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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 97 (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.
THE DERIVATION OF HYBRIDIZABLE DISCONTINUOUS GALERKIN METHODS FOR STOKES FLOW
"... Abstract. 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 ..."
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Cited by 36 (5 self)
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Abstract. 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.
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.
A Weak Galerkin mixed finite element method for secondorder elliptic problems
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To CG or to HDG: A Comparative Study
 J SCI COMPUT (2012) 51:183–212
, 2012
"... 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 ..."
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Cited by 11 (3 self)
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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
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.
Stabilized continuous and discontinuous Galerkin techniques for
"... Abstract. We design stabilized methods based on the variational multiscale decomposition of Darcy’s problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain ..."
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Cited by 9 (4 self)
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Abstract. We design stabilized methods based on the variational multiscale decomposition of Darcy’s problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic the different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L2norm. Key words. Darcy’s problem, stabilized finite element methods, characteristic length scale, orthogonal subgrid scales AMS subject classifications. 65N30, 35Q30 1. Introduction. Darcy’s
An adaptive shockcapturing HDG method for compressible flows presented at AIAA Conference
, 2011
"... We introduce a hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the compressible Euler equations with shock waves. By locally condensing the approximate conserved variables the HDG method results in a final system involving only the degrees of freedom of the approximate ..."
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Cited by 6 (3 self)
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We introduce a hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the compressible Euler equations with shock waves. By locally condensing the approximate conserved variables the HDG method results in a final system involving only the degrees of freedom of the approximate traces of the conserved variables. The HDG method inherits the geometric flexibility and highorder accuracy of discontinuous Galerkin methods, and offers a significant reduction in the computational cost. In order to treat compressible fluid flows with discontinuities, the HDG method is equipped with an artificial viscosity term based on an extension of existing artificial viscosity methods. Moreover, the artificial viscosity can be used as an indicator for adaptive grid refinement to improve shock profiles. Numerical results for subsonic, transonic, supersonic, and hypersonic flows are presented to demonstrate the performance of the proposed approach. I.
A Conservative and monotone mixedhybridized finite element approximation of transport problems
 in heterogeneous domains, Computer Methods in Applied Mechanics and Engineering
, 2010
"... In this article, we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties. The model in each subdomain consists of a partial differential equation with diffusive, convective and reactive terms, the couplin ..."
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Cited by 5 (0 self)
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In this article, we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties. The model in each subdomain consists of a partial differential equation with diffusive, convective and reactive terms, the coupling between each subdomain being realized through an interface transmission condition of Robin type. The numerical approximation of the problem in the two–dimensional case is carried out through a dual mixed–hybridized finite element method with numerical quadrature of the mass flux matrix. The resulting method is a conservative finite volume scheme over triangular grids, for which a discrete maximum principle is proved under the assumption that the mesh is of Delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface. The stability, accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in Biology, Electrophysiology and Neuroelectronics. Key words: Transport phenomena, heterogeneous problems, mixed–hybridized finite
A mixed method for the biharmonic problem based on a system of firstorder equations
 SIAM J. Numer. Anal
"... Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is rewritten as a system of four firstorder equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to ..."
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Cited by 3 (0 self)
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Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is rewritten as a system of four firstorder equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation. For k ≥ 1 a projection of the primal variable error superconverges with order k+3 while the error itself converges with order k+ 1 only. This fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k + 3. We provide numerical experiments that validate our theoretical results.