On 2D elliptic discontinuous Galerkin methods

Sherwin, Spencer J.; Kirby, Robert M.; Peiró, Joaquim; Taylor, Robert L.; Zienkiewicz, O.C.

doi:10.1002/nme.1466

Cited by 39 publications

(39 citation statements)

References 24 publications

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“…In [29] a block diagonal preconditioner for linear problems based upon a domain decomposition method with static condensation is seen to achieve an optimal convergence where the convergence rate of the implicit iterative solver is independent of the number of DOF. Static condensation was also successfully applied to a direct solver in [42]. Multigrid approaches where convergence acceleration is achieved through the use of coarse levels constructed by reducing the polynomial degree (p-multigrid) or using coarser grids with fewer elements (h-multigrid) have also been proposed [18,25,36].…”

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confidence: 99%

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Renac¹,

Marmignon²,

Coquel³

2012

SIAM J. Sci. Comput.

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To cite this version:Florent Renac, Claude Marmignon, Frédéric Coquel.Time implicit high-order discontinuous galerkin method with reduced evaluation cost.SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (1) Abstract. An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A Von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.Key words. discontinuous Galerkin method, nonlinear convection-diffusion equation, implicitexplicit time discretization, pseudo-time integration AMS subject classifications. 65N30, 65N121. Introduction. Discontinuous Galerkin (DG) methods are high-order finite element discretizations and were introduced in the early 1970s for the numerical simulation of the first-order hyperbolic neutron transport equation [34,40]. The method was later extended to nonlinear convection dominated flow problems with the use of a Runge-Kutta method for the time integration [12,14]. In recent years, there has been a strong interest for these techniques in the field of computational fluid dynamics which has led to the introduction of discretization schemes for parabolic and purely elliptic equations. See for example [4,5,8,9,13,19,20,24,28,37,43] and references cited therein. For more details, the reader is referred to the analysis of existing discretizations in an unified framework developed by Arnold et al. [3] and to the overview of recent progress in DG methods for compressible flows [32]. The success of these methods lies in their high-order of accuracy and flexibility thanks to their high degree of locality. The stencil of most DG methods is compact and independent of the space discretization order. This means that the evaluation of the residual in a discretization element involves only this element and its immediate neighbours. The order of the numerical scheme depends on the degree of the approximated piecewise polynomials which can be easily increased. These properties make the DG method well suited to algorithm parallelization, hp-refinement, hp-multigrid, unstructured meshes, the applicati...

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confidence: 99%

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Renac¹,

Marmignon²,

Coquel³

2012

SIAM J. Sci. Comput.

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“…All the methods considered in the unifying analysis of DG methods for second-order elliptic problems in [1], which use polynomial approximations of degree k for both the potential and the flux, converge with the optimal order of k + 1 for the potential and with the suboptimal order of k for the flux. Since the classic continuous Galerkin method needs considerably less degrees of freedom, on the same mesh, and converges with exactly the same orders, the use of DG methods for second-order elliptic equations has been judged as not being particularly advantageous; see, for example, [15]. In [9], the HDG methods were introduced to address this criticism.…”

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An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

Cockburn¹,

Guzmán²,

Soon

et al. 2009

SIAM J. Numer. Anal.

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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 second-order 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.

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“…In both cases however the formulation is not adjoint consistent and one may not prove L 2 -convergence using the Nitsche trick. The question of how much the discontinuous Galerkin method needs to be stabilized was discussed in the case of mixed formulations of elliptic problems by Sherwin et al in [20] and by Marazzina in [16]. It was found that stabilization needs to be applied on the boundary of the domain only to assure well-posedness of the discrete system.…”

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Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems

Burman¹,

Stamm²

2009

SIAM J. Numer. Anal.

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Abstract. We consider DG-methods for 2nd order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric version of the DG-method are well-posed also without penalization of the interelement solution jumps provided boundary conditions are imposed weakly. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a discontinuous Galerkin method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and non-symmetric DG-method without stabilization. All these proposed methods share the feature that they conserve mass locally independent of the penalty parameter.Key words. Discontinuous Galerkin; elliptic equation; Crouzeix-Raviart approximation; interior penalty; local mass conservation.AMS subject classifications. 65M160, 65M151. Introduction. The discontinuous Galerkin method (DG) for (2n)th-order elliptic problems was introduced and analysed by Baker [2] with special focus on the fourth order case. In this work both the jumps of the solution and its gradient were penalized. Douglas and Dupont however announced already in [10] a forthcoming analysis for discontinuous approximations and penalty on the solution jumps. This analysis was then realized in the work of Wheeler [22], in the framework of collocation methods for discontinuous Galerkin and Arnold [1] in the framework of parabolic problems leading to the symmetric interior penalty DG-method (SIPG).During the nineties there was a strong development of discontinuous Galerkin methods for elliptic problems. Bassi and Rebay proposed a formulation for NavierStokes equations [3], introducing an auxiliary variable for the diffusive fluxes, this method was then analysed by Cockburn and co workers in [8,7] and more recently in the framework of Friedrichs systems by Ern and Guermond [12].Babuska, Baumann and Oden proposed a non-symmetric method for elliptic problems with a less stiff penalization term [17] (NIPG). This method was modified by Oden, Prudhomme and Romkes in [19] where they proposed a DG-method for reaction-diffusion equations with penalization of the diffusive fluxes to enhance stability for the non-symmetric version without perturbing the local massconservation properties. The DG-methods for second order elliptic problems have been further analysed in the works by Girault, Rivière and Wheeler [18] and Larson and Niklasson [15]. In the last reference the authors proved that in the non-symmetric case when using high order polynomial approximation the stabilization term may be dropped since control of the solution jumps can be recovered from the anti-symmetric part of the diffusion-operator using an inf-sup argument.For a review of discontinuous Galerkin methods for elliptic problems we refer to Arnold et al. [1] and for a review of stabilization mechanisms in discontinuous Galerkin methods we refer to Brezzi et al. [5].

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On 2D elliptic discontinuous Galerkin methods

Cited by 39 publications

References 24 publications

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems

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