Instance optimality of the adaptive maximum strategy

TL;DR

This paper proves that the adaptive finite element method with a maximum marking strategy is optimally efficient for solving Poisson's equation, balancing error and oscillation in a specific geometric setting.

Contribution

It establishes the instance optimality of the adaptive maximum strategy for finite element methods in a rigorous mathematical framework.

Findings

Proves instance optimality for the adaptive maximum strategy.

Applies to Poisson's equation with linear finite elements.

Uses conforming triangulations via newest vertex bisection.

Abstract

In this paper, we prove that the standard adaptive finite element method with a (modified) `maximum marking strategy' is `instance optimal' for the `total error', being the sum of the energy error and the oscillation. This result will be derived in the model setting of Poisson's equation on a polygon, linear finite elements, and conforming triangulations created by newest vertex bisection.