Instance-Optimal Goal-Oriented Adaptivity

Abstract

We consider an adaptive finite element method with arbitrary but fixed polynomial degree p ≥ 1, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [Diening et al, Found. Comput. Math. 16, 2016], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. Numerical experiments underline our theoretical findings.Date: July 31, 2019. 2010 Mathematics Subject Classification. 65N30, 41A25, 65N12, 65N50. Key words and phrases. Adaptive finite element method, goal-oriented algorithm, quantity of interest, maximum marking strategy, convergence, instance optimality.Acknowledgement: The authors acknowledge support through the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245), the special research program Taming complexity in PDE systems (grant SFB F65), and the stand-alone project Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005).

Several Proofs of Coerciveness of First-Order System Least-Squares Methods for General Second-Order Elliptic PDEs

“…In this section, we discuss the least-squares finite element methods for the second-order elliptic equation with an H −1 righthand side. Such a problem appears in many situations, for example, in the goal-oriented a posteriori error estimate [31].…”

Plain convergence of adaptive algorithms without exploiting reliability and efficiency

“…However, all these works employ the so-called Dörfler marking strategy proposed in [Dör96] to single out elements for refinement. Moreover, for the 2D Poisson problem, it has recently been shown that a modified maximum criterion does not only lead to optimal convergence rates, but even leads to instance optimal meshes [DKS16, KS16,IP20]. As the focus comes to other marking strategies, only plain convergence results are known and the essential works are [MSV08,Sie11].…”