Abstract
This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side f∈H −1(Ω). Prior work assumes almost exclusively that f∈L 2(Ω). New data indicators based on local H −1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dörfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N −s with 0<s≤1/2. In contrast to previous work, it is shown that it is not necessary to assume a compatible decay s<1/2 of the data estimator, but rather that this is automatically guaranteed by the approximability assumptions on the solution by adaptive meshes, without further assumptions on f; the borderline case s=1/2 yields an additional factor logN. Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates N −s with s≤1/2.
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Acknowledgements
This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contract W911NF-07-1-0185; the NSF Grants DMS-0915231, DMS-0807811, and DMS-1109325; the Agence Nationale de la Recherche (ANR) project ECHANGE (ANR-08-EMER-006); the excellence chair of the Fondation “Sciences Mathématiques de Paris” held by Ronald DeVore. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
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Communicated by Wolfgang Dahmen.
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Cohen, A., DeVore, R. & Nochetto, R.H. Convergence Rates of AFEM with H −1 Data. Found Comput Math 12, 671–718 (2012). https://doi.org/10.1007/s10208-012-9120-1
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DOI: https://doi.org/10.1007/s10208-012-9120-1