Homogenisation and approximation of solutions to multiscale problem

• Main Author: Khoo, Freddy Hong Shan
• Other Authors: Hoang Viet Ha
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2023
• Subjects: Science::Mathematics
• Online Access: https://hdl.handle.net/10356/166449
• Institution: Nanyang Technological University

This paper focuses on the homogenization problem of a steady-state diffusion problem in a heterogeneous medium, with the objective of describing the global behavior of a finely distributed medium whose constituents are periodic. Firstly, a two-scale elliptic equation was utilized, with $x$ representing the macroscopic scale and $\frac{x}{\varepsilon}$ representing the microscopic scale. Secondly, we defined the homogenisation problem and examine how we are able to obtain the homogenised matrix and equation to solve the original problem. Specifically, we will be looking at the method of asymptotic expansions. Thirdly, we went on to derive the Voigt-Reiss' inequalities using the variational principle. This helped to give us estimates for the homogenized matrix. Finally, we end our research by exploring the Maxwell Approximation. With these, we were able to gather information about the global behaviour of the heterogeneous medium by considering the effective homogenised medium.