2. Posterior error estimators#
2.1. Definition of a posteriori estimators#
In a posteriori estimation, the objective is not to find an estimate of the error function \(e\) but to determine an estimate of a measure of the error. Thus, the estimators that have been developed over the last thirty years provide a reliable and accurate estimate of the error in global standards, such as standards \({H}^{1}\), standards \({L}^{2}\) or even energy standards, in post-processing the finite elements solution. This choice of global standards is imposed by the bilinear form. When the form is positive and symmetric, as is the case here, this induces a dot product for which the associated norm is the energy norm; it is therefore natural to estimate the error in this norm.
The basic principle of this type of estimation is to use the approximate solution to estimate the discretization error. As opposed to a priori estimation, a posteriori estimates can only be made once the approximate solution has been calculated.
A \(\eta (h,{u}^{h},d)\) function is an a posteriori error estimate if:
where \(\parallel \cdot \parallel\) is a standard for travel fields, \(h\) the size of elements, is the size of the elements, \(d\) is a data set of the problem and \({u}^{h}\) is the approximate solution. Additionally, if \(\eta (h,d,{u}^{h})\) can be located in the form of:
then the quantities \({\eta }_{E}({u}^{h},d)\), elementary contributions to the estimation of the global error \(\eta (h,{u}^{h},d)\), are called local error indicators. They provide a basis for adapting meshes.
Numerous estimators exist and can be classified into three categories:
Error estimators based on balance faults from the work of Babuška and Rheinboldt [bib3];
Error indicators constructed from smoothed constraints from the work of Zienkiewicz and Zhu [bib4];
Error measures based on the concept of error in relation to behavior from the work of Ladevèze [bib5].
These various estimators, which are now well controlled, will be presented in more or less detail. A more detailed review can be found in [bib6] to [bib10].
2.2. Quality of estimators#
When an estimator is defined, it is necessary to question the criteria and the means by which to judge the performance of this estimator. In general, it is necessary to try to build an estimator whose asymptotic behavior (when the \(h\) size of the elements tends to zero) follows that of the error. This behavior results in the existence of two constants \({C}_{1}\) and \({C}_{2}\), depending on the problem data and on the discretization but not on the size of the elements, verifying the following relationship (Ladevèze & Pelle, 2004):
: label: EQ-None
{C} _ {1}etaleleparallel eparallelle {C} _ {2}eta
where \(\eta\) represents the estimate of the \(\parallel e\parallel\) measure of the error \(e\) for the domain in question. Thus, to judge the performance of estimators, there are various intrinsic criteria.
2.2.1. Efficiency Index#
The efficiency index \(\gamma\) is defined as the ratio between the error calculated by an estimator \({e}_{\text{estimée}}\) and the true error \({e}_{\text{vraie}}\):
: label: EQ-None
gamma =frac {{e} _ {text {estimated}}} {{e} _ {text {true}}}
Unless an analytical solution is available, the true error is calculated as the difference between a solution obtained on a very fine mesh (overkill solution) and the solution obtained on a given mesh. An efficiency index close to unity characterizes a good estimator. If this property is reached when the size of the elements approaches zero, the estimator is said to be asymptotically accurate. However, the estimate must be accurate enough (\(\gamma\) close to 1) for fairly coarse meshes in order to be usable for current engineering discretizations. Finally, it is desirable to overestimate the error (\(\gamma >1\)) in order to be able to be used as a stopping criterion in an adaptive process.
Systematic studies have been carried out by Strouboulis and Haque [bib11] and by Babuška [bib12] on various test cases having an analytical solution or for which a reference solution can be determined on a very fine mesh. Overall, it appears that the quality of an estimator depends on the topology of the mesh, the regularity of the solution and the regularity (flattening) of the elements.
The only global index \(\gamma\) does not make it possible to account for how the local error behaves: in some cases, the estimator may be satisfactory for this criterion without succeeding in locating the areas where the energy error is significant [bib13].
2.2.2. Robustness index#
In order to get rid of the local behavior of the estimator, another criterion was defined by Babuška and Rodriguez [bib14]. For a given triangulation \(T\), the local efficiency index \({\gamma }_{\omega }\) relating to the sub-domain \(\omega\) is defined by:
: label: EQ-None
{gamma} _ {omega} =frac {{e} _ {text {estimated}} ^ {omega}} {{e} _ {text {true}}} {text {true}}} ^ {omega}}
Babuška and Strouboulis showed that the asymptotic variation range of \({\gamma }_{\omega }\) [bib15] could be accessed numerically:
: label: EQ-None
0< {C} _ {text {inf}}} ^ {omega}the {omega} _ {omega}the {C} _ {text {sup}}} ^ {omega}} ^ {omega} <infty
The difference to 1 of \({\gamma }_{\omega }\) is measured by the index \({R}_{\omega }\) defined by:
: label: EQ-None
{R} _ {omega} =text {max}left{omega}left{12- {C} _ {text {sup}} ^ {omega} -+1- {C} _ {text {inf}}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^ {omega}} ^{ omega} }-frac {1} {{C} _ {text {inf}}} ^ {omega}} ===right}
The robustness index \(R\) is the largest \({R}_{\omega }\) obtained by varying the position of the cell \(\omega\) from its neighborhood on the meshes belonging to a triangulation class. This approach makes it possible to highlight the discretization error on a sub-domain but also the distortion of the elements of the mesh and the regularity of the solution. Although it has mathematical bases, this technique seems complicated to implement for complex meshes used in engineering.