利用有限元法求解Laplace方程

摘 要

本文针对Laplace方程讨论了数值计算方法中的有限元方法。分为两部分，前1部分首先论述了有限元法的原理、发展趋势和优点，重点的阐述了有限元法的解题思想和1般步骤，即由网格剖分（本文采用3角剖分）到最后线性方程组的数值求解，在这之前还引入了1些预备知识的介绍。然后利用Fortran语言将算法进行实现，并利用直接法（这里用Guass顺序消去法）求解有限元离散后得到的线性方程组；后1部分讲述了数值实验，主要包括了模型问题（变分问题和有限元方程）、流程图和1些数值实验结果的分析。最后还对本设计做了总结，包括了1些心得和体会以及需要改进的地方。

关 键 词：有限元法；Fortran；Laplace方程

Abstract

    This article discussed in the value computational method finite element method in view of the Laplace equation. The paper divides into two parts, the preceding part first elaborated the finite element method principle, the trend of development and the merit, the key elaboration finite element method problem solving thought and the general step, namely (this article used triangulation) from grid cutting in half to the final system of linear equations value solution, also has before this introduced some preparation knowledge introduction. Then carries on using the Fortran language the algorithm the realization, and (here after Guass order elimination) solves the system of linear equations using the direct method which the finite element separate obtains; The latter part narrated the value experiment, mainly has included the model question (variation question and finite element equation), the flow chart and some value experiment result analysis. The paper finally has also made the summary to this design, including the place which some attainments and the experience as well as needed to improve.

Keywords：Finite Element Method；Fortran；Laplace equation 中国大学排名