多尺度有限元方法的一些研究及应用
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摘要
在当代科学与工程计算领域,许多重要的实际问题是复杂的多尺度问题,比如复合材料的热(电)性质、多孔介质的流体分析、湍流现象、集成电路设计、化学反应的时间尺度等等。多尺度问题的复杂性和多样性催生了各种自有特色的多尺度计算方法。
本文利用求解局部偏微分方程获得的基函数具备良好的性态,该基函数由方程的微分算子性质所决定,可以自适应地将微观尺度的信息带入到宏观尺度。针对带多个尺度参数ε_k的强振荡变系数多尺度椭圆问题,重点研究多尺度有限元方法(MFEM)和单位分解法(PUM),大大减少了计算资源的消耗,并得到了精确的、收敛的数值结果。研究奇异摄动反应扩散问题的多尺度有限元解法,发现利用多尺度基改善了奇异摄动的边界层误差。我们给出了多尺度方法的数学理论,通过实验证明了该方法用有限的计算资源可以获得好的结果,并对算例结果做了相应的数值分析。
多尺度有限元法通过在宏观尺度上求解多尺度基函数的局部方程,将微观尺度的信息带入基函数中,使得宏观粗网格的解具有良好的计算精度,能够节约大量的计算资源.单位分解法是一种重要的无网格方法,利用不依赖于网格的单位分解函数和灵活的局部逼近空间,可以从全局上很好地逼近未知解。
本文获得了以下主要结果:
1.使用多尺度有限元法求解强振荡变系数问题,在系数可分离变量时限定基函数的扰动边界条件所得到的结果精度更好,而且L~2范数不会出现负阶收敛。
2.当系数非分离变量时,我们使网格边界以有理角或无理角穿越周期系数和非周期系数的场域,多尺度基扰动边界条件当ε_k→0时可以获得各方向上单元的完整信息因而有更好的求解精度;反之当ε_k(?)0且网格以无理角穿过场域时得到的单元信息不完全,这种情形下限定多尺度基线性边界条件才能获得更好的精度。
3.用矩形2阶再生插值的单位分解方案处理强振荡变系数问题,与经典有限元法相比证明了单位分解法的效率更高,且有2阶收敛率。
4.研究奇异摄动反应扩散问题,通过在边界层区域使用多尺度基函数、内部光滑区域使用标准线性基函数来构造多尺度有限元空间,探讨了适当厚度边界层区域的选取问题,使用多尺度有限元法在一致的粗网格获得了与ε大小无关的一致收敛结果。
文章的创新之处在于针对多尺度有限元法的多尺度基函数,精心设计了系统的实验方案研究不同的线性边界条件与扰动边界条件的效果,针对周期函数和非周期函数的场域考虑尺度参数ε_k遍历性的影响,给出了具体环境下限定何种基函数边界条件的原则。针对奇异摄动反应扩散问题,在合适的边界厚度区域使用多尺度基可以改善边界层误差,在一致粗网格上得到了ε的一致收敛。
文章的内容安排如下:
第一章简要介绍多尺度问题背景和各种多尺度计算方法,并统一文章中用到的符号。
第二章给出多尺度有限元法的理论基础,指出多尺度基的优点在于能够如实反映微分算子的性质如高阶振荡性,按尺度参数ε_k与网格步长h的大小关系分情形证明了多尺度解的H~1误差估计和L~2误差估计的收敛性定理。
第三章使用多尺度有限元法求解强振荡变系数问题,对多尺度基函数的子问题定义线性边界条件与扰动边界条件。分别考虑系数可分离变量与非分离变量的情况,通过系统地研究数值实验给出了多尺度有限元法基函数最佳边界条件的选取原则。
第四章针对强振荡变系数问题使用单位分解法,介绍该方法的理论基础,给出问题的矩形2阶再生插值的单位分解方案,通过实验验证了单位分解法处理变系数问题有良好的效果。
第五章研究奇异摄动反应扩散问题,在边界区域利用多尺度基可以改善边界层误差获得高精度,多尺度有限元法可以在一致的粗网格上得到2阶收敛的L~2范数误差,1阶收敛的能量范数误差。
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.
