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UNIFIED HYBRIDIZATION OF DISCONTINUOUS GALERKIN, MIXED AND CONTINUOUS GALERKIN METHODS FOR SECOND ORDER ELLIPTIC PROBLEMS
"... Abstract. 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 mixeddual finite element methods including hybridized mixed, continuous Galerkin, nonconforming and a new wide cla ..."
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Cited by 100 (18 self)
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Abstract. 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 mixeddual finite element methods including hybridized mixed, continuous Galerkin, nonconforming and a new wide class of hybridizable discontinuous Galerkin methods. The main feature of the methods in this framework is that their approximate solutions can be expressed in an elementbyelement fashion in terms of an approximate trace satisfying a global weak formulation. Since the associated matrix is 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 new, extremely localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom. 1.
A superconvergent LDGhybridizable Galerkin method for secondorder elliptic problems
 Math. Comp
"... Abstract. We identify and study an LDGhybridizable Galerkin method, which is not an LDGmethod, for secondorder elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuousGalerkinmethods using polynomials of degree k ≥ 0 for both the po ..."
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Cited by 37 (10 self)
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Abstract. We identify and study an LDGhybridizable Galerkin method, which is not an LDGmethod, for secondorder elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuousGalerkinmethods using polynomials of degree k ≥ 0 for both the potential as well as the flux, the order of convergence in L2 of both unknowns is k + 1. Moreover, both the approximate potential as well as its numerical trace superconverge in L2like norms, to suitably chosen projections of the potential, with order k+2. This allows the application of elementbyelement postprocessing of the approximate solution which provides an approximation of the potential converging with order k+2 in L2. The method can be thought to be in between the hybridized version of the RaviartThomas and that of the BrezziDouglasMarini mixed methods. 1.
A Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems,” submitted
 SIAM J. for Numerical Analaysis
, 2006
"... Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is the ..."
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Cited by 34 (14 self)
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Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility.
A PROJECTIONBASED ERROR ANALYSIS OF HDG METHODS
"... Abstract. 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 dis ..."
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Cited by 28 (7 self)
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Abstract. 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 secondorder elliptic problem. 1.
Local Discontinuous Galerkin Methods for HighOrder TimeDependent Partial Differential Equations
, 2010
"... Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, ..."
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Cited by 12 (1 self)
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Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, less restriction in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and local data structure and the resulting high parallel efficiency. In this paper, we give a general review of the local DG (LDG) methods for solving highorder timedependent partial differential equations (PDEs). The important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, is highlighted. Some of the applications of the LDG methods for highorder timedependent PDEs are also be discussed.
Analysis of a local discontinuous Galerkin method for linear timedependent fourthorder problems
 SIAM J. Numer. Anal
"... Abstract. We analyze a local discontinuous Galerkin (LDG) method for fourthorder timedependent problems. Optimal error estimates are obtained in one dimension and in multidimensions for Cartesian and triangular meshes. We extend the analysis to higher evenorder equations and the linearized Cahn ..."
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Cited by 9 (3 self)
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Abstract. We analyze a local discontinuous Galerkin (LDG) method for fourthorder timedependent problems. Optimal error estimates are obtained in one dimension and in multidimensions for Cartesian and triangular meshes. We extend the analysis to higher evenorder equations and the linearized CahnHilliard type equations. Numerical experiments are displayed to verify the theoretical results. 1.
An analysis of the embedded discontinuous Galerkin method for secondorder elliptic problems
 SIAM J. Numer. Anal
"... Abstract. 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 secondorder elliptic problems obtained from hybridizable discontinuous methods ..."
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Cited by 9 (4 self)
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Abstract. 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 secondorder 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 ≥ 1 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.
The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode
 J. Sci. Comput
"... equation in a resonant tunneling diode∗ ..."
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Local Discontinuous Galerkin Method with Reduced Stabilization for Diffusion Equations
, 2008
"... Abstract. We extend the results on minimal stabilization of Burman and Stamm [J. Sci. Comp., 33 (2007), pp. 183208] to the case of the local discontinuous Galerkin methods on mixed form. The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum. Stability in the ..."
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Cited by 2 (0 self)
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Abstract. We extend the results on minimal stabilization of Burman and Stamm [J. Sci. Comp., 33 (2007), pp. 183208] to the case of the local discontinuous Galerkin methods on mixed form. The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum. Stability in the form of a discrete infsup condition is proved and optimal convergence follows. Some numerical examples using high order approximation spaces illustrate the theory. AMS subject classifications: 65N30, 35F15 Key words: Local discontinuous Galerkin hFEM, interior penalty, diffusion equation.
AN ANALYSIS OF HDG METHODS FOR CONVECTION–DOMINATED DIFFUSION PROBLEMS
"... Abstract. We present the first a priori error analysis of the h–version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection–dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L2–error of the scalar variable converges with o ..."
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Cited by 1 (1 self)
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Abstract. We present the first a priori error analysis of the h–version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection–dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L2–error of the scalar variable converges with order k + 1/2 on general conforming quasi–uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L2–convergence order of k+ 1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results. 1.