论文标题
分形方程的有限差和有限元方法
Finite difference and finite element methods for partial differential equations on fractals
论文作者
论文摘要
在本文中，我们提出了计算分形上偏微分方程的解的数值程序。特别是，我们使用标准图拉普拉斯矩阵以及使用标准长度或区域度量得出的方程式的弱形式来考虑方程的强形式，并在分形集合的离散近似值上得出。然后，我们引入了一个数值过程，以使所获得的扩散归一化，即一种计算分形集合实际偏微分方程定义所需的重新规范化常数的方法。详细研究的一种特定情况是Sierpinski三角形中的Dirichlet问题的解决方案。还提供了其他例子，包括一棵非平面hata树。
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or area measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree.
