Search

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.

We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $u$ is the solution of a steady-state diffusion problem, the standard approximation $J(u_h)$ converges with order $h^{2k+1}$, where $u_h$ is the Hybridizable Discontinuous Galerkin approximation to $u$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $h^{4k}$, and can be computed by only using twice the computational effort needed to compute $J(u_h)$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $u$ or the functional $J(\cdot )$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods.

We analyze a technique to improve the spatial accuracy, by the single application at the end of the simulation of a local post-processing, for pure Lagrange–Galerkin (PLG) methods applied to evolutionary convection-diffusion (possibly pure convection/diffusion) equations with time-dependent domains. The post-processing technique is based on a simple convolution that extracts the ‘hidden accuracy’ of Galerkin schemes, and it is used and rigorously analyzed in a fully discrete context. We prove that, when applied to the numerical solution of PLG schemes, it improves the spatial accuracy in the $l^{\infty }(L^2(\varOmega ^0))$-norm from order $k+1$ to at least order $2k$, where $k$ is the degree of the polynomials defining the finite element space and $\varOmega ^0$ any interior region of the computational domain meshed with translation-invariant meshes. For pure convection, a spatial accuracy enhancement in the $l^{\infty }(L^2(\varOmega ^0))$-norm from order $k+1$ to order $2k+2$ is obtained by post-processing the numerical solution of PLG schemes. Numerical tests are presented that confirm these theoretical results.

In Chen et al. (J. Sci. Comput. 81(3): 2188–2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree $$k\geqslant 1$$ . In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3): 635–650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements affine-equivalent to a finite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for $$k\geqslant 0$$ by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

We propose the interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method for a class of scalar parabolic semilinear PDEs. The Interpolatory HDG method uses an interpolation procedure to efficiently and accurately approximate the nonlinear term. This procedure avoids the numerical quadrature typically required for the assembly of the global matrix at each iteration in each time step, which is a computationally costly component of the standard HDG method for nonlinear PDEs. Furthermore, the Interpolatory HDG interpolation procedure yields simple explicit expressions for the nonlinear term and Jacobian matrix, which leads to a simple unified implementation for a variety of nonlinear PDEs. For a globally-Lipschitz nonlinearity, we prove that the Interpolatory HDG method does not result in a reduction of the order of convergence. We display 2D and 3D numerical experiments to demonstrate the performance of the method.

We propose a new tool, which we call $M$-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an $M$-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit $M$-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings., 45 pages, 2 figures

In our earlier work (Cockburn et al. in J Sci Comput 79(3):1777–1800, 2019), we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method.

We present a class of high-order finite element methods that can conserve the linear and angular momenta as well as the energy for the equations of linear elastodynamics. These methods are devised by exploiting and preserving the Hamiltonian structure of the equations of linear elastodynamics. We show that several mixed finite element, discontinuous Galerkin , and hybridizable discontinuous Galerkin (HDG) methods belong to this class. We discretize the semidiscrete Hamiltonian system in time by using a symplectic integrator in order to ensure the symplectic properties of the resulting methods, which are called symplectic Hamiltonian finite element methods. For a particular semidiscrete HDG method, we obtain optimal error estimates and present, for the symplectic Hamiltonian HDG method, numerical experiments that confirm its optimal orders of convergence for all variables as well as its conservation properties.

We design and analyze the first hybridizable discontinuous Galerkin methods for stationary, third-order linear equations in one-space dimension. The methods are defined as discrete versions of characterizations of the exact solution in terms of local problems and transmission conditions. They provide approximations to the exact solution u u and its derivatives q := u ′ q:=u’ and p := u p:=u which are piecewise polynomials of degree k u k_u , k q k_q and k p k_p , respectively. We consider the methods for which the difference between these polynomial degrees is at most two. We prove that all these methods have superconvergence properties which allows us to prove that their numerical traces converge at the nodes of the partition with order at least 2 k + 1 2\,k+1 , where k k is the minimum of k u , k q k_u,k_q , and k p k_p . This allows us to use an element-by-element post-processing to obtain new approximations for u , q u, q and p p converging with order at least 2 k + 1 2k+1 uniformly. Numerical results validating our error estimates are displayed.

We define and analyze hybridizable discontinuous Galerkin methods for the Laplace-Beltrami problem on implicitly defined surfaces. We show that the methods can retain the same convergence and superconvergence properties they enjoy in the case of flat surfaces. Special attention is paid to the relative effect of approximation of the surface and that introduced by discretizing the equations. In particular, we show that when the geometry is approximated by polynomials of the same degree of those used to approximate the solution, although the optimality of the approximations is preserved, the superconvergence is lost. To recover it, the surface has to be approximated by polynomials of one additional degree. We also consider mixed surface finite element methods as a natural part of our presentation. Numerical experiments verifying and complementing our theoretical results are shown.

We present an explicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the acoustic wave equation. The method is fully explicit, high-order accurate in both space and time, and coincides with the classic discontinuous Galerkin (DG) method with upwinding fluxes for a particular choice of its stabilization function. This means that it has the same computational complexity as other explicit DG methods. However, just as its implicit version, it provides optimal convergence of order k + 1 for all the approximate variables including the gradient of the solution, and, when the time-stepping method is of order k + 2 , it displays a superconvergence property which allow us, by means of local postprocessing, to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k + 2 in the L 2 -norm for k ≥ 1 . These properties do not hold for all numerical fluxes. Indeed, our results show that, when the HDG numerical flux is replaced by the Lax–Friedrichs flux, the above-mentioned superconvergence properties are lost, although some are recovered when the Lax–Friedrichs flux is used only in the interior of the domain. Finally, we extend the explicit HDG method to treat the wave equation with perfectly matched layers. We provide numerical examples to demonstrate the performance of the proposed method.

We find new discrete $H^1$- and Poincar\'e-Friedrichs inequalities by studying the invertibility of the DG approximation of the flux for local spaces admitting M-decompositions. We then show how to use these inequalities to define and analyze new, superconvergent HDG and mixed methods for which the stabilization function is defined in such a way that the approximations satisfy new $H^1$-stability results with which their error analysis is greatly simplified. We apply this approach to define a wide class of energy-bounded, superconvergent HDG and mixed methods for the incompressible Navier-Stokes equations defined on unstructured meshes using, in 2D, general polygonal elements and, in 3D, general, flat-faced tetrahedral, prismatic, pyramidal and hexahedral elements., Comment: 22 pages

We revisit the hybridizable discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient.

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order $$-\alpha $$-? with $$-1

We provide a short overview of our recent work on the devising of hybridizable discontinuous Galerkin ( HDG ) methods for the Stokes equations of incompressible flow. First, we motivate and display the general form of the methods and show that they provide a well defined approximate solution for arbitrary polyhedral elements. We then discuss three different but equivalent formulations of the methods. Next, we describe a systematic way of constructing superconvergent HDG methods by using, as building blocks, the local spaces of superconvergent HDG methods for the Laplacian operator. This can be done, so far, for simplexes, parallelepipeds and prisms. Finally, we show how, by means of an elementwise computation, we can obtain divergence-free velocity approximations converging faster than the original velocity approximation when working with simplicial elements. We end by briefly discussing other versions of the methods, how to obtain HDG methods with H (div)-conforming velocity spaces, and how to extend the methods to other related systems. Several open problems are described.

In this paper, we show that by sending the normal stabilization function to infinity in the hybridizable discontinuous Galerkin methods previously proposed in [Comput. Methods Appl. Mech. Engrg. 199 (2010), 582–597], for Stokes flows, a new class of divergence-conforming methods is obtained which maintains the convergence properties of the original methods. Thus, all the components of the approximate solution, which use polynomial spaces of degree k k , converge with the optimal order of k + 1 k+1 in L 2 L^2 for any k ≥ 0 k \ge 0 . Moreover, the postprocessed velocity approximation is also divergence-conforming, exactly divergence-free and converges with order k + 2 k+2 for k ≥ 1 k\ge 1 and with order 1 1 for k = 0 k=0 . The novelty of the analysis is that it proceeds by taking the limit when the normal stabilization goes to infinity in the error estimates recently obtained in [Math. Comp., 80 (2011) 723–760].

We propose a new a posteriori error analysis of the variable-degree, hybridized version of the Raviart-Thomas method for second-order elliptic problems on conforming meshes made of simplexes. We establish both the reliability and efficiency of the estimator for the L 2 L_2 -norm of the error of the flux. We also find the explicit dependence of the estimator on the order of the local spaces k ≥ 0 k\ge 0 ; the only constants that are not explicitly computed are those depending on the shape-regularity of the simplexes. In particular, the constant of the local efficiency inequality is proven to behave like ( k + 2 ) 3 / 2 (k+{2})^{3/2} . However, we present numerical experiments suggesting that such a constant is actually independent of k k .

We present the first a priori error analysis of a technique that allows us to numerically solve steady-state diffusion problems defined on curved domains Ω by using finite element methods defined in polyhedral subdomains Dh ⊂ Ω. For a wide variety of hybridizable discontinuous Galerkin and mixed methods, we prove that the order of convergence in the L2-norm of the approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain Γ and that of the computational domain Γh is of order h. We also prove that the L 2-norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between Γ and Γh is of order h 5/4 but with only half an additional order when such a distance is of order h. Finally, we present numerical experiments confirming the theoretical results and showing that even when the distance between Γ and Γh is of order h, the above-mentioned projection of the error of the scalar variable can still superconverge with a full additional order.

