The Two-scale Finite Element Error Analysis For Elliptic Differential Equations With Damping Terms In Small Periodic Perforated Domain

In dealing with scientific theory and engineering calculation problems,the processing methods are often not single and they are multi-dimensional.For example,when dealing with the equivalent evaluation of microscopically fine-structure composites,mathematically-used differential equations can be used to solve problems.When dealing with these problems in the past,in order to meet the calculation accuracy requirements,a very fine mesh division is usually adopted,and the disadvantage is that the calculation amount is cumbersome and the amount is very large.The solution of approximate differential equations by using the higher-order asymptotic solution is a commonly method in physical performance analysis.This dissertation focuses on the elliptic equation and elastic equation with damping terms,and uses the two-scale finite element method for two-scale asymptotic expansion and error analysis.The specific chapter is listed as follows:The first chapter introduces the historical background of partial differential equations in composite materials,the basic knowledge applied in the thesis,the existing research results,and proposes methods for solving such problem.In the second chapter,we discuss the existence and uniqueness of the weak solution for the elliptic equation with damping in the small-period perforated domain firstly.Then based on the general framework of the two-scale method,we construct the two-scale asymptotic expansion for the solution to the model problem.Lastly,the two-scale finite element method in triangular meshing are designed and the posterior error are analyzed.In the third chapter,based on the method of Chapter two,we discuss the existence and uniqueness of the weak solution for the elastic equation with damping term,and analyze the two-scale asymptotic expansion,and corresponding asymptotic error.In the last chapter,based on the above analysis,summarizes the research results and puts forward some perspectives.