scholarly journals An a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for computing band gaps in photonic crystals

2012 ◽  
Vol 236 (18) ◽  
pp. 4810-4826 ◽  
Author(s):  
Stefano Giani
Keyword(s):  
Photonic Crystals ◽  
Discontinuous Galerkin Methods ◽  
Posteriori Error ◽  
Band Gaps ◽  
Error Estimator ◽  
Galerkin Methods ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Posteriori Error Estimator ◽  
A Posteriori Error

scholarly journals AN A POSTERIORI ERROR ESTIMATOR FOR hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC EIGENVALUE PROBLEMS

2012 ◽  
Vol 22 (10) ◽  
pp. 1250030 ◽  
Author(s):  
STEFANO GIANI ◽  
EDWARD J. C. HALL
Keyword(s):  
Discontinuous Galerkin Methods ◽  
Eigenvalue Problems ◽  
Posteriori Error ◽  
Error Estimator ◽  
Galerkin Methods ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Elliptic Eigenvalue Problems

In this paper we present a residual-based a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. In particular, we use as a model problem the Laplace eigenvalue problem on bounded domains in ℝd, d = 2, 3, with homogeneous Dirichlet boundary conditions. Analogous error estimators can be easily obtained for more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of the residual-based error estimator also for non-convex domains and use numerical experiments to show that, under an hp-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non-smooth eigenfunctions.

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.

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