Convergence rates of AFEM with [H.sup.-1] data - Cohen, Albert; DeVore, Ronald; Nochetto, Ricardo H (Foundations of computational mathematics, 2012)
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.