复合材料带记忆项的导热方程的多尺度模型与算法研究

Our purpose in this project is to develop effective and reliable multi-scale numerical algorithms for solving integro-differential equations with rapidly oscillating coefficients. The basic idea is to convert the original equations to a steady state problem by Laplace transform with respect to the temporal variable. By applying multi-scale analysis on transformed equations, we finally obtain the approximate solution for the original problem via the inverse Laplace transform. To achieve this goal, we will firstly study the multi-scale asymptotic solution of the underlying problem. The related convergence results and error estimates are then obtained by using the homogenization method and multi-scale asymptotic method. Finally, we will propose an integrated multi-scale algorithm which could capture the local behavior of solution with moderate costs. We hope completion of our project will contribute to the theory, numerical implementations and applications of heat transfer equations in composite materials with a memory term.

本项目将Laplace变换技术和经典的多尺度渐近分析方法相结合，对周期复合材料带记忆项的导热方程（抛物型或双曲型积分微分方程）设计一套有效和可靠的多尺度数值算法，并给出相关的理论分析和程序实现。主要研究内容有：(1)研究周期复合材料带记忆项的导热方程的多尺度渐近解；（2)研究相关的收敛性估计；（3）设计出多尺度有限元数值算法并编写相应的计算程序。主要研究目标是：针对周期复合材料带记忆项的导热方程，设计出一套具有严格的数学基础及能用于解决实际问题的多尺度数值算法。

本项目将Laplace变换技术和经典的多尺度渐近分析方法相结合，研究了周期复合材料带记忆项的导热方程（抛物型或双曲型积分微分方程）的多尺度分析和数值算法。取得的研究成果有：(1)对具有快速振荡系数的双曲型积分-微分方程建立了多尺度渐近展开式，为设计多尺度有限元数值算法提供了理论依据；(2) 给出了求解具有快速振荡系数双曲型积分-微分方程的多尺度有限元数值算法，编写了相应的计算程序，通过数值算例验证了算法的正确性和有效性；(3)对具有快速振荡系数抛物型积分-微分方程给出了多尺度方法相关的收敛定理。