| In modern scientific and engineering computation,many important practical problems are complicated and troublesome multiscale problems,such as composite materials with thermal/electrical conductivity,flow through the heterogeneous porous media,turbulent transport problem,designing of the integrate circuit,and time scale of the chemical reactions,etc..There come up with many kinds of multiscale computation methods because of the intrinsic complexity of the multiscale problems.This paper acquires the optimal multiscale base functions by solving the local partial differential equations,these base functions are adapted to the local property of the differential operator and can capture the small scales information onto the large scales.We put emphases on studying the multiscale finite element method(MFEM) and the partition of unity method(PUM) for the rapidly oscillating coefficients multiscale elliptic problem with several scale parametersε_k,the methods can greatly reduce the costing of the computer resources and obtain accurate and convergent numerical results.And we investigate the MFEM for solving the singularly perturbed reaction-diffusion problem,find that the method improves the boundary layer error with the help of the multiscale bases.We give the mathematical theories of our multiscale methods,and through numerical experiments we approve that the methods have the abilities to obtain good results with just costing a few handy computer resources,and achieve the numerical analysis for the experiment results.The multiscale finite element method is capable of accurately capturing the large scale behaviors of the solution by resolving the local PDEs for the multiscale base functions,which extract the small scale information of the solution,as a consequence the method acquires good accuracy on coarse meshes and saves plenty of computer resources.The partition of unity method is an important kind of meshless methods,it is made up of two parts,that is,the partition of unity function which may be independent of the grid,plus the agile local approximation space, thus can approximate the global unknown more precisely.In this paper we achieve the following main results.1.We study the MFEM for solving the elliptic problem with rapidly oscillating coefficients,when the coefficients are variables separated,the results of the MFEM with the oscillatory bases are more precise and don't show negative convergence rate for the L~2 norm.2.When the coefficients are not variables separated,we make the mesh grids cut across the periodic and nonperiodic coefficients' fields under rational or irrational angle,the multiscale bases with the oscillatory boundary condition are able to capture the entire period information from each direction in the case ofε_k→0, thus may get better accuracy.On the contrary whenε_k(?)0 and the grids go through the fields under irrational angle,the inadequate element information by the oscillatory bases is inferior to those by the linear ones,in this case we use the linear multiscale bases may gain preciser results.3.We propose the rectangular second order reproducing interpolation partition of unity for the rapidly oscillating coefficients problem,comparing with the FEM we demonstrate the high efficiency of the PUM and obtain the second order convergence rate.4.We investigate the singular perturbed reaction-diffusion problem,and construct the multiscale finite element space by enriching the multiscale bases in the boundary layers domain plus the standard linear bases in the inner smooth domain, and discuss the issue of the optimal width of boundary layers domain,we can acquire independentlyε-uniform convergence on the uniform coarse meshes by the MFEM.The innovations in this paper are represented that we design the systematic numerical experiments elaborately on studying the linear and the oscillatory boundary conditions for the multiscale bases in the MFEM,consider the ergodicity of the scale parametersε_k in the periodic and non-periodic cocfficients' fields,and present the guideline for choosing of the optimal bases boundary condition under different cases.For the singular perturbed reaction-diffusion problem,we use the multiscale bases in the appropriate boundary domains can improve the boundary layer error and on the uniform coarse meshes we obtainε-uniform convergence.The paper is organized as follows.In chapter one we introduce the multiscale problem backgrounds and existing kinds of multiscale computation methods,and unify the denotations used in this paper.In chapter two the theoretical foundation of the multiscale finite element method is provided,we point out that the multiscale bases can reflect the property of the differential operator,for instance highly oscillating property,and prove the convergence theorems of H~1 error estimate and L~2 error estimate for the multiscale solution according to different cases between the scale parametersε_k and the coarse mesh size h.In chapter three we study the MFEM for solving the elliptic problem with rapidly oscillating coefficients,define the linear boundary condition and the oscillatory boundary condition of the multiscale bases for the local PDEs.Considering the separated variables and the non-separated variables coefficients respectively, we present the guideline for defining the multiscale bases optimal boundary of the MFEM through systematic numerical studies.In chapter four we apply the PUM for solving the rapidly oscillating coefficients elliptic problem and provide the theories of this method,we propose the rectangular second order reproducing interpolation partition of unity,through numerical experiment we demonstrate the efficiency of the PUM.In the last chapter five the two-dimensional singular perturbed reactiondiffusion problem is studied,in the appropriate boundary domains the multiscale bases can improve the boundary layer error and obtain high accuracy,thus on the uniform coarse meshes the MFEM acquires second order convergence rate for the L~2 norm error and first order convergence rate for the energy norm error,respectively.
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