The Multiscale Finite Volume Method For Elliptic PDE With Oscillating Coefficients

In this paper we consider the elliptic PDE with oscillating coefficients. We adopt heterogeneous multiscale method (HMM) to solve the problem. HMM consists of two components : selection of a macroscopic solver on a macroscale grid, and estimating the missing macroscale data by solving locally the fine scale problem. So the key of this method is how to select a macroscopic solver and how to estimate the missing macroscale data. A kind of finite volume method(FVM) that considers the effects of numerical integration is used as the macroscopic solver in HMM. Then we get the method named HMM-FVM. We present the error estimate and show some numerical examples to confirm that HMM-FVM has good convergent order.