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Volume 3, Issue 3
Reliable and Efficient Averaging Techniques as Universal Tool for a Posteriori Finite Element Error Control on Unstructured Grids

Carsten Carstensen

Int. J. Numer. Anal. Mod., 3 (2006), pp. 333-347.

Published online: 2006-03

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  • Abstract

The striking simplicity of averaging techniques in a posteriori error control of finite element methods as well as their amazing accuracy in many numerical examples over the last decade have made them an extremely popular tool in scientific computing. Given a discrete stress or flux $P_h$ and a post-processed approximation $A(p_h)$, the a posteriori error estimator reads $\eta_A := ||p_h - A(p_h)||$. There is not even a need for an equation to compute the estimator $\eta_A$ and hence averaging techniques are employed everywhere. The most prominent example is occasionally named after Zienkiewicz and Zhu, and also called gradient recovery but preferably called averaging technique in the literature.
The first mathematical justification of the error estimator $\eta_A$ as a computable approximation of the (unknown) error $||p - p_h||$ involved the concept of superconvergence points. For highly structured meshes and a very smooth exact solution $p$, the error $||p - A(p_h)||$ of the post-processed approximation $Ap_h$ may be (much) smaller than $||p - p_h||$ of the given $p_h$. Under the assumption that $||p - A(p_h)||$= h.o.t. is in relative terms sufficiently small, the triangle inequality immediately verifies reliability, i.e.,
                                   $|| p-p_h ||  \leq C_{rel} \eta_A + $h.o.t.,
and efficiency, i.e.,
                                    $\eta_A \leq C_{eff} || p-p_h || +$ h.o.t.,
of the averaging error estimator $\eta_A$. However, the required assumptions on the symmetry of the mesh and the smoothness of the solution essentially contradict the use of adaptive grid refining when $p$ is singular and the proper treatment of boundary conditions remains unclear.
This paper aims at an actual overview on the reliability and efficiency of averaging a posteriori error control for unstructured grids. New aspects are new proofs of the efficiency of all averaging techniques and for all problems.

  • AMS Subject Headings

65N30

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-3-333, author = {Carstensen , Carsten}, title = {Reliable and Efficient Averaging Techniques as Universal Tool for a Posteriori Finite Element Error Control on Unstructured Grids}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2006}, volume = {3}, number = {3}, pages = {333--347}, abstract = {

The striking simplicity of averaging techniques in a posteriori error control of finite element methods as well as their amazing accuracy in many numerical examples over the last decade have made them an extremely popular tool in scientific computing. Given a discrete stress or flux $P_h$ and a post-processed approximation $A(p_h)$, the a posteriori error estimator reads $\eta_A := ||p_h - A(p_h)||$. There is not even a need for an equation to compute the estimator $\eta_A$ and hence averaging techniques are employed everywhere. The most prominent example is occasionally named after Zienkiewicz and Zhu, and also called gradient recovery but preferably called averaging technique in the literature.
The first mathematical justification of the error estimator $\eta_A$ as a computable approximation of the (unknown) error $||p - p_h||$ involved the concept of superconvergence points. For highly structured meshes and a very smooth exact solution $p$, the error $||p - A(p_h)||$ of the post-processed approximation $Ap_h$ may be (much) smaller than $||p - p_h||$ of the given $p_h$. Under the assumption that $||p - A(p_h)||$= h.o.t. is in relative terms sufficiently small, the triangle inequality immediately verifies reliability, i.e.,
                                   $|| p-p_h ||  \leq C_{rel} \eta_A + $h.o.t.,
and efficiency, i.e.,
                                    $\eta_A \leq C_{eff} || p-p_h || +$ h.o.t.,
of the averaging error estimator $\eta_A$. However, the required assumptions on the symmetry of the mesh and the smoothness of the solution essentially contradict the use of adaptive grid refining when $p$ is singular and the proper treatment of boundary conditions remains unclear.
This paper aims at an actual overview on the reliability and efficiency of averaging a posteriori error control for unstructured grids. New aspects are new proofs of the efficiency of all averaging techniques and for all problems.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/905.html} }
TY - JOUR T1 - Reliable and Efficient Averaging Techniques as Universal Tool for a Posteriori Finite Element Error Control on Unstructured Grids AU - Carstensen , Carsten JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 333 EP - 347 PY - 2006 DA - 2006/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/905.html KW - a posteriori error estimate, efficiency, finite element method, gradient recovery, averaging operator, mixed finite element method, non-conforming finite element method. AB -

The striking simplicity of averaging techniques in a posteriori error control of finite element methods as well as their amazing accuracy in many numerical examples over the last decade have made them an extremely popular tool in scientific computing. Given a discrete stress or flux $P_h$ and a post-processed approximation $A(p_h)$, the a posteriori error estimator reads $\eta_A := ||p_h - A(p_h)||$. There is not even a need for an equation to compute the estimator $\eta_A$ and hence averaging techniques are employed everywhere. The most prominent example is occasionally named after Zienkiewicz and Zhu, and also called gradient recovery but preferably called averaging technique in the literature.
The first mathematical justification of the error estimator $\eta_A$ as a computable approximation of the (unknown) error $||p - p_h||$ involved the concept of superconvergence points. For highly structured meshes and a very smooth exact solution $p$, the error $||p - A(p_h)||$ of the post-processed approximation $Ap_h$ may be (much) smaller than $||p - p_h||$ of the given $p_h$. Under the assumption that $||p - A(p_h)||$= h.o.t. is in relative terms sufficiently small, the triangle inequality immediately verifies reliability, i.e.,
                                   $|| p-p_h ||  \leq C_{rel} \eta_A + $h.o.t.,
and efficiency, i.e.,
                                    $\eta_A \leq C_{eff} || p-p_h || +$ h.o.t.,
of the averaging error estimator $\eta_A$. However, the required assumptions on the symmetry of the mesh and the smoothness of the solution essentially contradict the use of adaptive grid refining when $p$ is singular and the proper treatment of boundary conditions remains unclear.
This paper aims at an actual overview on the reliability and efficiency of averaging a posteriori error control for unstructured grids. New aspects are new proofs of the efficiency of all averaging techniques and for all problems.

Carsten Carstensen. (2019). Reliable and Efficient Averaging Techniques as Universal Tool for a Posteriori Finite Element Error Control on Unstructured Grids. International Journal of Numerical Analysis and Modeling. 3 (3). 333-347. doi:
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