Convergence of a nonconforming multiscale finite element method

The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients. This is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator. Our previous study reveals that the leading order error in this approach is caused by the "resonant sampling," which leads to large error when the mesh size is close to the small scale of the continuous problem. Similar difficulty also arises in numerical upscaling methods. An oversampling technique has been introduced to alleviate this difficulty [T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189]. A consequence of the oversampling method is that the resulting finite element method is no longer conforming. Here we give a detailed analysis of the nonconforming error. Our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale. We show that the cell resonance error is of lower order. Our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation

We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(Є^2/h^2). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems

DSSY 비순응유한요소와 멀티스케일 방법에 대한 적용

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 신동우.We first consider nonparametric DSSY nonconforming quadrilateral element introduced. The element satisfies the mean value property on each edge and shows optimal convergence for second-order elliptic problems. We estimate the effect of numerical integration on finite element method and construct new quadrature formula for DSSY element. It is shown that only three nodes are enough to get optimal convergence for second-order elliptic problems. Numerical results are presented to compare new quadrature formula with usual Gaussian quadrature rules. Next we study the nonconforming generalized multiscale finite element method(GMsFEM). The framework of GMsFEM is organized and every process of constructing nonconforming GMsFE spaces is presented in detail. GMsFE spaces consist of two ingredient. First one is the offline function space, a spectral decomposition of the snapshot space which is used to approximate the solution. Other one is the moment function space, which is used to impose continuity between local offline function spaces. Numerical results are presented based on nonparametric DSSY nonconforming element. In last chapter, an algebraic multiscale finite element method is investigated. Suppose that the coefficient and the source term of second-order elliptic problems are not available, and we only know the microscale linear system. We try to construct macroscale linear systems only using the algebraic information on the components of microscale systems. One-dimensional case is examined in detail following GMsFEM framework, and two dimensional case is also presented using the DSSY nonconforming finite element space.본 학위논문에서는 일반적인 사각형에서 정의되는 비모수적 DSSY 비순응유한요소공간을 고려한다. 1장에서는 유한요소법을 이용해 이차 타원형 문제를 해결할 때 수치 적분법이 해의 수렴속도에 작용하는 효과를 분석한다. 최적의 수렴 속도를 변화시키지 않는 수치 적분법의 충분 조건을 구하고, 이를 이용해 DSSY 유한요소에 적합한 새로운 구적법 공식을 고안한다. 단 3개의 점만을 이용해 최적의 수렴 속도를 얻을 수 있음을 보이고 다양한 수치적 결과들을 제시한다. 2장에서는 비모수적 DSSY 비순응유한요소공간을 적용한 일반화된 멀티스케일 비순응유한요소법을 연구한다. 일반화된 멀티스케일 유한요소공간은 두 개의 함수공간으로 구성된다. 첫 번째는 offline 함수공간으로 국소적 조화 문제를 풀어 얻어지는 snapshot 함수공간에 스펙트럼 분해를 적용하여 얻어진다. 두 번째는 moment 함수공간으로 국소적으로 얻어진 offline 함수들 간의 연속성을 부여하는 데 이용된다. 이러한 논의와 함께 1장에서 고안한 구적법 공식을 적용한 수치적 결과들을 제시한다. 3장에서는 대수적 멀티스케일 방법을 소개한다. 이차 타원형 문제의 계수와 소스 항을 모르는 상태에서 단지 미시적 스케일의 선형 시스템만 알고 있을 때, 이 시스템의 구성 성분에 대한 대수적 정보만을 바탕으로 거시적 스케일의 선형 시스템을 건설한다. 먼저 일차원 문제를 구체적으로 분석하고 이차원 문제를 일반화된 멀티스케일 비순응유한요소법을 이용하여 연구한다. 수치적 결과들을 보여준다.Chapter 1. Nonparametric DSSY Nonconforming Quadrilateral Element 1 1.1 Introduction 1 1.2 Quadrilateral nonconforming elements 2 1.2.1 The Rannacher-Turek element and the DSSY element 3 1.2.2 Nonparametric DSSY quadrilateral element 4 1.3 A new intermediate space $\bar{K}$ for nonparametric DSSY element 9 1.3.1 The Meng et al. approach 9 1.3.2 A class of nonparametric DSSY elements on $\bar{K}$ 12 1.4 Construction of quadrature formula 17 1.4.1 Effect of numerical integration on FEM 17 1.4.2 Quadrature formula on $\bar{K}$ 28 1.5 Numerical results 32 Chapter 2 Nonconforming Generalized Multiscale Finite Element Method 39 2.1 Introduction 39 2.2 Framework of nonconforming generalized multiscale finite element methods 40 2.2.1 Preliminaries 40 2.2.2 Framework of nonconforming GMsFEM 42 2.3 Construction of multiscale finite element spaces 43 2.3.1 Snapshot function space $V^{\text{snap}}$ 44 2.3.2 Offline function space $V^{\text{off}}$ 45 2.3.3 Moment function space $\mathcal{M}_H$ 46 2.3.4 Nonconforming GMsFE spaces $V^H$ and $V^{H,0}$ 48 2.4 Error analysis 50 2.5 Numerical results 52 Chapter 3 Algebraic Multiscale Method 57 3.1 Introduction 57 3.2 Preliminaries 59 3.3 Algebraic Multiscale Method 64 3.3.1 Algebraic formulation of stiffness matrix 64 3.3.2 Multiscale solution 70 3.4 Error analysis 70 3.5 Numerical results 73 3.5.1 Known Coefficient Case 73 3.5.2 Random Coefficient Case 75 3.6 2D case 82 3.6.1 Implementation of the DSSY nonconforming element 83 3.6.2 Construction of multiscale finite element spaces 89 3.6.3 Numerical results 92 국문 초록 103Docto

Expanded mixed multiscale finite element methods and their applications for flows in porous media

We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity and Lagrange multipliers. We use multiscale basis functions for the both velocity and gradient of pressure. In the expanded mixed MsFEM framework, we consider both cases of separable-scale and non-separable spatial scales. We specifically analyze the methods in three categories: periodic separable scales, $G$- convergence separable scales, and continuum scales. When there is no scale separation, using some global information can improve accuracy for the expanded mixed MsFEMs. We present rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes both conforming and nonconforming expanded mixed MsFEM. Numerical results are presented for various multiscale models and flows in porous media with shales to illustrate the efficiency of the expanded mixed MsFEMs.Comment: 33 page