Quasi-Optimality of an AFEM for General Second Order Elliptic PDE

Abstract

Abstract In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b ∈ [ L ∞ ⁢ ( Ω ) ] d {b\in[L^{\infty}(\Omega)]^{d}} and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H 1 {H^{1}} -regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments.