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On the method of functional equations and the performance of designualrized boundary element methods

机译：关于函数方程的方法和设计边界元方法的性能

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摘要

A correspondence is made between the reciprocal relation for linear elliptic partial differential equations and the Riesz integral representation. The former relates the boundary distributions and appropriate normal fluxes of two arbitrary solutions, and the latter expresses and continuous linear functional in terms of an integral involving a representing function. When sufficient regularity conditions are met, the representing function is identified with the unknown boundary distribution. In principle, the representing function may be expressed in terms of the images of a compete set of orthonormal basis functions with known normal fluxes, as suggested by Kupradze [Kupradze VD. On the approximate solution of problems in mathematical physics. Russ Math Surv 1967; 22:59-107]; in practice, the representing function is computed by solving integral equations using boundary element methods. The basic procedure involves expressing the representing function in terms of finite-element or other basis functions, and requiring the satisfaction of the reciprocal relationship with a suitable set of test functions such as Green's functions and their dipoles.
机译：线性椭圆型偏微分方程的倒数关系与Riesz积分表示形式相对应。前者将两个任意解的边界分布和适当的法向通量联系起来，后者将包含表示函数的积分表示为连续线性函数。当满足足够的规则性条件时，代表函数将被识别为未知的边界分布。原则上，代表函数可以用具有已知法线通量的正交基函数竞争集合的图像表示，如Kupradze [Kupradze VD。关于数学物理问题的近似解。 Russ Math Surv 1967； 22：59-107];在实践中，代表函数是通过使用边界元方法求解积分方程来计算的。基本过程涉及用有限元或其他基础函数来表示表示函数，并要求使用一组适当的测试函数（例如格林函数及其偶极子）来满足倒数关系。

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