Nochetto, Ricardo H. Pointwise a posteriori error estimates for elliptic problems on highly graded meshes. (English) Zbl 0920.65063 Math. Comput. 64, No. 209, 1-22 (1995). Summary: Pointwise a posteriori error estimates are derived for linear second-order elliptic problems over general polygonal domains in 2D. The analysis carries over regardless of convexity, accounting even for slit domains, and applies to highly graded unstructured meshes as well. A key ingredient is a new asymptotic a priori estimate for regularized Green’s functions. The estimators lead always to upper bounds for the error in the maximum norm, along with lower bounds under very mild regularity and nondegeneracy assumptions. The effect of both point and line singularities is examined. Three popular local estimators for the energy norm are shown to be equivalent, when suitably interpreted, to those introduced here. Cited in 41 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 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations Keywords:maximum norm; equivalence; point and line singularities; error estimates; second-order elliptic problems; polygonal domains Software:PLTMG PDFBibTeX XMLCite \textit{R. H. Nochetto}, Math. Comput. 64, No. 209, 1--22 (1995; Zbl 0920.65063) Full Text: DOI References: [1] Ivo Babuška, Ricardo Durán, and Rodolfo Rodríguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements, SIAM J. Numer. Anal. 29 (1992), no. 4, 947 – 964. · Zbl 0759.65069 [2] I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. 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