Convergence Rates of AFEM with <i>H</i>⁻¹ Data

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 2 . 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 log N . Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates N − s with s ≤1/2.