利用有限元法求解Laplace方程

利用有限元法求解Laplace方程

摘 要

本文針對Laplace方程討論了數值計算方法中的有限元方法。論文分為兩部分，前1部分首先論述了有限元法的原理、發展趨勢和優點，重點的闡述了有限元法的解題思想和1般步驟，即由網格剖分（本文采用3角剖分）到最后線性方程組的數值求解，在這之前還引入了1些預備知識的介紹。然后利用Fortran語言將算法進行實現，并利用直接法（這里用Guass順序消去法）求解有限元離散后得到的線性方程組；后1部分講述了數值實驗，主要包括了模型問題（變分問題和有限元方程）、流程圖和1些數值實驗結果的分析。論文最后還對本設計做了總結，包括了1些心得和體會以及需要改進的地方。

關 鍵 詞：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

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