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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.
SUPERCONVERGENT DISCONTINUOUS GALERKIN METHODS FOR SECONDORDER ELLIPTIC PROBLEMS
"... Abstract. We identify discontinuous Galerkin methods for secondorder elliptic problems in several space dimensions having superconvergence properties similar to those of the RaviartThomas and the BrezziDouglasMarini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potent ..."
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Cited by 30 (9 self)
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Abstract. We identify discontinuous Galerkin methods for secondorder elliptic problems in several space dimensions having superconvergence properties similar to those of the RaviartThomas and the BrezziDouglasMarini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L 2 with the optimal order of k + 1, and that the approximate potential and its numerical trace superconverge, in L 2like norms, to suitably chosen projections of the potential, with order k + 2. We also apply elementbyelement postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across interelement boundaries, to converge in L 2 with order k + 1, and to have a divergence converging in L 2 also with order k+1. The new approximate potential is proven to converge with order k+2in L 2. Numerical experiments validating these theoretical results are presented. 1.
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.
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
OPTIMAL CONVERGENCE OF THE ORIGINAL DG METHOD FOR THE TRANSPORTREACTION EQUATION ON SPECIAL MESHES
, 2006
"... Abstract. We show that the approximation given by the original discontinuous Galerkin method for the transportreaction equation in d space dimensions is optimal provided the meshes are suitably chosen: the L 2norm of the error is of order k + 1 when the method uses polynomials of degree k. These m ..."
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Cited by 9 (2 self)
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Abstract. We show that the approximation given by the original discontinuous Galerkin method for the transportreaction equation in d space dimensions is optimal provided the meshes are suitably chosen: the L 2norm 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 only required to be made of simplexes each of which has a unique outflow face. We also find a new, elementbyelement postprocessing of the derivative in the direction of the flow which superconverges with order k + 1.
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 NUMERICAL STUDY ON THE WEAK GALERKIN METHOD FOR THE HELMHOLTZ EQUATION
"... Abstract. A weak Galerkin (WG) method is introduced and numerically tested for the Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite element formulation is symmetric and parameter free. ..."
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Cited by 5 (3 self)
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Abstract. A weak Galerkin (WG) method is introduced and numerically tested for the Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and nonconvex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.
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.