A posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements

Friedhelm Schieweck

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 3, page 489-503
  • ISSN: 0764-583X

Abstract

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For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited that the nonconforming finite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many known nonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for the post-processing. We demonstrate that the idea to use a computed conforming approximation of the nonconforming solution can be applied also to derive an a posteriori error estimate for a linear functional of the solution which represents some quantity of physical interest.

How to cite

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Schieweck, Friedhelm. "A posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements." ESAIM: Mathematical Modelling and Numerical Analysis 36.3 (2010): 489-503. <http://eudml.org/doc/194113>.

@article{Schieweck2010,
abstract = { For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited that the nonconforming finite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many known nonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for the post-processing. We demonstrate that the idea to use a computed conforming approximation of the nonconforming solution can be applied also to derive an a posteriori error estimate for a linear functional of the solution which represents some quantity of physical interest. },
author = {Schieweck, Friedhelm},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori error estimates; nonconforming finite elements; post-processing.; a posteriori error estimate; post-processing; second order elliptic equation},
language = {eng},
month = {3},
number = {3},
pages = {489-503},
publisher = {EDP Sciences},
title = {A posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements},
url = {http://eudml.org/doc/194113},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Schieweck, Friedhelm
TI - A posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 3
SP - 489
EP - 503
AB - For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited that the nonconforming finite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many known nonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for the post-processing. We demonstrate that the idea to use a computed conforming approximation of the nonconforming solution can be applied also to derive an a posteriori error estimate for a linear functional of the solution which represents some quantity of physical interest.
LA - eng
KW - A posteriori error estimates; nonconforming finite elements; post-processing.; a posteriori error estimate; post-processing; second order elliptic equation
UR - http://eudml.org/doc/194113
ER -

References

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