Adaptive 2D IGA boundary element methods

TL;DR

This paper develops and analyzes adaptive isogeometric boundary element methods for 2D weakly-singular integral equations, demonstrating their optimal convergence and superiority over standard methods through numerical experiments.

Contribution

It introduces new a posteriori error estimators for collocation and weighted-residual IGA boundary element methods, enabling adaptive refinement with NURBS and proving their effectiveness.

Findings

Adaptive IGA BEM achieves optimal convergence rates.

Adaptive strategy outperforms standard boundary element methods.

Numerical results confirm reliability and efficiency of proposed estimators.

Abstract

We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error estimator for Galerkin IGA boundary element methods, the present work focuses more on collocation and weighted- residual error estimators, which provide reliable upper bounds for the energy error. Our analysis allows piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. We formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments show that the proposed adaptive strategy leads to optimal convergence, and related IGA boundary element methods are superior to standard boundary element methods with piecewise polynomials.