精确有限元法

Exact Finite Element Method

本文提出构造有限单元的新方法——精确有限元法.它可以求解在任意边界条件下任意变系数正定或非正定偏微分方程。文中给出它的收敛性证明和计算偏微分方程的一般格式。用精确元法所得到的单元是一个非协调元,单元之间的相容条件容易处理.与相同自由度普通有限元相比,由精确元法所得到的解的高阶导数具有较高的收敛精度.文末给出数值算例,所得到的结果均收敛于精确解,并有较好的数值精度.

In this paper, a new method, exact element method for constructing finite element, is presented. It can be applied to solve nonpositive definite or positive definite partial differential equation with arbitrary variable coefficient under arbitrary boundary condition. Its convergence is proved and its united formula for solving partial differential equation is given. By the present method, a noncompatible element can be obtained and the compatibility conditions between elements can be treated very easily. Comparing the exact element method with the general finite element method with the same degrees of freedom, the high convergence rate of the high order derivatives of solution can be obtained. Three numerical examples are given at the end of this paper, which indicate all results can converge to exact solution and have higher numerical precision.