Solving Elliptic Differential Equation With Adaptive Finite Element Method

Adaptive finite element method is one of the most important methods for solving partial differential equations nowadays.Based on the conventional finite element method,it improves the numerical solution process through error estimation and adaptive adjustment strategy.It is a highly efficient and reliable numerical solution method.Speaking of the origin of the finite element method,the paper introduced the development and application of the conventional finite element method,explained the necessity of adding the adaptive process to the finite element method,described in detail the implementation steps of the adaptive finite element method.The effectiveness of the adaptive finite element method is proved by the numerical test.The first chapter of the article introduces the origin of the adaptive finite element method in detail,which is improved by mathematicians based on the finite element method.Then in the second chapter,the basic knowledge of the finite element analysis is introduced,and the symbol marks are unified.The third chapter is based on the four basic steps of adaptive finite element method(solve-estimate-mark-refine),and introduces the main operations in each step in detail.The application of them is explained respectively,and two numerical examples,including crack problem and heat flux reconstruction,are introduced to illustrate the efficiency of the adaptive finite element method.In the fourth chapter,the application of adaptive finite element method in engineering field is introduced,and the prospect of its development is also prospected.