Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

TL;DR

This paper develops a reliable a posteriori error estimator for adaptive isogeometric boundary element methods solving weakly-singular integral equations, enabling optimal mesh refinement and convergence.

Contribution

It introduces the first adaptive algorithm for IGABEM with proven error bounds and demonstrates optimal convergence through numerical experiments.

Findings

The error estimator provides both upper and lower bounds for the BEM error.

The adaptive algorithm effectively guides local mesh refinement.

Numerical results confirm optimal convergence rates.

Abstract

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence.