scholarly journals Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods

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

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

2021 ◽  
pp. 1-37
Author(s):  
Silvia Bertoluzza ◽  
Ilaria Perugia ◽  
Daniele Prada
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Experiments ◽  
Discontinuous Galerkin Methods ◽  
Convergence Rates ◽  
Galerkin Methods ◽  
Poisson Problem ◽  
Optimal Convergence Rates ◽  
Stability And Error Analysis

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.

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

2014 ◽  
Vol 14 (1) ◽  
pp. 71-87 ◽  
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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