Prudhomme, S.; Nobile, F.; Chamoin, L.; Oden, J. T. Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. (English) Zbl 1049.65123 Numer. Methods Partial Differ. Equations 20, No. 2, 165-192 (2004). Authors’ abstract: We analyze a subdomain residual error estimator for finite element approximations of elliptic problems. It is obtained by solving local problems on patches of elements in weighted spaces and provides an upper bound on the energy norm of the error when the local problems are solved in sufficiently enriched discrete spaces. A guaranteed lower bound on the error is also derived by a simple postprocess of the solutions to the local problems. Numerical tests show very good effectivity indices for both the upper and lower bounds and a strong reliability of this estimator even for coarse meshes. Reviewer: Pavel Burda (Praha) Cited in 21 Documents MSC: 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs Keywords:a posteriori error estimates; subdomain residual error estimator; finite element method; elliptic problem; numerical examples PDFBibTeX XMLCite \textit{S. Prudhomme} et al., Numer. Methods Partial Differ. Equations 20, No. 2, 165--192 (2004; Zbl 1049.65123) Full Text: DOI References: [1] and A posteriori error estimation in finite element analysis, John Wiley and Sons, New York, 2000. · doi:10.1002/9781118032824 [2] Babu?ka, Int J Numer Methods Engrg 12 pp 1597– (1978) [3] Ladevèze, SIAM J Numer Anal 20 pp 485– (1983) [4] Ainsworth, Numer Math 65 pp 23– (1993) [5] Carstensen, SIAM J Sci Comput 21 pp 1465– (2000) [6] Morin, Math Comp 72 pp 1067– (2002) [7] Computer Analysis of Error Estimation in Finite Element Computations for Elliptic and Parabolic Problems, Ph.D. Dissertation, Texas A&M University, 2001. [8] and Eléments Finis: Théorie, Applications, Mise en Oeuvre, Springer-Verlag, Berlin, 2002. [9] The Finite Element Method for Elliptic Problems, North-Holland, New York, 1978. [10] and Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems, TICAM Report 02-34. [11] Babu?ka, Comput Methods Appl Mech Engrg 176 pp 51– (1999) [12] Strouboulis, Int J Numer Methods Engrg 47 pp 427– (2000) [13] Zienkiewicz, Int J Numer Methods Engrg 33 pp 1331– (1992) [14] Zienkiewicz, Int J Numer Methods Engrg 33 pp 1365– (1992) [15] Bank, Math Comp 44 pp 283– (1985) [16] Demkowicz, Comput Methods Appl Mech Engrg 46 pp 217– (1984) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.