Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case

Larson, Mats G.; Niklasson, A. Jonas

doi:10.1007/s00211-004-0528-7

Cited by 25 publications

(27 citation statements)

References 11 publications

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“…In fact, many papers reported observing optimal convergence rates when odd polynomial approximations are used; see for example [4], [3] and [1]. Moreover, two different papers prove that these method, in fact, converge in an optimal way if uniform meshes are used in one dimension; see [6] and [5]. Here we give examples of meshes in one dimension for which we clearly observe suboptimal convergence rates for both the Oden-Babuska-Baumann method [3] and the non-symmetric interior penalty Galerkin (NIPG) method [7].…”

Section: Introductionmentioning

confidence: 99%

Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations

Guzmán

Rivière

2008

J Sci Comput

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Section: Introductionmentioning

confidence: 99%

Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations

Guzmán

Rivière

2008

J Sci Comput

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“…[2,10,12]). Nonsymmetric examples are the celebrated Baumann and Oden DG formulation (BODG [3]), the stabilized DG formulation (SDG [14]), the nonsymmetric interior penalty DG formulation (NIPDG [13]), and the family of formulations considered by Larson and Niklasson (LNDG [9]). …”

Section: ] [[U]][[v]] [[∂ N U]][[∂ N V]]mentioning

confidence: 99%

“…Moreover, for nonsymmetric formulations the error converges suboptimally in the L 2 (Ω)-norm for even-degree broken polynomial spaces. It has been conjectured that this behavior emanates from the nonsymmetry of the formulation; see [3,9,10].…”

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confidence: 99%

An ${H^1(\mathcal{P}^{\mathsf{h})}$‐Coercive Discontinuous Galerkin Formulation for the Poisson Problem: 1D Analysis

Zee¹,

Brummelen²,

Borst³

2006

SIAM J. Numer. Anal.

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Abstract. Coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite-element discretizations: By the classical Lax-Milgram theorem, any conforming discretization of a coercive variational problem is stable; i.e., discrete approximations are well-posed and possess unique solutions, irrespective of the specifics of the underlying approximation space. Based on the prototypical one-dimensional Poisson problem, we establish in this work that most concurrent discontinuous Galerkin formulations for second-order elliptic problems represent instances of a generic conventional formulation and that this generic formulation is noncoercive. Consequently, all conventional discontinuous Galerkin formulations are a fortiori noncoercive, and typically their well-posedness is contingent on approximation-space-dependent stabilization parameters. Moreover, we present a new symmetric nonconventional discontinuous Galerkin formulation based on element Green's functions and the data local to the edges. We show that the new discontinuous Galerkin formulation is coercive on the broken Sobolev space H 1 (P h ), viz., the space of functions that are elementwise in the H 1 Sobolev space. The coercivity of the new formulation is supported by calculations of discrete inf-sup constants, and numerical results are presented to illustrate the optimal convergence behavior in the energy-norm and in the L 2 (Ω)-norm.Key words. finite element method, discontinuous Galerkin, elliptic problems, coercivity AMS subject classifications. 65N30, 65N12 DOI. 10.1137/05063057X1. Introduction. The recent renewal of interest in discontinuous Galerkin (DG) methods for second-order elliptic boundary value problems can be attributed to twofold reasons. First, DG methods provide robust finite-element discretizations for hyperbolic conservation laws, as the interelement discontinuities enable an extension of Godunov's method for finite-volume methods. However, to extend these techniques to singularly perturbed elliptic problems, an appropriate treatment for the elliptic part of the operator is required. Second, the absence of interelement-continuity constraints renders DG methods ideally suited for hp adaptivity, e.g., based on a posteriori error estimation; see, for instance, [1,8]. A comprehensive overview of the historical development of DG methods is provided in [7].A framework for analyzing DG formulations for elliptic problems has recently been erected in [2]. Although the analysis in [2] clarifies basic properties of the different formulations, it does not seem to warrant a clear preference. The literature on DG methods for elliptic problems is dominated by formulations that possess edge terms composed of linear combinations of the jumps and averages of the test and trial functions and their normal derivatives. That is, denoting by u and v the test and trial functions, and by [[ · ]] and {·} the jump and average of (·) at an interelement edge, these formulations contain terms conforming to {∂ n u} [[v]

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“…However, numerical experiments [7,23] reveal that the nonsymmetric discontinuous Galerkin methods, including the Oden-Babuška-Baumann (OBB) formulation [7,17], the nonsymmetric interior penalty Galerkin (NIPG) method [19], and the incomplete interior penalty Galerkin (IIPG) method [12,22,23], are generally not optimal in the L 2 norm. An optimal-order error estimate in the L 2 norm has been proved for the NIPG and IIPG methods with odd degree polynomials for onedimensional elliptic problems with a uniform partition [16].…”

Section: Introductionmentioning

confidence: 99%

A Family of Characteristic Discontinuous Galerkin Methods for Transient Advection-Diffusion Equations and Their Optimal-Order L^2 Error Estimates

Wang

Al‐Lawatia³

et al. 2009

CiCP

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Abstract. We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations, including the characteristic NIPG, OBB, IIPG, and SIPG schemes. The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods. An optimal-order error estimate in the L 2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG, IIPG, and SIPG scheme. Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG, OBB, IIPG, and SIPG schemes in the context of advection-diffusion equations.

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Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case

Cited by 25 publications

References 11 publications

Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations

Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations

An ${H^1(\mathcal{P}^{\mathsf{h})}$‐Coercive Discontinuous Galerkin Formulation for the Poisson Problem: 1D Analysis

A Family of Characteristic Discontinuous Galerkin Methods for Transient Advection-Diffusion Equations and Their Optimal-Order L^2 Error Estimates

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