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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.

We present the first a priori error analysis of a class of space-discretizations by Hybridizable Discontinuous Galerkin (HDG) methods for the time-dependent Maxwell's equations introduced in Comput. Methods Appl. Mech. Engrg., vol. 396, paper. No. 114969, 27 pages, 2022. The distinctive feature of these discretizations is that they display a discrete version of the Hamiltonian structure of the original Maxwell's equations. This is why they are called "Hamiltonian" HDG methods. Because of this, when combined with symplectic time-marching methods, the resulting methods display an energy that does not drift in time. We provide a single analysis for several of these methods by exploiting the fact that they only differ by the choice of the approximation spaces and the stabilization functions. We also introduce a new way of discretizing the static Maxwell's equations in order to define the initial condition in a manner consistent with our technique of analysis. Finally, we present numerical tests to validate our theoretical orders of convergence and to explore the convergence properties of the method in situations not covered by our analysis.

We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.

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 devise the first symplectic Hamiltonian hybridizable discontinuous Galerkin (HDG) methods for the acoustic wave equation. We discretize in space by using a Hamiltonian HDG scheme, that is, an HDG method which preserves the Hamiltonian structure of the wave equation, and in time by using symplectic, diagonally implicit and explicit partitioned Runge–Kutta methods. The fundamental feature of the resulting scheme is that the conservation of a discrete energy, which is nothing but a discrete version of the original Hamiltonian, is guaranteed. We present numerical experiments which indicate that the method achieves optimal approximations of order k + 1 in the L 2 -norm when polynomials of degree k ≥ 0 and Runge–Kutta time-marching methods of order k + 1 are used. In addition, by means of post-processing techniques and by increasing the order of the Runge–Kutta method to k + 2 , we obtain superconvergent approximations of order k + 2 in the L 2 -norm for the displacement and the velocity. We also present numerical examples that corroborate that the methods conserve energy and that they compare favorably with dissipative HDG schemes, of similar accuracy properties, for long-time simulations.

We introduce and analyze the first energy-conservative hybridizable discontinuous Galerkin method for the semidiscretization in space of the acoustic wave equation. We prove optimal convergence and superconvergence estimates for the semidiscrete method. We then introduce a two-step fourth-order-in-time Stormer-Numerov discretization and prove energy conservation and convergence estimates for the fully discrete method. In particular, we show that by using polynomial approximations of degree two, convergence of order four is obtained. Numerical experiments verifying that our theoretical orders of convergence are sharp are presented. We also show experiments comparing the method with dissipative methods of the same order.

We show that two widely used Galerkin formulations for second-order elliptic problems provide approximations which are actually superclose, that is, their difference converges faster than the corresponding errors. In the framework of linear elasticity, the two formulations correspond to using either the stiffness tensor or its inverse the compliance tensor. We find sufficient conditions, for a wide class of methods (including mixed and discontinuous Galerkin methods), which guarantee a supercloseness result. For example, for the HDG $$_{k}$$ method using polynomial approximations of degree $${k>0}$$ , we find that the difference of approximate fluxes superconverges with order $${k+2}$$ and that the difference of the scalar approximations superconverges with order $${k+3}$$ . We provide numerical results verifying our theoretical results.

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where $$J(\cdot )$$ is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if $$u_h$$ is the approximation given by the continuous Galerkin method with piecewise polynomials of degree $$k>0$$ , then, as a direct consequence of its property of Galerkin orthogonality, the functional $$J(u_h)$$ converges to J(u) with a rate of order $$h^{2k}$$ . We show how to define approximations to J(u), with a computational effort about twice of that of computing $$J(u_h)$$ , which converge with a rate of order $$h^{4k}$$ . The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for $$k=1,2,3$$ in one-space dimension and for $$k=1,2$$ in two-space dimensions, which show that $$J(u_h)$$ converges to J(u) with order $$h^{2k+1}$$ and that the new approximations converges with order $$h^{4k}$$ . The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of $$J(u_h)$$ .

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 apply the concept of an M -decomposition in the framework of steady-state diffusion problems to construct local spaces defining superconvergent hybridizable discontinuous Galerkin methods as well as their companion sandwiching mixed methods in ℝ 3 with tetrahedral, pyramidal, prismatic, and hexahedral elements.

