Multiscale Asymptotic Expansion For Composite Materials Heat Transfer Problems And Mixed Finite Element Method

This article demonstates the research to the composite material heat transfer problems. At first, using asymptotic expansion and homogenization, we give two different asymptotic expansions. In the end, we give an anisotropic mixed finite element analysis for the parabolic equation.This article mainly has the following contents:Firstly, using asymptotic expansion and homogenization,we discuss the heat transfer problem in the periodic composite materials, and obtain the asympotic expansion of the parabolic equation with oscillation coefficients. With semi-group operator theory, we prove that the asymptotic solution has better convergence in the space L²(0, T; H¹(Ω)) ifΩ∈R² is a smooth demain. So this asymptotic expansion is reasonable.Secondly, we present a kind of heat transfer problems asymptotic expansion for some symmetrical small cyclical composite materials. It is different from traditional multi-scale methods. It changes the problem of periodic boundary for cell Q in H_per¹(Q) into homogenous boundary problem.Then Using this method is easy to construct the conforming finite element space in numerical solution. On the other hand, the traditional multi-scale asymptotic solution doesn't satisfy the boundary conditions of the original problem. The new asymptotic expansion is not only meet the original physical boundary conditions ,but also maintaining a certain convergent order. Therefore, it will be accepted by the engineers.In the end, we present the parabolic equation mixed finite element anisotropic analysis. We give error estimate of the semi-discrete scheme. This new element has the anisotropic characteristic. It has relieved the regular condition and is better used.