An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

2019 ◽  
Vol 40 (2) ◽  
pp. 951-975 ◽  
Author(s):  
Dietrich Braess ◽  
Astrid S Pechstein ◽  
Joachim Schöberl
Keyword(s):  
Error Bound ◽  
Biharmonic Equation ◽  
Moment Tensor ◽  
Galerkin Approximation ◽  
Symmetric Tensor ◽  
Posteriori Error ◽  
A Posteriori ◽  
Tensor Fields ◽  
A Posteriori Error ◽  
The Moment

Abstract We develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan–Herrmann–Johnson formulation, which directly provides an equilibrated moment tensor.

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.

scholarly journals Quantifying the Impact of Material-Model Error on Macroscale Quantities-of-Interest Using Multiscale a Posteriori Error-Estimation Techniques

MRS Advances ◽  
2016 ◽  
Vol 1 (40) ◽  
pp. 2789-2794 ◽  
Author(s):  
Judith A. Brown ◽  
Joseph E. Bishop
Keyword(s):  
Elastic Properties ◽  
Error Bound ◽  
Error Estimation ◽  
Posteriori Error ◽  
A Posteriori Error Estimation ◽  
Material Behavior ◽  
A Posteriori ◽  
Posteriori Error Estimation ◽  
A Posteriori Error ◽  
Weld Properties

ABSTRACTAn a posteriori error-estimation framework is introduced to quantify and reduce modeling errors resulting from approximating complex mesoscale material behavior with a simpler macroscale model. Such errors may be prevalent when modeling welds and additively manufactured structures, where spatial variations and material textures may be present in the microstructure. We consider a case where a <100> fiber texture develops in the longitudinal scanning direction of a weld. Transversely isotropic elastic properties are obtained through homogenization of a microstructural model with this texture and are considered the reference weld properties within the error-estimation framework. Conversely, isotropic elastic properties are considered approximate weld properties since they contain no representation of texture. Errors introduced by using isotropic material properties to represent a weld are assessed through a quantified error bound in the elastic regime. An adaptive error reduction scheme is used to determine the optimal spatial variation of the isotropic weld properties to reduce the error bound.

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