# Adaptive isogeometric boundary element methods with local smoothness control

## TL;DR

This paper introduces an adaptive isogeometric boundary element method that locally refines boundary partitions and controls NURBS smoothness, achieving optimal convergence rates for 2D Laplacian problems.

## Contribution

It presents a novel adaptive algorithm that adjusts both local mesh refinement and NURBS smoothness, enhancing isogeometric analysis capabilities.

## Key findings

- Proves linear convergence of the adaptive method.
- Achieves optimal algebraic convergence rates.
- Numerical experiments confirm theoretical predictions.

## Abstract

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.