new approximate potential is proven to converge with order k +2 in L2. Numerical experiments validating these theoretical results are presented.
We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-orderelliptic problemsin severalspace dimensions with remarkable convergence properties. Unlike all other known discontinuousGalerkin methods using polynomialsof degreek � 0 for both the potential as well as the flux, the order of convergence in L2 of both unknowns is k + 1. Moreover, both the approximate potential as well as its numerical trace superconverge in L2-like norms, to suitably chosen projections of the potential, with order k + 2. This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potentialconverging with order k+2 in L2. The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.
We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG method applied to a model second-order elliptic problem.
iFEM: An integrated finite element methods package in MATLAB
- Long Chen
Creating sparse finite-element matrices in MATLAB
- Tim Davis