On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation
On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation
复制标题
关于泊松方程狄利克雷问题的有限差分类似物的表述
DOI：
10.1007/bf01386325
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发表时间：
1962
影响因子：
2.1
通讯作者：
B. Hubbard
中科院分区：
文献类型：
--
作者：
J. Bramble;B. Hubbard
在泊松方程Dirichlet问题的近似求解中，已有许多有限差分方法。对于近似问题的公式化来说，具有有限差分解与精确解的偏差的某种度量是很重要的。关于一类问题的收敛性问题，COURANT，FRIEDRICHS和LEWY [8]曾作过讨论。在GERSCHENIN [5]中，给出了一类椭圆型方程Dirichlet问题解的有限差分逼近的收敛阶的估计。他的方法是基于tinite差异模拟的最大值原理。在1933年的一份说明中，科拉茨[11]提出了一定的边界近似，并使用GERSCHENGIN的技术，表明这种近似引起了0（h 2）估计的截断误差。估计都GERSCHENIN和COLLATZ假设的知识界某些高阶导数的解决方案的狄利克雷问题。
In the approximate solution of the Dirichlet problem for Poisson's equation many finite difference methods have been proposed. It is important for the formulation of the approximating problem to have some measure of the deviation of the finite difference solution from the exact solution. The question of convergence itself for a class of problems has been discussed by COURANT, FRIEDRICHS and LEWY [8 I.In t930 GERSCHGORIN [5] gave a method for obtaining an estimate of the order of convergence of the finite difference approximation to the solution of the Dirichlet problem for a class of elliptic equations. His method was based on a maximum principle for the tinite difference analogue. In a note in 1933 COLLATZ [11 proposed a certain boundary approximation and, using the techniques of GERSCHGORIN, showed that this approximation gives rise to an 0 (h 2) estimate for the truncation error. The estimates of both GERSCHGORIN and COLLATZ assume the knowledge of bounds for certain higher derivatives of the solution of the Dirichlet problem.