The Bassi Rebay 1 scheme is a special case of the Symmetric Interior Penalty formulation for discontinuous Galerkin discretisations with Gauss–Lobatto points

2018 ◽  
Vol 363 ◽  
pp. 1-10 ◽  
Author(s):  
Juan Manzanero ◽  
Andrés M. Rueda-Ramírez ◽  
Gonzalo Rubio ◽  
Esteban Ferrer
Keyword(s):  
Discontinuous Galerkin ◽  
Interior Penalty ◽  
Penalty Formulation ◽  
Special Case

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

2021 ◽  
Vol 36 (6) ◽  
pp. 313-336
Author(s):  
Ronald H. W. Hoppe ◽  
Youri Iliash
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Interior Penalty ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Flux Function

Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

A two-energies principle for the biharmonic equation and ana posteriorierror estimator for an interior penalty discontinuous Galerkin approximation

2018 ◽  
Vol 52 (6) ◽  
pp. 2479-2504 ◽  
Author(s):  
Dietrich Braess ◽  
R.H.W. Hoppe ◽  
Christopher Linsenmann
Keyword(s):  
Discontinuous Galerkin ◽  
Biharmonic Equation ◽  
Moment Tensor ◽  
Galerkin Approximation ◽  
Direct Consequence ◽  
Mixed Formulation ◽  
Error Estimator ◽  
A Posteriori ◽  
Reliability Estimate ◽  
Interior Penalty

We consider ana posteriorierror estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator.

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