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首页> 美国卫生研究院文献>Proceedings. Mathematical Physical and Engineering Sciences >A numerical technique for linear elliptic partial differential equations in polygonal domains
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A numerical technique for linear elliptic partial differential equations in polygonal domains

机译：多边形域中线性椭圆型偏微分方程的数值技术

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Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.
机译：线性椭圆偏微分方程（PDE）解的积分表示可以使用格林定理获得。但是，这些表示涉及边界上的Dirichlet值和Neumann值，并且对于适当摆放的边界值问题（BVP），这些函数之一是未知的。第二作者在90年代后期提出了一种新的求解线性和可积分非线性PDE的BVP的变换方法，通常称为统一变换（或Fokas变换）。对于线性椭圆PDE，可以将该方法视为Green函数法的类似方法，但现在它是在复杂的Fourier平面而非物理平面中公式化的。它采用在傅立叶平面中也形成的两个全局关系，将Dirichlet和Neumann边界值耦合在一起。这些关系可用于根据给定的边界数据来表征未知的边界值，从而为确定Dirichlet到Neumann映射提供了一种优雅的方法。可以将统一变换的数值实现视为在物理平面中公式化的众所周知的边界积分方法在傅立叶平面中的对应项。对于此实现，必须选择（i）扩展未知函数的适当基础和（ii）适当的一组复杂值（我们称为并置点），以在其中评估全局关系。在这里，通过采用各种示例，我们提供了有关如何进行上述选择的简单指南。此外，我们提供了选择搭配点的具体规则，以使关联的线性系统矩阵的条件数保持较低。

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