Error Estimates Of Multiscale Modeling For Elliptic Problems

Many multiscale methods need to solve an array of microscale models over smalldomains when estimating macroscopic behavior of solutions. Modeling error arisesdue to the mismatch between the local microstructures and the artiftial boundarycondition over the domains for solving the microscale problems. In this paper, weprovide a detailed estimate for the modeling error of elliptic problems with highlyoscillatory coe?cients. Four di?erent boundary conditions are considered: Dirichletboundary condition, Periodic boundary condition, Nuemann boundary conditionand a specified Dirichlet-Nuemann mixed boundary condition. Our analysis revealsthat all the modeling errors come to the ratio of the small scaleεand the size ofthe local domain.In chapter 1, we begin with the history and current situation of multiscalemethods for solving elliptic problems with highly oscillatory coe?cients. In chapter2, we give a brie?y review of HMM and an overall procedure for solving ellipticproblems is issued. Chapter 3 deals with the homogenization theory. we introducethe multiscale asymptotic expansion technique in deriving homogenized equation.In chapter 4, we mainly discuss the modeling error under the four di?erent boundaryconditions.