AI 中文总结
本研究针对拉普拉斯-狄利克雷问题的边界元离散化，提出泛函型误差估计器，证明其与残差估计器局部等价，结合网格细化策略实现最优收敛，经数值实验验证了该自适应算法的有效性。
AI 中文摘要
在本研究中，我们推导了拉普拉斯-狄利克雷问题边界元离散化产生的位势误差的泛函上界。这些上界基于边界顶点面片上的局部辅助问题，所得后验误差估计器被证明与已广泛研究的残差误差估计器局部等价。该等价性结果使我们能够证明泛函后验误差估计器的R-线性收敛性，结合合适的网格细化策略，可确立位势误差及泛函误差估计器关于边界元数量以最优速率收敛。数值实验验证了理论结果，并说明了由所提泛函误差估计器驱动的相关自适应算法的实际性能。
英文摘要
In the present work, we derive functional upper bounds for the potential error arising from boundary element discretizations of the Laplace-Dirichlet problem. These bounds are based on local auxiliary problems on patches of boundary vertices and the resulting a posteriori error estimator is shown to be locally equivalent to the well-studied residual error estimator. This equivalence result allows us to prove R-linear convergence of the functional a posteriori error estimator and, together with a suitable mesh-refining strategy, to establish that the potential error as well as the functional error estimator converge with optimal rates with respect to the number of boundary elements. Numerical experiments affirm the theoretical findings and illustrate the practical performance of the related adaptive algorithm driven by the proposed functional error estimator.