本文利用求解局部偏微分方程获得的基函数具备良好的性态,该基函数由方程的微分算子性质所决定,可以自适应地将微观尺度的信息带入到宏观尺度。针对带多个尺度参数ε_k的强振荡变系数多尺度椭圆问题,重点研究多尺度有限元方法(MFEM)和单位分解法(PUM),大大减少了计算资源的消耗,并得到了精确的、收敛的数值结果。研究奇异摄动反应扩散问题的多尺度有限元解法,发现利用多尺度基改善了奇异摄动的边界层误差。我们给出了多尺度方法的数学理论,通过实验证明了该方法用有限的计算资源可以获得好的结果,并对算例结果做了相应的数值分析。
多尺度有限元法通过在宏观尺度上求解多尺度基函数的局部方程,将微观尺度的信息带入基函数中,使得宏观粗网格的解具有良好的计算精度,能够节约大量的计算资源.单位分解法是一种重要的无网格方法,利用不依赖于网格的单位分解函数和灵活的局部逼近空间,可以从全局上很好地逼近未知解。
本文获得了以下主要结果:
1.使用多尺度有限元法求解强振荡变系数问题,在系数可分离变量时限定基函数的扰动边界条件所得到的结果精度更好,而且L~2范数不会出现负阶收敛。
2.当系数非分离变量时,我们使网格边界以有理角或无理角穿越周期系数和非周期系数的场域,多尺度基扰动边界条件当ε_k→0时可以获得各方向上单元的完整信息因而有更好的求解精度;反之当ε_k(?)0且网格以无理角穿过场域时得到的单元信息不完全,这种情形下限定多尺度基线性边界条件才能获得更好的精度。
3.用矩形2阶再生插值的单位分解方案处理强振荡变系数问题,与经典有限元法相比证明了单位分解法的效率更高,且有2阶收敛率。
4.研究奇异摄动反应扩散问题,通过在边界层区域使用多尺度基函数、内部光滑区域使用标准线性基函数来构造多尺度有限元空间,探讨了适当厚度边界层区域的选取问题,使用多尺度有限元法在一致的粗网格获得了与ε大小无关的一致收敛结果。
文章的创新之处在于针对多尺度有限元法的多尺度基函数,精心设计了系统的实验方案研究不同的线性边界条件与扰动边界条件的效果,针对周期函数和非周期函数的场域考虑尺度参数ε_k遍历性的影响,给出了具体环境下限定何种基函数边界条件的原则。针对奇异摄动反应扩散问题,在合适的边界厚度区域使用多尺度基可以改善边界层误差,在一致粗网格上得到了ε的一致收敛。
文章的内容安排如下:
第一章简要介绍多尺度问题背景和各种多尺度计算方法,并统一文章中用到的符号。
第二章给出多尺度有限元法的理论基础,指出多尺度基的优点在于能够如实反映微分算子的性质如高阶振荡性,按尺度参数ε_k与网格步长h的大小关系分情形证明了多尺度解的H~1误差估计和L~2误差估计的收敛性定理。
第三章使用多尺度有限元法求解强振荡变系数问题,对多尺度基函数的子问题定义线性边界条件与扰动边界条件。分别考虑系数可分离变量与非分离变量的情况,通过系统地研究数值实验给出了多尺度有限元法基函数最佳边界条件的选取原则。
第四章针对强振荡变系数问题使用单位分解法,介绍该方法的理论基础,给出问题的矩形2阶再生插值的单位分解方案,通过实验验证了单位分解法处理变系数问题有良好的效果。
第五章研究奇异摄动反应扩散问题,在边界区域利用多尺度基可以改善边界层误差获得高精度,多尺度有限元法可以在一致的粗网格上得到2阶收敛的L~2范数误差,1阶收敛的能量范数误差。
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.
引文
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