# The saturation assumption yields optimal convergence of two-level adaptive BEM

## TL;DR

This paper proves that the saturation assumption ensures optimal linear convergence of adaptive boundary element methods for certain integral equations, driven by a two-level error estimator.

## Contribution

It establishes that the saturation assumption guarantees optimal algebraic convergence rates for adaptive BEM, a novel theoretical result.

## Key findings

- Adaptive BEM converges to zero error under local mesh refinement.
- Saturation assumption implies linear convergence with optimal rates.
- Error estimators effectively guide mesh refinement.

## Abstract

We consider the convergence of adaptive BEM for weakly-singular and hypersingular integral equations associated with the Laplacian and the Helmholtz operator in 2D and 3D. The local mesh-refinement is driven by some two-level error estimator. We show that the adaptive algorithm drives the underlying error estimates to zero. Moreover, we prove that the saturation assumption already implies linear convergence of the error with optimal algebraic rates.