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**Validation of a posteriori error estimators by numerical approach.**
*(English)*
Zbl 0811.65088

The authors appear to have a threefold objective in this presentation. In the first place, they present an objective validation methodology for a posteriori error estimators for finite element solutions. The methodology is numerical and can be carried out by application of a computer program and without access to its source language code. They summarize the theory underlying their methodology. Their second objective is to present a review of several of the more popular a posteriori error estimators. Finally, they apply their methodology to those estimators in a manner suggestive of commercial applications.

The validation methodology is applied to a given elliptic boundary value problem in two dimensions and a finite element mesh. One of the interior mesh elements is chosen as representative and becomes the focus of the methodology. A small number of “layers” of elements surrounding the focus element is chosen and the resulting collection of elements imbedded into a unit square. The remainder of the square not already covered by mesh elements is meshed in any convenient fashion. This meshed unit square is regarded as covering all space by periodic extension.

Suppose the finite element is constructed using polynomials of degree no larger than \(p\). A Taylor series argument can be used to show that the largest part of the error in the solution is, in general, a polynomial of degree \(p+1\). Each periodic polynomial of degree \(p+1\) can be regarded as the exact solution of an appropriately chosen inhomogeneous boundary value problem. For this problem, the finite element solution can be found, its exact error computed, and a comparison made with the a posteriori estimate. The least advantageous of the comparisons made for all polynomials of degree \(p+1\) is the one used to evaluate the quality of the error estimate.

The authors present a review of several types of error estimates. Among these are: (1) “implicit element-residual” estimators based on so- called “bubble spaces” of polynomials; (2) several distinct “equilibrated residual” estimators based on the idea that the residual in the interior of an element should be balanced by jumps in the residual along the element edges; and (3) a “super-convergent patch-recovery” estimator due to O. C. Zienkiewicz and J. Z. Zhu [ibid. 33, No. 7, 1331-1364 and 1365-1382 (1992; Zbl 0769.73084 and Zbl 0769.73085)].

The authors present the results of a study of these estimators using their validation methodology. They consider a large variety of equations, including the Laplace equation, the orthotropic heat equation, and the equations of isotropic elasticity. They consider a variety of elements, including triangular and quadrilateral elements of various degrees of accuracy. They also consider a variety of mesh element shapes, some almost pathological. The paper includes nineteen tables presenting their results.

This paper is complete and well presented. It would be an ideal point of departure for anyone interested in theory and application of methods for a posteriori error estimation.

The validation methodology is applied to a given elliptic boundary value problem in two dimensions and a finite element mesh. One of the interior mesh elements is chosen as representative and becomes the focus of the methodology. A small number of “layers” of elements surrounding the focus element is chosen and the resulting collection of elements imbedded into a unit square. The remainder of the square not already covered by mesh elements is meshed in any convenient fashion. This meshed unit square is regarded as covering all space by periodic extension.

Suppose the finite element is constructed using polynomials of degree no larger than \(p\). A Taylor series argument can be used to show that the largest part of the error in the solution is, in general, a polynomial of degree \(p+1\). Each periodic polynomial of degree \(p+1\) can be regarded as the exact solution of an appropriately chosen inhomogeneous boundary value problem. For this problem, the finite element solution can be found, its exact error computed, and a comparison made with the a posteriori estimate. The least advantageous of the comparisons made for all polynomials of degree \(p+1\) is the one used to evaluate the quality of the error estimate.

The authors present a review of several types of error estimates. Among these are: (1) “implicit element-residual” estimators based on so- called “bubble spaces” of polynomials; (2) several distinct “equilibrated residual” estimators based on the idea that the residual in the interior of an element should be balanced by jumps in the residual along the element edges; and (3) a “super-convergent patch-recovery” estimator due to O. C. Zienkiewicz and J. Z. Zhu [ibid. 33, No. 7, 1331-1364 and 1365-1382 (1992; Zbl 0769.73084 and Zbl 0769.73085)].

The authors present the results of a study of these estimators using their validation methodology. They consider a large variety of equations, including the Laplace equation, the orthotropic heat equation, and the equations of isotropic elasticity. They consider a variety of elements, including triangular and quadrilateral elements of various degrees of accuracy. They also consider a variety of mesh element shapes, some almost pathological. The paper includes nineteen tables presenting their results.

This paper is complete and well presented. It would be an ideal point of departure for anyone interested in theory and application of methods for a posteriori error estimation.

Reviewer: Myron Sussman (Bethel Park)

### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74S05 | Finite element methods applied to problems in solid mechanics |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35K05 | Heat equation |

74B05 | Classical linear elasticity |

### Keywords:

implicit element-residual estimators; bubble spaces of polynomials; equilibrated residual estimators; super-convergent patch-recovery estimator; validation methodology; a posteriori error estimators; elliptic boundary value problem; finite element; Laplace equation; heat equation; equations of isotropic elasticit
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\textit{I. Babuška} et al., Int. J. Numer. Methods Eng. 37, No. 7, 1073--1123 (1994; Zbl 0811.65088)

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