Abstract
Abstract We consider a linear elliptic PDE and a quadratic goal functional. The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand. We show that the marking strategy proposed in [M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal. 54 (2016), 3, 1423–1448] for a linear goal functional is also optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates.
Highlights
The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand
Anal. 54 (2016), no. 3, 1423–1448] for a linear goal functional is optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates
Goal-oriented adaptivity is more important in practice than standard adaptivity and has attracted much interest in the mathematical literature; see, e.g
Summary
Let Ω ⊂ Rd, d ≥ 2, be a bounded Lipschitz domain. For given f ∈ L2(Ω) and f ∈ [L2(Ω)]d, we consider a general linear elliptic partial differential equation. While standard adaptivity aims to approximate the exact solution u ∈ H1(Ω) at optimal rate in the energy norm (see, e.g., [7, 11, 12, 23, 25] for some seminal contributions and [14] for the present model problem), goal-oriented adaptivity aims to approximate, at optimal rate, only the functional value G(u) ∈ R ( called quantity of interest in the literature). There are only few works which aim for a mathematical understanding of optimal rates for goal-oriented adaptivity; see [4, 15, 16, 20] While the latter works consider only linear goal functionals, the present work aims to address, for the first time, optimal convergence rates for goal-oriented adaptivity with a nonlinear goal functional. Optimal Convergence Rates for Goal-Oriented FEM assume that our primary interest is not in the unknown solution u ∈ H1(Ω), but only in the functional value.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
