改进广义移动最小二乘近似的无网格法

无网格法是求解微分方程定解问题的一种新数值方法.移动最小二乘近似只要求近似函数在各节点处的误差的平方和最小,对近似函数导数的误差没有任何约束.而广义移动最小二乘近似要求近似函数及其导数在所有节点处的误差的平方和最小.为了降低计算工作量,本文构造了要求近似函数在全部节点处和任意阶导数在部分节点处误差的平方和最小的改进广义移动最小二乘近似.数值计算显示本文提供的方法关于函数值和各阶导数值都具有很高的精度.

Meshless Method with Modifying Generalized Moving Least Squares Approximation

Meshless method is a new numerical method on problem for determining solution of differential equation.The moving least squares approximation makes only require least squares approximation with regard to functional value on all nodes.It makes no require for the residual of derivative approximation.However,the generalized moving least squares approximation makes require least squares approximation with regard to functional and its derivative value on all nodes.For the sake of decrease the computing time,modifying generalized moving least squares approximation was constructed under adding the residual of high orders derivative only on the portion nodes.The numerical examples show that the modifying generalized moving least squares approximation has high accuracy not only for function value but also first or higher orders derivatives.