Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence

  • Authors:

  • Randolph E. Bank;Jinchao Xu

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Numerical Analysis
  • Year:
  • 2003

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Abstract

In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant uI. We then analyze a postprocessing gradient recovery scheme, showing that $Q_h\nabla u_h$ is a superconvergent approximation to $\nabla u$. Here Qh is the global L2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators.