Analysis And Calculation Of Discontinuous And Combined Mul- Tiscale Finite Element Methods

The research on the multiscale model problems (including the multiscale problems with singularities) has a very wide range of applications in science and engineering. In this paper, we present the multiscale discontinuous Galerkin methods (including multiscale discontinuous finite element method and multi-scale discontinuous Petrov-Galerkin method); the combined finite element and oversampling multiscale Petrov-Galerkin method; multiscale discontinuous finite volume element method for the multiscale model problems, respectively.In Chapter 2, we consider the multiscale discontinuous Galerkin method (Ms-DGM). DG methods have several advantages such as curved boundaries, highly nonuniform and unstructured meshes. Moreover, DG methods admit good lo-cal conservation properties. MsDPGM is a couple of multiscale method and DG method. The main idea is to use the discontinuous Galerkin method based on the oversamping multiscale finite element space to approximate the multiscale solutions. The error estimates and some numerical experiments are presented to demonstrate the efficiency of our numerical method.In Chapter 3, we present the combined finite element and oversampling multi-scale Petrov-Galerkin method (FE-OMsPGM) for the multiscale elliptic problems with singularities. For example, in the simulation of subsurface flow, singularities lie in the porous media with channelized features, or in near-well regions since the solution behaves like the Green function. The basic idea of FE-OMsPGM is separate the computational domain into problematic part and common part, where the problematic part contains singularities. Then we utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and use the Petrov-Galerkin version of oversampling multiscale finite element method (OMsPGM) on a coarse mesh of the common part. The transmission condition across the FE-OMsPG interface is treated by the penal-ty technique. The FE-OMsPGM takes advantages of the FEM and OMsPGM, which uses much less DOFs than the standard FEM and may be more accurate than the OMsPGM for problems with singularities. The theoretical analysis and numerical experiments are given. Some numerical results are presented to show the accuracy and efficiency of the FE-OMsPGM.In Chapter 4, we consider the discontinuous finite element volume method (DFVEM). FVEM is a formulation with the mass conservation property, which is widely used in computational fluid dynamics. DFVEM takes the advantages of DG and FVEM. We construct a new DFVEM based on the new control volume. The main purpose of this Chapter is to propose the multiscale discontinuous finite element volume method for solving the multiscale model problem. The error analysis and corresponding numerical tests are given to show the accuracy of this method.In Chapter 5, we consider the multiscale discontinuous finite element vol-ume method (MsDFVEM) for the multiscale problem. MsDFVEM is a couple of multiscale method and DFVEM. The main idea is to use the discontinuous finite volume element method based on the oversampling multiscale finite element space to approximate the multiscale solutions. This method has the advantage of cap-turing the small scale effects on the large one, and numerical conservative on the coarse mesh. MsDFVEM is the small perturbation of MsDPGM. Therefore, we only estimate the perturbation, and in turn use the existing results of MsDPGM to obtain the convergence of MsDFVEM.