Convergence and optimality of adaptive least squares finite element methods

Carsten Carstensen, Eun Jae Park

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in H (div), the data resolution error measures the L 2 norm of the right-hand side f minus the piecewise polynomial approximation II f without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.

Original languageEnglish
Pages (from-to)43-62
Number of pages20
JournalSIAM Journal on Numerical Analysis
Issue number1
Publication statusPublished - 2015 Jan 1

Bibliographical note

Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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