In this paper, we provide a projection-based analysis of the hversion of the hybridizable discontinuous Galerkin methods for convectiondiffusion equations on semimatching nonconforming meshes made of simplexes; the degrees of the piecewise polynomials are allowed to vary from element to element. We show that, for approximations of degree k on all elements, the order of convergence of the error in the diffusive flux is k+ 1 and that that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k > 0. We also show that, for the variable-degree case, the projection of the error in the scalar variable is h-times the projection of the error in the vector variable, provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes. These results hold for any (bounded) irregularity index of the nonconformity of the mesh. Moreover, our analysis can be extended to hypercubes.

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 method’s 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.

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δ x \Delta x only. For example, when polynomials of degree k k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k + 1 / 2 k+1/2 in the L 2 L^2 -norm, whereas the post-processed approximation is of order 2 k + 1 2k+1 ; if the exact solution is in L 2 L^2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1 / 2 k+1/2 in L 2 ( Ω 0 ) L^2(\Omega _0) , where Ω 0 \Omega _0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

In this paper we propose a new general framework for the construction and the analysis of discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes-Franca type residual-based stabilizations. © 2005 Elsevier B.V. All rights reserved.

International audience; We build a bridge between the hybrid high-order (HHO) and the hybridizable discontinuous Galerkin (HDG) methods in the setting of a model diffusion problem. First, we briefly recall the construction of HHO methods and derive some new variants. Then, by casting the HHO method in mixed form, we identify the numerical flux so that the HHO method can be compared to HDG methods. In turn, the incorporation of the HHO method into the HDG framework brings up new, efficient choices of the local spaces and a new, delicate construction of the numerical flux ensuring optimal orders of convergence on meshes made of general shape-regular polyhedral elements. Numerical experiments comparing two of these methods are shown.

We provide a projection-based analysis of a large class of finite element methods for second order elliptic problems. It includes the hybridized version of the main mixed and hybridizable discontinuous Galerkin methods. The main feature of this unifying approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary, locally defined projection and of the local spaces defining the methods. Sufficient conditions for the optimal convergence of the approximate flux and the superconvergence of an element-by-element postprocessing of the scalar variable are obtained. New mixed and hybridizable discontinuous Galerkin methods with these properties are devised which are defined on squares, cubes and prisms.

University of Minnesota Ph.D. dissertation. May 2010. Major: Mathematics. Advisor: Prof. Bernardo Cockburn. 1 computer file (PDF); viii, 67 pages. Ill. (some col.)

In this paper, we give the first a priori error analysis of the hybridizable discontinuous Galerkin (HDG) methods for Timoshenko beams. The analysis is based on the use of a projection especially designed to fit the structure of the numerical traces of the HDG method. This property allows to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection. The study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element. Surprisingly, and unlike any other discontinuous Galerkin method, this can be done without assuming any positivity property of the stabilization function of the HDG method. We apply this approach to HDG methods using polynomials of degree k ≥ 0 in all the unknowns, and show that the projection of the error in each of them superconverges with order k + 2 when k ≥ 1 and converges with order 1 for k = 0. As a result, we show that the HDG methods converge with optimal order k + 1 for all the unknowns, and that they are free from shear locking. Finally, we show that all the numerical traces converge with order 2k + 1. Numerical experiments validating these results are shown.

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.

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.

We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain "bubble matrices", which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order k + 1 in both the displacement and the stress, and that a postprocessed displacement approximation converging at order k + 2 can be computed element by element. We also show that the globally coupled degrees of freedom can be reduced by hybridization to those of a displacement approximation on the element boundaries.

We introduce hybridization and postprocessing techniques for the Raviart-Thomas approximation of second-order elliptic eigenvalue problems. Hybridization reduces the Raviart-Thomas approximation to a condensed eigenproblem. The condensed eigenproblem is nonlinear, but smaller than the original mixed approximation. We derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates. An element-by-element postprocessing technique to improve accuracy of computed eigenfunctions is also presented. We prove that a projection of the error in the eigenspace approximation by the mixed method (of any order) superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions. Numerical experiments using a square and an L-shaped domain illustrate the theoretical results.

