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.

中文翻译：

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具有局部平滑控制的自适应等几何边界元方法

在等几何分析的框架中，我们考虑了与二维拉普拉斯算子相关的超奇异积分方程的伽辽金边界元离散化。我们提出并分析了一种自适应算法，该算法可以局部细化边界划分，此外，还可以控制 NURBS ansatz 函数跨元素的平滑度。特别是与先前的工作不同，该算法可以增加和减少局部平滑度属性，从而充分利用等几何分析的潜力。我们证明了新的自适应策略导致具有最优代数率的线性收敛。数值实验证实了理论结果。一个简短的附录评论了弱奇异积分方程的类似结果。