Spectrally accurate Nyström-solver error bounds for 1-D Fredholm integral equations of the second kind
Spectrally accurate Nyström-solver error bounds for 1-D Fredholm integral equations of the second kind
复制标题
第二类一维 Fredholm 积分方程的光谱精确 Nyström 求解器误差范围
DOI：
10.1016/j.amc.2017.07.034
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
M. Kelmanson
中科院分区：
文献类型：
--
作者：
Abigail I. Fairbairn;M. Kelmanson
我们提出的理论基础和计算实现的分析预测的误差界的近似解的一维Fredholm积分方程的第二类。通过渐近估计的近上确界算子范数，易于实现的公式计算的误差界显式地使用仅基于Nyström的方法在不同的正交多项式的根或极值的节点的分布的数值解。尽管预测的界限要求noa priorinformation的确切解决方案，他们被验证是光谱准确的比较后，从各种测试问题的数值解，一些选择是具有挑战性的近似方法，与已知的解决方案的显式计算误差。潜在的限制理论进行了讨论，但这些都没有出现在数值计算。
We present the theory underlying and computational implementation of analytical predictions of error bounds for the approximate solution of one-dimensional Fredholm integral equations of the second kind. Through asymptotic estimates of near-supremal operator norms, readily implementable formulae for the error bounds are computed explicitly using only the numerical solution of Nyström-based methods on distributions of nodes at the roots or extrema of diverse orthogonal polynomials. Despite the predicted bounds demanding noa prioriinformation about the exact solution, they are validated to be spectrally accurate upon comparison with the explicit computational error accruing from the numerical solution of a variety of test problems, some chosen to be challenging to approximation methods, with known solutions. Potential limitations of the theory are discussed, but these are shown not to arise in the numerical computations.