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Unified analysis of discontinuous Galerkin methods for elliptic problems
 SIAM J. Numer. Anal
, 2001
"... Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment ..."
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Cited by 525 (31 self)
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Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
An A Priori Error Analysis Of The Local Discontinuous Galerkin Method For Elliptic Problems
, 2000
"... . In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin 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 L 2 norm of the gradient and ..."
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Cited by 96 (25 self)
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. In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin 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 L 2 norm of the gradient and the L 2 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 \Gamma1 are taken, the order of convergence of the potential increases to k + 1. The optimality of these theoretical results are tested in a series of numerical experiments on two dimensional domains. Key words. Finite elements, discontinuous Galerkin methods, elliptic problems AMS subject classifications. 65N30 1. Introduction. In this paper, we present the first a priori error analysis of the Local Discontinuous Galerkin (LDG) method for the following classical model elliptic problem: \Gamma\Deltau = f in\Omega ; u ...
Discontinuous Galerkin methods for elliptic problems
 In Discontinuous Galerkin Methods (Newport, RI, 1999), Lecture Notes Computational Science Engineering
, 2000
"... Abstract. We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the varian ..."
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Cited by 55 (7 self)
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Abstract. We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the socalled interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others. 1
Preconditioning in H(div) and Applications
 Math. Comp
, 1998
"... . We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I \Gamma grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational fo ..."
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Cited by 47 (5 self)
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. We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I \Gamma grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems. 1. Introduction The Hilbert space H(div) consists of squareintegr...
Preconditioning discrete approximations of the ReissnerMindlin plate model
 Math. Modelling Numer. Anal
, 1997
"... Abstract. We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner–Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear sys ..."
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Cited by 38 (10 self)
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Abstract. We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner–Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral condition number independent of both the mesh size h and the plate thickness t. We further discuss how this preconditioner may be implemented and then apply it to efficiently solve this indefinite linear system. Although the mixed formulation of the Reissner–Mindlin problem has a saddlepoint structure common to other mixed variational problems, the presence of the small parameter t and the fact that the matrix in the upper left corner of the partition is only positive semidefinite introduces new complications.
PRECONDITIONING DISCRETIZATIONS OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
, 2009
"... This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be c ..."
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Cited by 26 (4 self)
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This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be considered. In particular, parameter dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several examples and models which have been discussed in the literature previously. However, here each example is discussed with reference to a more unified abstract approach.
Convergence and optimality of adaptive mixed finite element methods
, 2009
"... The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved ..."
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Cited by 18 (5 self)
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The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.
Preconditioning Methods for Local Discontinuous Galerkin Discretizations
 SIAM J. SCI. COMPUT
, 2002
"... A multilevel interior penalty method is used as an efficient preconditioner for the Schur complement of the local discontinuous Galerkin (LDG) discretization of Poisson's problem. Then, this is used in a blocktriangular preconditioner of the LDG saddle point system. The block preconditioner i ..."
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Cited by 8 (2 self)
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A multilevel interior penalty method is used as an efficient preconditioner for the Schur complement of the local discontinuous Galerkin (LDG) discretization of Poisson's problem. Then, this is used in a blocktriangular preconditioner of the LDG saddle point system. The block preconditioner is of the same eciency as the Schur complement version. Finally, the block preconditioner is extended to the discretization of Stokes' problem by the LDG method. Again, the preconditioned saddle point problem can be solved in about as many steps as the Schur complement. The influence of several parameters on the performance of these methods is investigated.
Interior penalty discontinuous approximations of elliptic problems
, 2000
"... Abstract. This paper studies an interior penalty discontinuous approximation of elliptic problems on non–matching grids. Error analysis, interface domain decomposition type preconditioners, as well as numerical results illustrating both, discretization errors and condition number estimates of the pr ..."
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Cited by 5 (2 self)
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Abstract. This paper studies an interior penalty discontinuous approximation of elliptic problems on non–matching grids. Error analysis, interface domain decomposition type preconditioners, as well as numerical results illustrating both, discretization errors and condition number estimates of the problem and reduced forms of it, are presented. 1.