论文标题
二维圆盘状系统通过一维减少的近似解：使用有限元素方法通过绿色功能形式主义的方法
Approximate solution of two dimensional disc-like systems by one dimensional reduction: an approach through the Green function formalism using the Finite Elements Method
论文作者
论文摘要
我们在二维磁盘样系统中为普通二阶PDE提供了一项综合研究，并通过找到有效的一维系统的绿色功能来近似其解决方案。详细阐述了形式主义后，我们建议通过绿色函数的傅立叶扩展来确保精确的解决方案，这需要为绿色模式的无限计数系统解决一个无限可计数的系统，在最简单的情况下，它产生了无源的绿色分布。我们在非可分离系统$ - $上提出了结果，其解决方案无法通过对环和盘几何形状的常规变量分离技术$ - $获得的结果，并展示产生的一维傅立叶模式如何潜在地产生近乎过度的解决方案。数值解决方案将通过使用FDM或FEM与衍生物的三分模块近似值获得有限分化。与已知的确切解决方案相比，我们的结果达到了估计的数值相对错误，低于$ 10^{ - 6} $。
We present a comprehensive study for common second order PDE's in two dimensional disk-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails to solve an infinitely countable system of differential equations for the Green-Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems$-$or such whose solution cannot be obtained by the usual variable separation technique$-$on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using FDM or FEM with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below $10^{-6}$.