We apply the concept of an M -decomposition introduced in Part I to systematically construct local spaces defining superconvergent hybridizable discontinuous Galerkin methods, and their companion sandwiching mixed methods. This is done in the framework of steady-state diffusion problems for the h - and p -versions of the methods for general polygonal meshes in two-space dimensions.

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 propose the first hybridizable discontinuous Galerkin method for the $p$-Laplacian equation. When using polynomials of degree $k \geq 0$ for the approximation spaces of $u$, $\boldsymbol{\nabla} u$, and $|\boldsymbol{\nabla} u|^{p-2}\boldsymbol{\nabla} u$, the method exhibits optimal $k+1$ order of convergence for all variables in $L^1$- and $L^p$-norms in our numerical experiments. For $k \geq 1$, an elementwise computation allows us to obtain a new approximation $u_h^*$ that converges to $u$ with order $k+2$. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of $u$ on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of...

We present a class of embedded discontinuous Galerkin (EDG) methods for numerically solving the Euler equations and the Navier-Stokes equations. The essential ingredients are a local Galerkin projection of the underlying governing equations at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the approximate trace, a judicious choice of the numerical flux to provide stability and consistency, and a global jump condition that weakly enforces the single-valuedness of the numerical flux to arrive at a global formulation in terms of the numerical trace. The EDG methods are thus obtained from the hybridizable discontinuous Galerkin (HDG) method by requiring the approximate trace to belong to smaller approximation spaces than the one in the HDG method. In the EDG methods, the numerical trace is taken to be continuous on a suitable collection of faces, thus resulting in an even smaller number of globally coupled degrees of freedom than in the HDG method. On the other hand, the EDG methods are no longer locally conservative. In the framework of convection-diffusion problems, this lack of local conservativity is reflected in the fact that the EDG methods do not provide the optimal convergence of the approximate gradient or the superconvergence for the scalar variable for diffusion-dominated problems as the HDG method does. However, since the HDG method does not display these properties in the convection-dominated regime, the EDG method becomes a reasonable alternative since it produces smaller algebraic systems than the HDG method. In fact, the resulting stiffness matrix has a similar sparsity pattern as that of the statically condensed continuous Galerkin (CG) method. The main advantage of the EDG methods is that they are generally more stable and robust than the CG method for solving convection-dominated problems. Numerical results are presented to illustrate the performance of the EDG methods. They confirm that, even though the EDG methods are not locally conservative, they are a viable alternative to the HDG method in the convection-dominated regime.

We establish the contraction property between consecutive loops of adaptive hybridizable discontinuous Galerkin methods for the Poisson problem with homogeneous Dirichlet condition. The contractive quantity is the sum of the square of the L2-norm of the flux error, which is not even monotone, and a two-parameter scaled error estimator, which quantifies both the lack of H(div,Ω)-conformity and the deviation from a gradient of the approximate flux. A distinctive and novel feature of this analysis, which enables comparison between two nested meshes, is the lifting of trace residuals from inter-element boundaries to element interiors.

We show that the staggered discontinuous Galerkin (SDG) method (Kim et al. in SIAM J Numer Anal 51:3327---3350, 2013) for the Stokes system of incompressible fluid flow can be obtained from a new hybridizable discontinuous Galerkin (HDG) method 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, as a consequence, the SDG method immediately acquires three new properties all inherited from this limiting HDG method, namely, its efficient implementation (by hybridization), its superconvergence properties, and its postprocessing of the velocity. In particular, the postprocessing of the velocity is $$\varvec{H}(\mathrm {div})$$H(div)-conforming, weakly divergence-free and converges with order $$k+2$$k+2 where $$k>0$$k>0 is the polynomial degree of the approximations.

