# zbMATH — the first resource for mathematics

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.

##### 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
Full Text:
##### References:
  Babuška, SIAM J. Numer. Anal. 15 pp 736– (1978)  Babuška, Int. J. numer. methods eng. 12 pp 1597– (1978)  Babuška, Comput. Methods Appl. Mech Eng. 17/18 pp 519– (1979)  and , ’Reliable error estimation and mesh adaptation for the finite element method’, in (ed.), Computational Methods in Nonlinear Mechanics, North Holland, Amsterdam, 1983, pp. 67-108.  Babuška, SIAM J. Numer. Anal. 18 pp 565– (1981)  and , ’A posteriori error estimates and adaptive techniques for the finite element method’, Technical Note BN-968, Institute for Physical Science and Technology, University of Maryland, College Park, 1981.  Kelly, Int. J. numer. methods eng. 19 pp 1593– (1983)  Bank, Math. Camp. 44 pp 283– (1985)  Babuška, Comput. Methods Appl. Mech. Eng. 61 pp 1– (1987)  Babuška, Finite Elements in Analysis and Design 3 pp 341– (1987)  Oden, Comput. Methods Appl. Mech. Eng. 77 pp 113– (1989)  ’A posteriori estimation of errors in h-p finite element methods for linear elliptic boundary value problems’, M. Sc. Thesis University of Texas at Austin, Austin, Texas, 1989.  Verfürth, Numer. Math. 55 pp 309– (1989)  ’A posteriori error estimation and adaptive mesh refinement techniques’, preprint. 1992.  Baranger, RAIRO Math. Model. Numer. Anal. 25 pp 31– (1991)  Ainsworth, Burner. Math. 60 pp 429– (1992)  Ainsworth, Numer. Math. 65 pp 23– (1993)  Dur??n, Numer. Math. 62 pp 297– (1992)  Baehmann, Int. J. numer. methods eng. 34 pp 979– (1992)  Ainsworth, Comput. Methods Appl. Mech. Eng. 101 pp 73– (1992)  Ainsworth, Computers and Mathematics with Applications 26 pp 75– (1993)  and , ’A posteriori error estimation for h- p approximations in elastostatics’, J. Appl. Numer. Anal., (in press.) · Zbl 0802.73072  Strouboulis, Comput. Methods Appl. Mech. Eng. 97 pp 399– (1992)  Strouboulis, Comput. Methods Appl. Mech. Eng. 100 pp 359– (1992)  Bank, Comput. Methods Appl. Mech. Eng. 82 pp 323– (1990)  and , ’A posteriori error estimates based on hierarchical bases’, preprint, 1992.  Ladeveze, SIAM J. Numer. Anal. 20 pp 485– (1983)  Kelly, Int. J. numer. methods eng. 20 pp 1491– (1984)  Kelly, Int. J. numer. methods eng. 24 pp 1921– (1987)  Kelly, Comput. Struct. 31 pp 63– (1989)  Ohtsubo, Int. J. numer. methods eng. 29 pp 223– (1990)  Ohtsubo, Int. J. numer. methods eng. 34 pp 969– (1992)  Ohtsubo, Int. J. numer. methods eng. 33 pp 1755– (1992)  Ladeveze, Comput. Methods Appl. Mech. Eng. 94 pp 303– (1992)  Zienkiewicz, Int. J. numer. methods eng. 24 pp 337– (1987)  Rank, Commun. Appl. Numer. Methods 3 pp 243– (1987)  Ainsworih, Int. J. numer. methods eng. 28 pp 2161– (1989)  and , ’Some results using stress projectors for error indication’ in , , (eds.), Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia, 1989, pp. 83-99  Babuška, Int. J. numer. methods eng. 36 pp 539– (1993)  Durán, Numer. Math. 59 pp 107– (1991)  Durán, SIAM J. Numer. Anal. 29 pp 78– (1992)  Zienkiewicz, Int. J. numer. methods eng. 33 pp 1331– (1992)  Zienkiewicz, Int. J. numer. methods eng. 33 pp 1365– (1992)  Wiberg, Int. J. numer. methods eng. 36 pp 2703– (1993)  Zienkiewicz, Comput. Methods Appl. Mech. Eng. 101 pp 207– (1992)  Zienkiewicz, Comnun. Numer. Methods Eng. 9 pp 251– (1993)  Rodriguez, Int. J. Numer. Methods in PDEs  ’A review of a posteriori error estimation and adaptive mesh-refinement techniques’, Technical Report, Institüt für Angewandte Mathmatik der Universität Zürich, 1993.  Szabo, Comput. Methods Appl. Mech. Eng. 55 pp 181– (1986)  Diaz, Comput. Methods Appl. Mech. Eng. 41 pp 29– (1983)  Demkowicz, Comput. Methods Appl. Mech. Eng. 53 pp 67– (1985)  Eriksson, Math. Camp. 50 pp 361– (1988)  Johnson, Comput. Methods Appl. Mech. Eng. 101 pp 143– (1992)  Babu, SIAM J. Numer. Anal. 29 pp 947– (1992)  Babuška, Finite Elements in Analysis and Design 11 pp 285– (1992)  Babuška, Comput Methods Appl. Mech. Eng. 101 pp 97– (1992)  and , ’A model study of the quality of a posteriori estimators for linear elliptic problems, Part Ia: Error estimation in the interior of patchwise uniform grids of triangles’, Technical Note BN-1147, Institute for Physical Science and Technology, University of Maryland, College Park, May. 1993.  and , ’Some studies of simple error estimators, Part I–The Philosophy’, Finite Element News, Issue No, 4, 38-42 (1992).  and , ’Some studies of simple error estimators. Part II. Problem I–Convergence characteristics of the error estimators’, Finite Element News, Issue No. 5, 36-41 (1992).  ’Basic error estimates for elliptic problems’, in and (eds.), Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 17-351. · Zbl 0875.65086  ’Comparaison de modeles de milieux continus’, These, Universite P. et M. Curie, Paris, 1975.  ’Sur le Controle de la Qualite des Maillages Elements Finis’, These de Doctorate de l’Universite, Paris 6, Cachan, France.  Ladeveze, Eng. Comp. 9 pp 69– (1991)  and , ’Finite Element Analysis’, Wiley, New York, 1991.  , and , ’Pollution error in the h-version of the finite-element method and a posteriori error estimation’, in preparation.  ’Local behavior in finite element methods’, in and (eds.), Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 357-522. · Zbl 0875.65089  , and , ’Study of superconvergence by a computer-hased approach, Superconvergence of the gradient in finite element solutions of Laplace’s and Poissort’s equations’, Technical Note BN-1155, Institute for Physical Science and Technology, University of Maryland, College Park, November, 1993.  Johnson, Numer. Math. 30 pp 103– (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.