To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. Sobolev spaces are the correct framework in which most partial derivative equations may be stated and solved. These functional spaces are built on integration and measure theory. Hence, this chapter in functional analysis is a mandatory theoretical cornerstone for the definition of the finite element method. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs of the main results from integration and measure theory.
翻译:为了获得对执行有限元素方法的数字模拟程序校正的最大信心,人们必须正式确定数学概念和结果,以便确定该方法的健全性。 Sobolev空间是说明和解决大多数部分衍生方程式的正确框架。这些功能空间建立在整合和计量理论之上。因此,功能分析中的本章是确定有限元素方法的强制性理论基石。本文件的目的是为正式证据界提供非常详细的笔纸证明,说明整合和计量理论的主要结果。