| In this dissertation we develop new numerical method with homogenization to solve elliptic partial differential equations in irregular shaped composite materials.; From the fact that composite material have more than one scale involved, it is very difficult to find numerical solutions of the multi-scale phenomenons in composite materials it is often of main interest to find the global behavior of these multi-scale problems. Moreover, the singularities near the crack tips pollute the accuracy of numerical solutions. To overcome these difficulties two methods are employed: One is homogenization which can capture the global behavior of the multi-scale phenomenon, the other one is the Method of Auxiliary Mapping (MAM) which can effectively handle the singularities. Our numerical experiments demonstrate that our approaches are successful to obtain the macroscopic behavior of the multi-scale problems with singularities.; When the periodicity condition is violated at a boundary, an additive Schwartz iteration is employed to use advantages of homogenization theory. Numerical experiments show that this coupled method achieves the similar results which can be found when a periodicity condition is satisfied.; We also analyze the properties of the homogenized coefficients. |
