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On the Nystroem discretization of integral equations on planar curves with corners

机译:关于带拐角的平面曲线上积分方程的Nystroem离散

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The Nystrom method can produce ill-conditioned systems of linear equations when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplace's equation. We explain the origin of this instability and show that a straightforward modification to the Nystrom scheme, which renders it mathematically equivalent to Galerkin discretization, corrects the difficulty without incurring the computational penalty associated with Galerkin methods. We also present the results of numerical experiments showing that highly-accurate solutions of integral equations on domains with corners can be obtained, irrespective of whether their solutions exhibit bounded or unbounded singularities, assuming that proper discretizations are used.
机译:当将Nystrom方法应用于带角域上的积分方程时,它会产生病态线性方程组。在由拉普拉斯方程的Neumann问题引起的积分方程的简单情况下,已经可以看到这种缺陷。我们解释了这种不稳定性的根源,并表明对Nystrom方案的直接修改(在数学上等效于Galerkin离散化)可以纠正这一难题,而不会引起与Galerkin方法相关的计算损失。我们还提供了数值实验的结果,结果表明,假定使用适当的离散化,无论其解是否显示有界奇点或无界奇点,都可以获得具有拐角域上积分方程的高精度解。

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