泊松方程的有限差分方法及快速实现

The Finite Difference Method of Poisson Equation and Fast Realization

针对d维(d=1,2,3)带有Dirichlet边界的泊松方程,设计一类快速求解方法。首先采用有限差分方法将方程离散,利用Kronecker积的性质将离散后的方程进行矩阵分解,进而应用快速离散正弦变换(DST)方法进行有效求解。数值实验结果表明,该方法可快速求解d维泊松方程,并验证了其准确性和有效性。

For the d-dimensional(d=1,2,3)Poisson equation with Dirichlet boundary,a fast solution method is developed in this paper.The method uses the finite difference method to discretize the equations,and uses the properties of the Kronecker product to decompose the discretized equations into a matrix,and then uses the fast discrete sine transform(DST)method to solve effectively.This method can construct a fast numerical method to solve the d-dimensional Poisson equation.Numerical experiments verify the accuracy and effectiveness of the method.