A p-robust polygonal discontinuous Galerkin method with minus one stabilization

In this paper, we introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree [Formula: see text]. The stabilization is obtained by penalizing, in each mesh element [Formula: see text], a residual in the norm of the dual of [Formula: see text]. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a [Formula: see text]-explicit stability and error analysis, proving [Formula: see text]-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.

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Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods

Journal of Computational Mathematics ◽

10.4208/jcm.2003-m2018-0223 ◽

2021 ◽

Vol 39 (2) ◽

pp. 283-310

Author(s):

Qingguo Hong & Jinchao Xu

Keyword(s):

Error Analysis ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Methods ◽

Uniform Stability ◽

Galerkin Methods ◽

Stability And Error Analysis

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Discontinuous Galerkin methods on hp-anisotropic meshes II: a posteriori error analysis and adaptivity

Applied Numerical Mathematics ◽

10.1016/j.apnum.2008.12.008 ◽

2009 ◽

Vol 59 (9) ◽

pp. 2179-2194 ◽

Cited By ~ 24

Author(s):

Emmanuil H. Georgoulis ◽

Edward Hall ◽

Paul Houston

Keyword(s):

Error Analysis ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Methods ◽

Posteriori Error ◽

Galerkin Methods ◽

A Posteriori ◽

A Posteriori Error Analysis ◽

A Posteriori Error ◽

Anisotropic Meshes ◽

Posteriori Error Analysis

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DISCONTINUOUS GALERKIN METHODS FOR THE MULTI-DIMENSIONAL VLASOV–POISSON PROBLEM

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820251250042x ◽

2012 ◽

Vol 22 (12) ◽

pp. 1250042 ◽

Cited By ~ 22

Author(s):

BLANCA AYUSO DE DIOS ◽

JOSÉ A. CARRILLO ◽

CHI-WANG SHU

Keyword(s):

Discontinuous Galerkin ◽

Discontinuous Galerkin Methods ◽

Mixed Finite Element ◽

Galerkin Approximation ◽

Mixed Finite Element Method ◽

Optimal Error Estimates ◽

Galerkin Methods ◽

Poisson System ◽

Optimal Error ◽

Poisson Problem

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov–Poisson system. The schemes are constructed by combining a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

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Pre-asymptotic error analysis of hp-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2015.06.007 ◽

2015 ◽

Vol 70 (5) ◽

pp. 917-933 ◽

Cited By ~ 7

Author(s):

Lingxue Zhu ◽

Yu Du

Keyword(s):

Error Analysis ◽

Helmholtz Equation ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Methods ◽

Wave Number ◽

Galerkin Methods ◽

Large Wave ◽

Asymptotic Error ◽

Large Wave Number ◽

Asymptotic Error Analysis

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A New Generalization of the P1 Non-Conforming FEM to Higher Polynomial Degrees

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2016-0031 ◽

2017 ◽

Vol 17 (1) ◽

pp. 161-185 ◽

Cited By ~ 1

Author(s):

Mira Schedensack

Keyword(s):

Adaptive Algorithm ◽

Numerical Experiments ◽

Convergence Rates ◽

A Priori ◽

Projection Property ◽

Polynomial Degree ◽

Poisson Problem ◽

Optimal Convergence Rates ◽

New Formulation ◽

Three Space

AbstractThis paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

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A Remark on the A Posteriori Error Analysis of Discontinuous Galerkin Methods for the Obstacle Problem

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2013-0015 ◽

2014 ◽

Vol 14 (1) ◽

pp. 71-87 ◽

Cited By ~ 5

Author(s):

Thirupathi Gudi ◽

Kamana Porwal

Keyword(s):

Error Analysis ◽

Discontinuous Galerkin ◽

Obstacle Problem ◽

Discontinuous Galerkin Methods ◽

Auxiliary Problem ◽

Posteriori Error ◽

Error Estimator ◽

Galerkin Methods ◽

A Posteriori ◽

Posteriori Error Analysis

Abstract. We revisit the a posteriori error analysis of discontinuous Galerkin methods for the obstacle problem derived in [Math. Comput. (2013), DOI 10.1090/S0025-5718-2013-02728-7]. Under a mild assumption on the trace of obstacle, we derive a reliable a posteriori error estimator which does not involve min/max functions. A key in this approach is an auxiliary problem with discrete obstacle. Applications to various discontinuous Galerkin finite element methods are presented. Numerical experiments show that the new estimator obtained in this article performs better.

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