具有待定形函数的有限元线法

有限元线法是一种优良的半解析法，但其解函数存在解析方向和离散方向上的精度不相称的问题。本文提出了一种基于有限元线法的双向解析法——具有待定形函数的有限元线法。该方法设形函数为未知待定并借助能量原理对形函数进行解析性求解，使解函数在两个方向上的精度达到相称，从而大幅度提升了解答的整体质量和精度。文中，以二维Poisson方程问题为例，具体给出了待定形函数的有限元线法的算法和算例，用以表明本法的可行性和有效性。

FEMOL with undetermined shape functions

While FEMOL is a general and powerful semi-analytical method for BVPs, its solutions in the discrete direction and the analytical direction do not behave equally well. The present paper proposed a bi-directional semi-analytical method based on FEMOL, i.e. FEMOL with Undetermined Shape Functions. With the aid of energy theorem, the optimization approach, by solving an additional ODE problem after obtaining a FEMOL solution, greatly enhances the shape functions, and hence tremendously improves the overall solution quality and equalizes the accuracies in the two directions. Detailed formulation and numerical examples for two-dimensional Poisson equation were given in the paper to demonstrate the feasibility and effectiveness of the method.