Convergence Rates of AFEM with H ?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 ???2. The main contribution is to build algorithms that hold for a general right-hand side f?H ?1(?). Prior work assumes almost exclusively that f?L 2(?). New data indicators based on local H ?1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Drfler 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.
rates h ?1 data adaptivity contraction cardinality approximation classes 65n30 65n15 41a25 65n50 65y20