An abstract analysis of optimal goal-oriented adaptivity

TL;DR

This paper develops an abstract framework for optimal goal-oriented adaptivity applicable to finite and boundary element methods, extending previous results to a broader class of second-order linear elliptic PDEs.

Contribution

It introduces a generalized abstract framework that encompasses standard discretizations for second-order linear elliptic PDEs, broadening the scope of adaptive methods analysis.

Findings

Framework covers standard discretizations of elliptic PDEs

Generalizes previous adaptivity results beyond Poisson equation

Applicable to finite and boundary element methods

Abstract

We provide an abstract framework for optimal goal-oriented adaptivity for finite element methods and boundary element methods in the spirit of [Carstensen et al., Comput. Math. Appl. 67 (2014)]. We prove that this framework covers standard discretizations of general second-order linear elliptic PDEs and hence generalizes available results [Mommer & Stevenson, SIAM J. Numer. Anal. 47 (2009); Becker et al., SIAM J. Numer. Anal. 49 (2011)] beyond the Poisson equation.