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.

中文翻译：

---

弱奇异积分方程的自适应 IGA 边界元方法的可靠且高效的后验误差估计

我们考虑二维第一类弱奇异积分方程的伽辽金边界元法（BEM）。我们分析了一些残差型后验误差估计器，它为未知的 Galerkin BEM 误差提供了下限和上限。所需的假设很弱，并且允许边界的分段平滑参数化、局部网格细化以及相关的标准分段多项式以及 NURBS。特别是，我们的分析对等几何分析（IGABEM）框架中的自适应边界元法做出了第一个贡献，为此我们制定了一种自适应算法来控制局部网格细化和结的多重性。数值实验强调了理论发现，并表明所提出的自适应策略可以实现最优收敛。