A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

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.

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

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2012.02.007 ◽

2012 ◽

Vol 236 (18) ◽

pp. 4810-4826 ◽

Cited By ~ 4

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

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

Russian Journal of Numerical Analysis and Mathematical Modelling ◽

10.1515/rnam-2021-0026 ◽

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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Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

Numerische Mathematik ◽

10.1007/s00211-019-01095-x ◽

2019 ◽

Vol 144 (3) ◽

pp. 585-614

Author(s):

Joscha Gedicke ◽

Arbaz Khan

Keyword(s):

Discontinuous Galerkin ◽

Eigenvalue Problems ◽

Convergence Rates ◽

A Priori ◽

Posteriori Error ◽

Error Estimator ◽

A Posteriori ◽

A Posteriori Error Estimator ◽

Posteriori Error Estimator ◽

A Posteriori Error

AbstractIn this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient. We finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator.

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