# Instance-optimal goal-oriented adaptivity

## TL;DR

This paper introduces a goal-oriented adaptive finite element method with proven instance optimality, effectively balancing primal and dual errors to improve computational efficiency in solving PDEs.

## Contribution

It proposes a new goal-oriented adaptive algorithm driven by an edge-based residual error estimator and proves its instance optimality.

## Key findings

- The algorithm achieves instance optimality with respect to the combined primal and dual errors.
- Numerical experiments confirm the theoretical optimality results.
- The method effectively bounds goal-error by the product of primal and dual total errors.

## Abstract

We consider an adaptive finite element method with arbitrary but fixed polynomial degree $p \ge 1$, 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. More precisely, the goal-error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.