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首页> 外文期刊>Engineering analysis with boundary elements >A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems
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A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

机译：3D潜在问题的无网格Galerkin边界节点方法的后验误差估计和自适应过程

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摘要

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L~2-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, 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 technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.
机译：Galerkin边界节点方法（GBNM）是仅边界的无网格方法，将边界积分方程的变分公式与移动最小二乘近似相结合。本文介绍了后验误差估计的数学推导以及针对3D潜在问题的GBNM的自适应细化过程。详细开发了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布置而获得的数值解之间的差异而制定的。另一个是基于GBNM解本身与其L〜2正交投影之间的差异制定的投影误差估计器。建立了两种误差估计器的可靠性和效率。即，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。引入了一种定位技术，以适应积分算子的非局部属性，以获取所需的局部和可计算的后验误差指标。给出了相应的自适应无网格过程的收敛性分析结果。具有高奇异性的数值示例说明了理论结果，并表明所提出的自适应程序简单，有效且高效。

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