647 research outputs found
DSSY λΉμμμ νμμμ λ©ν°μ€μΌμΌ λ°©λ²μ λν μ μ©
νμλ
Όλ¬Έ(λ°μ¬)--μμΈλνκ΅ λνμ :μμ°κ³Όνλν μ리과νλΆ,2020. 2. μ λμ°.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.λ³Έ νμλ
Όλ¬Έμμλ μΌλ°μ μΈ μ¬κ°νμμ μ μλλ λΉλͺ¨μμ DSSY λΉμμμ νμμ곡κ°μ κ³ λ €νλ€. 1μ₯μμλ μ νμμλ²μ μ΄μ©ν΄ μ΄μ°¨ νμν λ¬Έμ λ₯Ό ν΄κ²°ν λ μμΉ μ λΆλ²μ΄ ν΄μ μλ ΄μλμ μμ©νλ ν¨κ³Όλ₯Ό λΆμνλ€. μ΅μ μ μλ ΄ μλλ₯Ό λ³νμν€μ§ μλ μμΉ μ λΆλ²μ μΆ©λΆ μ‘°κ±΄μ ꡬνκ³ , μ΄λ₯Ό μ΄μ©ν΄ DSSY μ νμμμ μ ν©ν μλ‘μ΄ κ΅¬μ λ² κ³΅μμ κ³ μνλ€. λ¨ 3κ°μ μ λ§μ μ΄μ©ν΄ μ΅μ μ μλ ΄ μλλ₯Ό μ»μ μ μμμ 보μ΄κ³ λ€μν μμΉμ κ²°κ³Όλ€μ μ μνλ€.
2μ₯μμλ λΉλͺ¨μμ DSSY λΉμμμ νμμ곡κ°μ μ μ©ν μΌλ°νλ λ©ν°μ€μΌμΌ λΉμμμ νμμλ²μ μ°κ΅¬νλ€. μΌλ°νλ λ©ν°μ€μΌμΌ μ νμμ곡κ°μ λ κ°μ ν¨μ곡κ°μΌλ‘ ꡬμ±λλ€. 첫 λ²μ§Έλ offline ν¨μ곡κ°μΌλ‘ κ΅μμ μ‘°ν λ¬Έμ λ₯Ό νμ΄ μ»μ΄μ§λ snapshot ν¨μ곡κ°μ μ€ννΈλΌ λΆν΄λ₯Ό μ μ©νμ¬ μ»μ΄μ§λ€. λ λ²μ§Έλ moment ν¨μ곡κ°μΌλ‘ κ΅μμ μΌλ‘ μ»μ΄μ§ offline ν¨μλ€ κ°μ μ°μμ±μ λΆμ¬νλ λ° μ΄μ©λλ€. μ΄λ¬ν λ
Όμμ ν¨κ» 1μ₯μμ κ³ μν ꡬμ λ² κ³΅μμ μ μ©ν μμΉμ κ²°κ³Όλ€μ μ μνλ€.
3μ₯μμλ λμμ λ©ν°μ€μΌμΌ λ°©λ²μ μκ°νλ€. μ΄μ°¨ νμν λ¬Έμ μ κ³μμ μμ€ νμ λͺ¨λ₯΄λ μνμμ λ¨μ§ λ―Έμμ μ€μΌμΌμ μ ν μμ€ν
λ§ μκ³ μμ λ, μ΄ μμ€ν
μ κ΅¬μ± μ±λΆμ λν λμμ μ 보λ§μ λ°νμΌλ‘ κ±°μμ μ€μΌμΌμ μ ν μμ€ν
μ 건μ€νλ€. λ¨Όμ μΌμ°¨μ λ¬Έμ λ₯Ό ꡬ체μ μΌλ‘ λΆμνκ³ μ΄μ°¨μ λ¬Έμ λ₯Ό μΌλ°νλ λ©ν°μ€μΌμΌ λΉμμμ νμμλ²μ μ΄μ©νμ¬ μ°κ΅¬νλ€. μμΉμ κ²°κ³Όλ€μ 보μ¬μ€λ€.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 for nonparametric DSSY element 9
1.3.1 The Meng et al. approach 9
1.3.2 A class of nonparametric DSSY elements on 12
1.4 Construction of quadrature formula 17
1.4.1 Effect of numerical integration on FEM 17
1.4.2 Quadrature formula on 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 44
2.3.2 Offline function space 45
2.3.3 Moment function space 46
2.3.4 Nonconforming GMsFE spaces and 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
κ΅λ¬Έ μ΄λ‘ 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, - 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
Generalized Multiscale Finite Element Methods for problems in perforated heterogeneous domains
Complex processes in perforated domains occur in many real-world
applications. These problems are typically characterized by physical processes
in domains with multiple scales (see Figure 1 for the illustration of a
perforated domain). Moreover, these problems are intrinsically multiscale and
their discretizations can yield very large linear or nonlinear systems. In this
paper, we investigate multiscale approaches that attempt to solve such problems
on a coarse grid by constructing multiscale basis functions in each coarse
grid, where the coarse grid can contain many perforations. In particular, we
are interested in cases when there is no scale separation and the perforations
can have different sizes. In this regard, we mention some earlier pioneering
works [14, 18, 17], where the authors develop multiscale finite element
methods. In our paper, we follow Generalized Multiscale Finite Element Method
(GMsFEM) and develop a multiscale procedure where we identify multiscale basis
functions in each coarse block using snapshot space and local spectral
problems. We show that with a few basis functions in each coarse block, one can
accurately approximate the solution, where each coarse block can contain many
small inclusions. We apply our general concept to (1) Laplace equation in
perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes
equations in perforated domain. Numerical results are presented for these
problems using two types of heterogeneous perforated domains. The analysis of
the proposed methods will be presented elsewhere
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