Plain convergence of adaptive algorithms without exploiting reliability and efficiency

TL;DR

This paper proves that adaptive finite and boundary element algorithms converge to zero error without relying on traditional reliability and efficiency estimates, broadening applicability to non-local operators.

Contribution

It introduces a new convergence analysis framework for adaptive algorithms that does not depend on reliability and efficiency estimates, applicable to non-local problems.

Findings

Adaptive algorithms drive error estimators to zero.

The analysis applies to non-local operators like fractional Laplacian.

Convergence is proven under general structural assumptions.

Abstract

We consider h-adaptive algorithms in the context of the finite element method (FEM) and the boundary element method (BEM). Under quite general assumptions on the building blocks SOLVE, ESTIMATE, MARK, and REFINE of such algorithms, we prove plain convergence in the sense that the adaptive algorithm drives the underlying a posteriori error estimator to zero. Unlike available results in the literature, our analysis avoids the use of any reliability and efficiency estimate, but only relies on structural properties of the estimator, namely stability on non-refined elements and reduction on refined elements. In particular, the new framework thus covers also problems involving non-local operators like the fractional Laplacian or boundary integral equations, where (discrete) efficiency is (currently) not available.