Nonconforming Elements in the Finite Element Method with Penalty

Babuška, Ivo; Zlámal, Miloš

doi:10.1137/0710071

Cited by 214 publications

(129 citation statements)

References 8 publications

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“…In this paper we mainly focus on consistent and stable DG methods. In Section 7 some discussions and numerical results are included for the two non-consistent methods BZ [5] and BMMPR2 [11] (see Tab. 1, third part) which are characterized by a super penalty stabilisation term.…”

Section: Discontinuous Galerkin Methods For the Model Problemmentioning

confidence: 99%

“…To conclude this section, we present a numerical experiment carried out with the super penalty method proposed by Babuška and Zlámal in [5]. This method is not consistent and to ensure stability (in some appropriate norm, see [4] for details) the jumps must be penalised with a power of h depending on the polynomial approximation degree h .…”

Section: Symmetric Methodsmentioning

confidence: 99%

“…At the end of this section, we present some experiments carried out with the super penalty method proposed by Babuška-Zlámal in [5].…”

Section: Symmetric Methodsmentioning

confidence: 99%

“…However, we have also included some numerical tests with our two-level Schwarz method when discretising with the super penalty method proposed in [5] which is not consistent hence to which our analysis does not apply. Nevertheless, a dramatic reduction on the number of iterates needed for convergence and on the condition number for the preconditioned system is observed.…”

Section: Introductionmentioning

confidence: 99%

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Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Antonietti

Ayuso

2007

ESAIM: M2AN

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Abstract. We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345-357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.Mathematics Subject Classification. 65N30, 65N55.

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Section: Discontinuous Galerkin Methods For the Model Problemmentioning

confidence: 99%

Section: Symmetric Methodsmentioning

confidence: 99%

“…At the end of this section, we present some experiments carried out with the super penalty method proposed by Babuška-Zlámal in [5].…”

Section: Symmetric Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Antonietti

Ayuso

2007

ESAIM: M2AN

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“…This, when ∇P (K) ⊂ Σ(K), can be seen as an extension of the Babuška-Zlámal IP method [3] to second order elliptic problems, when α e is chosen as in (3.2). If instead, α e is chosen as in (3.3), we obtain the penalty formulation proposed in [12].…”

Section: Numerical Fluxes Independent Of σ Hmentioning

confidence: 99%

Discontinuous Galerkin Methods for Elliptic Problems

Arnold

Brezzi²,

Cockburn

et al. 2000

Lecture Notes in Computational Science and Engineering

169

136

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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 so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others.

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DiscontinuousGalerkin Methods for Computational Fluid Dynamics

Cockburn

2004

Encyclopedia of Computational Mechanics

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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.

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Nonconforming Elements in the Finite Element Method with Penalty

Cited by 214 publications

References 8 publications

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Discontinuous Galerkin Methods for Elliptic Problems

DiscontinuousGalerkin Methods for Computational Fluid Dynamics

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