Asymptotically exact a posteriori error estimators, part II: General unstructured grids

Randolph E. Bank, Jinchao Xu

Research output: Contribution to journalArticlepeer-review

92 Scopus citations


In Part I of this work [SIAM J. Numer. Anal., 41 (2003), pp. 2294-2312], we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In this work, we consider superconvergence for general unstructured but shape regular meshes. We develop a postprocessing gradient recovery scheme for the finite element solution u h, inspired in part by the smoothing iteration of the rnultigrid method. This recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact. Next, we use the superconvergent gradient to approximate the Hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms. We provide several numerical examples illustrating the effectiveness of our procedures.

Original languageEnglish (US)
Pages (from-to)2313-2332
Number of pages20
JournalSIAM Journal on Numerical Analysis
Issue number6
StatePublished - 2003

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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