Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension

Celiker, Fatih; Cockburn, Bernardo

doi:10.1090/s0025-5718-06-01895-3

Cited by 97 publications

(40 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, we show that this happens if we take, on each simplex 6) where e τ K is an arbitrary but fixed face of K and τ K is a strictly positive real number. Since the local penalization parameter τ is nonzero only on a single face of each simplex, we call this LDG-H method the single-face hybridizable DG method; for simplicity, we are going to refer to the method under consideration by the SF-H k method.…”

Section: Introductionmentioning

confidence: 92%

“…Since the local penalization parameter τ is nonzero only on a single face of each simplex, we call this LDG-H method the single-face hybridizable DG method; for simplicity, we are going to refer to the method under consideration by the SF-H k method. It is interesting to note two of the minimal dissipation DG methods considered in [6], in the framework of a study of superconvergence properties of DG methods for one-dimensional steady-state convection-diffusion problems, happen to be an SF-H method. The first is called the md-DG method (see Table 1 in [6]) and is obtained, in our notation, by taking on each interior node x i ,…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

2008

Self Cite

View full text Add to dashboard Cite

Abstract. We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree k ≥ 0 for both the potential as well as the flux, the order of convergence in L 2 of both unknowns is k + 1. Moreover, both the approximate potential as well as its numerical trace superconverge in L 2 -like norms, to suitably chosen projections of the potential, with order k + 2. This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order k +2 in L 2 . The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.

show abstract

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

2008

Self Cite

View full text Add to dashboard Cite

show abstract

“…[30]. There have been several DG methods suggested in literature to solve the problem, including the method originally proposed by Bassi and Rebay [4] for compressible Navier-Stokes equations, its generalization called the local discontinuous Galerkin (LDG) methods introduced in [19] by Cockburn and Shu and further studied in [6,7,12,15]; as well as the method introduced by Baumann-Oden [5,27]. Also in the 1970s, Galerkin methods for elliptic and parabolic problems using discontinuous finite elements, called the interior penalty (IP) methods, were independently introduced and studied; see, e.g., [1,3,34].…”

Section: Introductionmentioning

confidence: 99%

The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems

Liu¹,

Yan²

2009

SIAM J. Numer. Anal.

182

154

View full text Add to dashboard Cite

Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47 (1) (2009), . In this work, we show that higher order (k ≥ 4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all p k elements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one-and two-dimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.

show abstract

“…We choose for application the NIPG because here a supercloseness result is known. For convection-diffusion equations in 1d several supercloseness results using numerical traces and possible postprocessing methods are known, see [4,15].…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Recovery of Discontinuous Galerkin

Franz

Tobiska

Zarin

2009

JCM

View full text Add to dashboard Cite

A new recovery operator P : Q disc n (T ) → Q disc n+1 (M) for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh T into a higher order polynomial space on a macro mesh M. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local L2-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.Mathematics subject classification: 65N12, 65N15, 65N30.

show abstract

Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension

Cited by 97 publications

References 28 publications

A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems

A New Approach to Recovery of Discontinuous Galerkin

Contact Info

Product

Resources

About