Multiscale Discontinuous Finite Volume Element Method

Finite Volume Element Method(FVEM)is an important numerical method for solving partial differential equations.It can be referred as the generalized finite difference method or Petrov-Galerkin method.The key of the FVEM is the choice of the control volumes,the trial function space and the test function space.Discontinuous Finite Volume Element Method(DFVEM)is a combination of FVEM and Discontinuous Galerkin method(DGM),which does not require the continuity of the approximation functions across the inter-element boundary.In this paper,we consider the numerical simulation of the multiscale elliptic problem with oscillating coefficients.Due to the oscillating feature of the problem,DFVEM can not solve the multiscale problem efficiently.The oversampling multiscale basis functions can capture the multiscale information of the solution.Then,we apply the oversampling multiscale basis functions into the DFVEM for the multiscale elliptic problem.This method is denoted by Multiscale Discontinuous Finite Volume Element Method(MsDFVEM).MsDFVEM is a disturbation of the Multiscale Discontinuous Petrov-Galerkin Method(MsDPGM).A rigorous theoretical analysis of MsDFVEM is carried out for the mul-tiscale elliptic problems with periodic coefficients.We use the theoretical results of MsDPGM and the estimation of the disturbation to obtain the error estimate of MsD-FVEM.Numerical experiments are given to verify the accuracy and efficiency of our method.