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Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

机译：超奇异积分方程的自适应无网格Galerkin边界节点方法

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摘要

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.
机译：本文针对超奇异边界积分方程的无网格Galerkin边界节点方法，开发了自适应细化技术。推导出两种类型的误差估计器。一种是摄动误差估计器，它是基于使用两个连续的节点排列获得的数值解之间的差异而制定的。另一个是投影误差估计器，它是根据数值解本身及其投影之间的差异制定的。这些误差估计量被证明具有能量范数中精确误差的恒定倍数的上限和下限。提出了一种本地化方案，以适应需要计算的局部误差指标的超奇异积分算子的非局部性质。理论上证明了自适应无网格技术的收敛性。为了证实理论结果并显示自适应技术的效率，提供了具有高奇异性的2D和3D数值示例。

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