Finite difference and finite element methods for partial differential equations on fractals

具有分形元素的间断伽辽金方法

A discontinuous Galerkin method with fractal elements

本文提出了一种用于分形边界域的间断伽辽金有限元方法，通过分形网格精确计算局部多项式基函数，并展示了该方法在泊松方程和狄利克雷特征值问题中的有效性。

Acoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximation

We study time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a planar screen (a "crack" in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset $Γ$ of the hyperplane $\mathbb{R}^{n-1}\times \{0\}$, on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, $Γ$ can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such $Γ$ we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regular…

一种受区域增长启发的求解常微分和偏微分方程的幂级数方法

A power series method for solving ordinary and partial differentials equations motivated by domain growth

本文提出一种受区域增长启发的幂级数方法，用于求解常微分与偏微分方程，并通过求解随机游走常微分方程组、非线性方程及波动、扩散和拟线性偏微分方程验证了其有效性。