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首页> 外文期刊>SIAM Journal on Numerical Analysis >SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS
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SUPERCONVERGENCE OF SOME PROJECTION APPROXIMATIONS FOR WEAKLY SINGULAR INTEGRAL EQUATIONS USING GENERAL GRIDS

机译：一般网格的弱奇异积分方程的某些投影逼近的超收敛

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摘要

This paper deals with superconvergence phenomena in general grids when projectionbased approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp,and confirm the superiority of the iterated Kantorovich scheme.
机译：当使用基于投影的逼近来求解具有弱奇异核的第二类Fredholm积分方程时，本文讨论了通用网格中的超收敛现象。考虑了Galerkin方法的四个变体。它们是经典的Galerkin方法，迭代的Galerkin方法，Kantorovich方法和迭代的Kantorovich方法。证明了如果积分方程的右侧不光滑，则迭代的Kantorovich逼近具有最佳的超收敛速度。所有误差估计都是针对没有任何均匀性或准均匀性条件的任意网格得出的，并根据数据进行公式化，而无需对解决方案进行任何额外假设。数值示例涉及控制恒星大气中光子传输的方程式。数值结果说明了在不同定理中提出的误差估计值非常精确的事实，并证实了迭代Kantorovich方案的优越性。

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