Research On Convergence Of Homogenization Problem Of Elliptic Equations

In recent years,the homogenization theory has become an important branch of partial differential equations and mathematical physics,and its improvement has also led to the development of related theories in other branches of mathematics.In this thesis,we study the convergence rates of the solution for the homogenization of elliptic equations.The basic methods such as scale analysis,asymptotic expansion,energy estimate,layer and co-layer estimate,and asymptotic estimate of Green function,are used to study the convergence rates ofH₀¹ and L².This thesis not only gives the relevant theory systematically,but also adds some essential understanding to the influence of each microscale on the convergence rate.At present,the results of the paper are still theoretical,but the conclusions and methods developed will have a wide range of applications in engineering,biomechanics,materials science and other fields.Firstly,the classical homogenization problem is studied.For the homogenization problem,the weak convergence rate is usually obtained by the multi-scale convergence method,but this paper innovatively uses the layer and co-layer estimae,together with the smooth operator and its basic properties,combined with the uniform regularity estimate of the solution,and finally establish the convergence rates ofH₀¹ and L².Then the repeated homogenization problem is studied.Compared with the classical homogenization problem,reiterated homogenization problem involves multiple microscopic scales and is more complex.Firstly,the homogenized problem is obtained by using the basic method of multi-scale and asymptotic expansion.Then the equation connect with oscillatory solutions and homogenized solutions is established by introducing correction functions.Next,the convergence rates ofH₀¹ and L²are obtained by the method of energy estimate.Finally,the L^∞error estimate is obtained by using the asymptotic estimate of the Green function.And finally,the summary.The contents,methods and main results of the thesis are summarized.For each kind of problem,the development direction of extensibility and the next work plan are put forward.