Plain convergence of goal-oriented adaptive FEM

TL;DR

This paper proves plain convergence of goal-oriented adaptive finite element methods under broad conditions, including cases without reliability and efficiency estimates, supported by theoretical analysis and numerical experiments.

Contribution

It introduces an abstract convergence analysis for goal-oriented adaptive FEM that applies even when traditional reliability and efficiency estimates are not available.

Findings

Convergence is established under local efficiency estimates.

Plain convergence is proven relying only on stability and reduction properties.

Numerical experiments confirm the theoretical results.

Abstract

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We present an abstract analysis for two different settings. First, we consider problems where a local discrete efficiency estimate holds. Second, we show plain convergence in a setting that relies only on structural properties of the error estimators, namely stability on non-refined elements as well as reduction on refined elements. In particular, the second setting does not require reliability and efficiency estimates. Numerical experiments underline our theoretical findings.