Discontinuous Galerkin Methods for Elliptic Problems

Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo; Marini, L. D.

doi:10.1007/978-3-642-59721-3_5

Cited by 169 publications

(141 citation statements)

References 20 publications

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“…Then the DG formulation associated with (1) reads [1,3]: ÿnd u ∈ H 1 ( h ) such that: and identify the di erent DG methods; = − 1 for symmetric DG; = 1 for the BaumannOden method; ¿0 for, respectively, the symmetric ( = − 1) and non-symmetric ( = 1) interior penalty DG method (i.e. IPG or NIPG, respectively).…”

Section: The Weak Formulation For the Convection-di Usion Equationmentioning

confidence: 99%

“…Because the fourth-order discretization (8), both for the Baumann-Oden DG method and for the NIPG method is coercive [3], in our block-relaxation analysis we use these methods for the discretization of the di usion term.…”

Section: We ÿNd Thatmentioning

confidence: 99%

“…DG methods were traditionally introduced for the solution of hyperbolic equations, as methods that have a natural cell-wise upwind character [1,2]. However, since renewed insights were obtained in their application to elliptic problems, DG methods gain in popularity [3][4][5][6], specially because of their convenient properties when combined with the hp-self-adaptive approach and with multigrid (MG) solvers [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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Two‐level multigrid analysis for the convection–diffusion equation discretized by a discontinuous Galerkin method

Raalte

Hemker

2005

Numerical Linear Algebra App

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SUMMARYIn this paper, we study a multigrid (MG) method for the solution of a linear one-dimensional convectiondi usion equation that is discretized by a discontinuous Galerkin method. In particular we study the convection-dominated case when the perturbation parameter, i.e. the inverse cell-Reynolds-number, is smaller than the ÿnest mesh size.We show that, if the di usion term is discretized by the non-symmetric interior penalty method (NIPG) with feasible penalty term, multigrid is su cient to solve the convection-di usion or the convection-dominated equation. Then, independent of the mesh-size, simple MG cycles with symmetric Gauss-Seidel smoothing give an error reduction factor of 0.2-0.3 per iteration sweep.Without penalty term, for the Baumann-Oden (BO) method we ÿnd that only a robust (i.e. cellReynolds-number uniform) two-level error-reduction factor (0.4) is found if the point-wise block-Jacobi smoother is used.

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Section: The Weak Formulation For the Convection-di Usion Equationmentioning

confidence: 99%

Section: We ÿNd Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two‐level multigrid analysis for the convection–diffusion equation discretized by a discontinuous Galerkin method

Raalte

Hemker

2005

Numerical Linear Algebra App

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“…Nitsche's method is closely related to some stabilized Lagrange multiplier techniques [29] which are used in Discontinuous Galerkin methods [1], Domain decomposition methods [9] and Mortar finite element methods [18,30]. From our point of view, this approach provides the most natural implementation of Dirichlet boundary conditions for meshfree methods.…”

Section: Boundary Conditionsmentioning

confidence: 99%

A Particle-Partition of Unity Method Part V: Boundary Conditions

Griebel

Schweitzer

2003

Geometric Analysis and Nonlinear Partial Differential Equations

100

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In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions. Here, the use of an almost forgotten method due to Nitsche from the 1970's allows us to overcome these problems at virtually no extra computational costs. The method is applicable to general point distributions and leads to positive definite linear systems. The results of our numerical experiments, where we consider discretizations with several million degrees of freedom in two and three dimensions, clearly show that we achieve the optimal convergence rates for regular and singular solutions with the (adaptive) h-version and (augmented) p-version.

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“…Therefore, it does not require the introduction of additional stabilization terms with associated parameters and, in contrast to the symmetric form of Nitsche's method, its performance does not depend on the accuracy of the variational estimate or the reliability and robustness of associated numerical algorithms. On the other hand, the non-symmetric Nitsche method leads to unsymmetric system matrices and its numerical analysis framework does not cover optimal convergence rates of the L 2 error [54][55][56][57][58]. This paper extends recent work [59][60][61] that demonstrated the potential of the non-symmetric Nitsche method for parameter-free analysis in the context of non-matching and non-boundary-fitted discretizations.…”

Section: Introductionmentioning

confidence: 86%

A parameter-free variational coupling approach for trimmed isogeometric thin shells

2016

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The non-symmetric variant of Nitsche's method was recently applied successfully for variationally enforcing boundary and interface conditions in non-boundary-fitted discretizations. In contrast to its symmetric variant, it does not require stabilization terms and therefore does not depend on the appropriate estimation of stabilization parameters. In this paper, we further consolidate the non-symmetric Nitsche approach by establishing its application in isogeometric thin shell analysis, where variational coupling techniques are of particular interest for enforcing interface conditions along trimming curves. To this end, we extend its variational formulation within Kirchhoff-Love shell theory, combine it with the finite cell method, and apply the resulting framework to a range of representative shell problems based on trimmed NURBS surfaces. We demonstrate that the non-symmetric variant applied in this context is stable and can lead to the same accuracy in terms of displacements and stresses as its symmetric counterpart. Based on our numerical evidence, the non-symmetric Nitsche method is a viable parameter-free alternative to the symmetric variant in elastostatic shell analysis.

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Discontinuous Galerkin Methods for Elliptic Problems

Cited by 169 publications

References 20 publications

Two‐level multigrid analysis for the convection–diffusion equation discretized by a discontinuous Galerkin method

Two‐level multigrid analysis for the convection–diffusion equation discretized by a discontinuous Galerkin method

A Particle-Partition of Unity Method Part V: Boundary Conditions

A parameter-free variational coupling approach for trimmed isogeometric thin shells

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