Convergence rates of AFEM with [H.sup.-1] data

• Published in: Foundations of computational mathematics Vol. 12; no. 5; p. 671
• Main Authors: Cohen, Albert, DeVore, Ronald, Nochetto, Ricardo H
• Format: Journal Article
• Language: English
• Published: Springer 01.10.2012
• Subjects: Convergence (Mathematics); Finite element method; France
• ISSN: 1615-3375
• DOI: 10.1007/s10208-012-9120-1

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 ⊂ [R.sup.2]. The main contribution is to build algorithms that hold for a general right-hand side f ∈ [H.sup.-1] (Ω). Prior work assumes almost exclusively that f ∈ [L.sup.2] (Ω). New data indicators based on local [H.sup.-1] norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dorfler 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.sup.-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 log N. Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates [N.sup.-s] with s [less than or equal to] 1/2.