In this paper, we introduce a hybridizable discontinuous Galerkin method for Stokes flow. The method is devised by using the discontinuous Galerkin methodology to discretize a velocity–pressure–gradient formulation of the Stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization. One of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries, thereby leading to a significant reduction in the size of the resulting matrix. Moreover, by using an augmented lagrangian method, the globally coupled unknowns are further reduced to the numerical trace of the velocity only. Another important feature is that the approximations of the velocity, pressure, and gradient converge with the optimal order of k + 1 in the L 2 -norm, when polynomials of degree k ⩾ 0 are used to represent the approximate variables. Based on the optimal convergence of the HDG method, we apply an element-by-element postprocessing scheme to obtain a new approximate velocity, which converges with order k + 2 in the L 2 -norm for k ⩾ 1 . The postprocessing performed at the element level is less expensive than the solution procedure. Numerical results are provided to assess the performance of the method.

We prove optimal convergence rates for the approximation provided by the original discontinuous Galerkin method for the transport-reaction problem. This is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport. Thus, if the method uses polynomials of degree $k$, the $L^2$-norm of the error is of order $k+1$. Moreover, we also show that, by means of an element-by-element postprocessing, a new approximate flux can be obtained which superconverges with order $k+1$.

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.

The embedded discontinuous Galerkin methods are obtained from hybridizable discontinuous Galerkin methods by a simple change of the space of the hybrid unknown. In this paper, we consider embedded methods for second-order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions. This change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous Galerkin method. It is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence. Indeed, we prove that if polynomials of degree $k\ge1$ are used for the original hybridizable discontinuous Galerkin method, its approximations to the scalar variable and its flux converge with order $k+2$ and $k+1$, respectively, whereas those of the corresponding embedded discontinuous Galerkin method converge with orders $k+1$ and $k$, respectively, only. We also provide numerical results comparing the relative efficiency of the methods.

We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equal-order approach is ensured by a pressure stabilization term. A simple element-by-element post-processing procedure is used to provide globally divergence-free velocity approximations. For small data, we prove the existence and uniqueness of discrete solutions and carry out an error analysis of the method. A series of numerical results are presented that validate our theoretical findings.

We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree k ≥ 0 for both the potential as well as the flux, the order of convergence in L 2 of both unknowns is k + 1. Moreover, both the approximate potential as well as its numerical trace superconverge in L 2 -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 potential converging with order k+2 in L 2 . 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 show that the approximation given by the original discontinuous Galerkin method for the transport-reaction equation in $d$ space dimensions is optimal provided the meshes are suitably chosen: the $L^2$-norm of the error is of order $k+1$ when the method uses polynomials of degree $k$. These meshes are not necessarily conforming and do not satisfy any uniformity condition; they are required only to be made of simplexes, each of which has a unique outflow face. We also find a new, element-by-element postprocessing of the derivative in the direction of the flow which superconverges with order $k+1$.

We study the approximation of nonsmooth solutions of the transport equation in one space dimension by approximations given by a Runge-Kutta discontinuous Galerkin method of order two. We take an initial datum, which has compact support and is smooth except at a discontinuity, and show that, if the ratio of the time step size to the grid size is less than $1/3$, the error at the time $T$ in the $L^2(\mathbb{R}\setminus\mathcal{R}_T)$-norm is the optimal order two when $\mathcal{R}_T$ is a region of size $O(T^{1/2}\,h^{1/2}\;\log{1/h})$ to the right of the discontinuity and of size $O(T^{1/3}\,h^{2/3}\;\log{1/h})$ to the left. Numerical experiments validating these results are presented.

We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two- and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families of exact sequences for triangles and three for squares, and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements. We apply the construction in three-space dimensions to rediscover two families of exact sequences for tetrahedra, three for cubes, and one for prisms; and to uncover four new families of exact sequences for pyramids, three for prisms, and one for cubes., Comment: 37pages

We analyze the so-called the minimal dissipation local discontinuous Galerkin method (MD-LDG) for convection---diffusion or diffusion problems. The distinctive feature of this method is that the stabilization parameters associated with the numerical trace of the flux are identically equal to zero in the interior of the domain; this is why its dissipation is said to be minimal. We show that the orders of convergence of the approximations for the potential and the flux using polynomials of degree k are the same as those of all known discontinuous Galerkin methods, namely, (k + 1) and k, respectively. Our numerical results verify that these orders of convergence are sharp. The novelty of the analysis is that it bypasses a seemingly indispensable condition, namely, the positivity of the above mentioned stabilization parameters, by using a new, carefully defined projection tailored to the very definition of the numerical traces.

In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convection-diffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov-Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at all the nodes of the mesh, provided that the traces are conservative, that is, provided they are single-valued. In particular, for is local discontinuous Galerkin method, we show that the superconvergence is order 2p + 1 when polynomials of degree at most p are used. Extensive numerical results verifying our theoretical results are displayed.