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We show, in the framework of steady-state diffusion boundary-value problems, that the staggered discontinuous Galerkin (SDG) method [SIAM J. Numer. Anal., 47 (2009), pp. 3820--3848] can be obtained from a hybridizable discontinuous Galerkin (HDG) method [SIAM J. Numer. Anal., 47 (2009), pp. 1319--1365] by setting its stabilization function to zero at some suitably chosen element faces and by letting it go to infinity at all the remaining others. We then show that this point of view allows the SDG method to immediately acquire new properties all inherited from the HDG methods, namely, their efficient implementation (by hybridization), their postprocessings, and their superconvergence properties.

We present a unified a posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic partial differential equations. The main feature of this analysis is that it reduces the task of establishing the reliability and efficiency of the estimator to the verification of three simple conditions. This approach allows us not only to derive new estimates for hybridizable discontinuous Galerkin methods but also to recover well-known a posteriori error estimates for mixed methods and for the continuous Galerkin method.

We present new a priori error analyses of the coupling of the Raviart--Thomas (RT) method and BEM as well as of the coupling of the hybridizable discontinuous Galerkin (HDG) method with BEM. The novel features of the analysis of the RT-BEM coupling are the superconvergence estimates for the scalar approximation of the interior problem and new bounds in weak and strong norms for the boundary variables. The analysis of the HDG-BEM coupling is the first analysis of this coupling and shows that the coupling provides approximations with the same convergence properties as those of the RT-BEM coupling.

We propose a projection-based a priori error analysis of a wide class of mixed and hybridizable discontinuous Galerkin methods for diffusion problems for which the mappings relating the elements to the reference elements are nonlinear. We show that if the local spaces on the reference elements satisfy suitable conditions, and if the mappings used to define the mesh and global spaces satisfy simple regularity and compatibility conditions, the methods provide optimally convergent approximations for both unknowns as well as superconvergent approximations for the scalar variable. A crucial feature of the analysis of the methods is the use of two new spaces of traces and two associated, suitably defined projections thanks to which the error analysis then becomes almost identical to that obtained by the authors in [Math. Comp., 81 (2012), pp. 1327--1353] where the case in which the mappings are affine is considered.

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.

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 $k+1$.

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

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 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.

The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations, a much needed quantity in many applications. We show how to overcome this shortcoming by using a two-step postprocessing. The first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous Galerkin methods. This computation is nonlocal in that it requires the solution of a symmetric positive definite system, but the system is well conditioned independently of mesh size, so it can be solved at asymptotically optimal cost. The second step is a local element-by-element postprocessing of the CG solution incorporating the result of the first step. This leads to a conservative flux approximation with continuous normal components. This postprocessing applies for the CG method in its standard form or for a hybridized version of it. We present the hybridized version since it allows easy handling of variable-degree polynomials and hanging nodes. Furthermore, we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the Raviart-Thomas mixed method of corresponding degree.

In this paper, we consider the so-called $hp$-version of discontinuous Galerkin methods for Timoshenko beams. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the beam. These methods are thus free from shear locking. We also prove that, when polynomials of degree $p$ are used, all the numerical traces superconverge with a rate of order $h^{2p+1}/p^{2p+1}$. Numerical experiments verifying the above-mentioned theoretical results are shown.

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart--Thomas and the Brezzi--Douglas--Marini methods of similar order are identical.

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For a class of shape regular meshes with hanging nodes we derive a priori estimates for the L2-norm of the errors in the velocities and the pressure. We show that optimal-order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k-1 for the pressure, for any $k\ge1$. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more efficient. Numerical experiments verifying these facts are displayed.

In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.

In this paper, we present the first a priori error analysis for the local discontinuous Galerkin (LDG) method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L2-norm of the gradient and the L2-norm of the potential are of order k and k+1/2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h-1 are taken, the order of convergence of the potential increases to k+1. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.