Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

TL;DR

This paper proves that an adaptive isogeometric boundary element method with a novel mesh-size function and knot multiplicity control converges with optimal algebraic rates for weakly-singular integral equations.

Contribution

It provides a rigorous proof of convergence and optimal rates for an adaptive IGA boundary element method, incorporating knot multiplicity and a new mesh-size function.

Findings

Convergence of the adaptive algorithm is mathematically established.

Optimal algebraic convergence rates are achieved.

New inverse estimates for NURBS in fractional Sobolev norms are developed.

Abstract

In a recent work, we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.