We introduce a new hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the Helmholtz equation over a wide range of wave frequencies. Our approach combines the HDG methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies. The phase-based HDG method is devised as follows. First, we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics. Second, we propose a novel scheme that combines the HDG method with ray tracing to compute multivalued solution of the eikonal equation. Third, we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase-based HDG method. And fourth, we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method. Numerical experiments presented show that optimal orders of convergence are achieved, that the number of degrees of freedom to achieve a given accuracy is independent of the wave number, and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element 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 provide a short introduction to the theory of M-decompositions in the framework of steady-state diffusion problems. This theory allows us to systematically devise hybridizable discontinuous Galerkin and mixed methods which can be proven to be superconvergent on unstructured meshes made of elements of a variety of shapes. The main feature of this approach is that it reduces such an effort to the definition, for each element K of the mesh, of the spaces for the flux, V (K), and the scalar variable, W(K), which, roughly speaking, can be decomposed into suitably chosen orthogonal subspaces related to the space traces on ∂K of the scalar unknown, M(∂K). We begin by showing how a simple a priori error analysis motivates the notion of an M-decomposition. We then study the main properties of the M-decompositions and show how to actually construct them. Finally, we provide many examples in the two-dimensional setting. We end by briefly commenting on several extensions including to other equations like the wave equation, the equations of linear elasticity, and the equations of incompressible fluid flow.

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.

The discontinuous Galerkin methods are locally conservative, high-order accurate, and robust methods that can easily handle elements of arbitrary shapes, irregular triangulations with hanging nodes, and polynomial approximations of different degrees in different elements. These properties, which render them ideal for hp-adaptivity in domains of complex geometry, have brought them to the main stream of computational fluid dynamics. In this paper, we study the properties of the DG methods as applied to a wide variety of problems, including linear, symmetric hyperbolic systems, the Euler equations of gas dynamics, purely elliptic problems, and the incompressible and compressible Navier–Stokes equations. In each instance, we discuss the main properties of the methods, display the mechanisms that make them work so well, and present numerical experiments showing their performance. Keywords: computational fluid dynamics; discontinuous finite elements

Summary We present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. We consider meshes made of polyhedral, shape-regular elements of arbitrary shape and show that, whenever piecewise-polynomial approximations of degree k≥0 are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k + 1/2, and the displacement converges with order k + 1. We also provide numerical results showing that the orders of convergence are actually sharp. Copyright © 2014 John Wiley & Sons, Ltd.

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.

It is now a well known fact that hybridizable discontinuous Galerkin for nonlinear elasticity may not converge to the exact solution if their inter-element jumps are not properly penalized or, equivalently, if the values of their stabilization function are not suitably chosen. For example, if their stabilization function is of order one, the method generates spurious oscillations in the region in which the elastic moduli tensor is indefinite. This phenomenon disappears when the values of the stabilization function are increased. Mixed methods display the same problem, as their stabilization function is identically zero. Here, we find explicit formulas for the lower bound of the values of the stabilization function which allow us to avoid this unpleasant phenomenon. Our numerical experiments show that, when we use polynomials of degree k > 0 for the approximate gradient, first-order Piola-Kirchhoff and displacement, all the approximations converge with the optimal order, k + 1 . They also show that, if we increase the polynomial degree of the local spaces by one and update the values of the stabilization function only for the elements for which the elasticity tensor is indefinite, we obtain that all approximate solutions converge with order k + 1 and that a local post processing of the displacement converges with order k + 2 .

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 present an adaptive spectral/discontinuous Galerkin (DG) method on curved elements to simulate high-frequency wavefronts within a reduced phase-space formulation of geometrical optics. Following recent work, the approach is based on the use of level sets defined by functions satisfying the Liouville equations in reduced phase-space and, in particular, it relies on the smoothness of these functions to represent them by rapidly convergent spectral expansions in the phase variables. The resulting (hyperbolic) system of equations for the coefficients in these expansions are then amenable to a high-order accurate treatment via DG approximations. In the present work, we significantly expand on the applicability and efficiency of the approach by incorporating mechanisms that allow for its use in scattering simulations and for a reduced overall computational cost. With regards to the former we demonstrate that the incorporation of curved elements is necessary to attain any kind of accuracy in calculations that involve scattering off non-flat interfaces. With regards to efficiency, on the other hand, we also show that the level-set formulation allows for a space p-adaptive scheme that under-resolves the level-set functions away from the wavefront without incurring in a loss of accuracy in the approximation of its location. As we show, these improvements enable simulations that are beyond the capabilities of previous implementations of these numerical procedures.

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 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 analyze the convergence of a multigrid algorithm for the Hybridizable Discontinuous Galerkin (HDG) method for diffusion problems. We prove that a non-nested multigrid V-cycle, with a single smoothing step per level, converges at a mesh independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it, and identify an abstract class of problems for which a nonnested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than one. Numerical experiments verifying our theoretical results are presented.

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 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.