Multiscale Model Reduction: Multiscale Finite Element Methods and Their Generalizations

Contents
Notations
1 Introduction
1.1 Challenges and motivation
1.1.1 Multiscale problems
1.1.2 Numerical challenges of solving multiscale problems
1.2 Multiscale model reduction concepts
1.2.1 Exemplary problems
1.2.2 Fine and coarse grids
1.2.3 Scale separation approaches
1.2.4 Multiscale finite element methods and some related methods
1.2.5 The need for a systematic multiscale model reduction approach
1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)
1.3.1 General idea of GMsFEM.
1.3.2 Multiscale basis functions and snapshot spaces
1.3.3 Reducing the degrees of freedom.
1.3.4 Constraint minimization concepts and mesh dependent convergence
1.3.5 The relation to upscaling and novel upscaled concepts
1.3.6 Adaptivity
1.3.7 Nonlinearities
1.3.8 Methodological ingredients of GMsFEM
1.3.9 High contrast and scale issues
1.3.10 Modeling with multiscale methods and applications
1.3.11 Efficient temporal discretizations with multiscale methods
1.3.12 Learning and multiscale methods
1.4 Relevant literature review
1.5 Overview of the content of the book
1.6 Objectives
2 Homogenization and numerical homogenization of linear equations
2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts
2.1.1 Elliptic equations with heterogeneous coefficients
2.1.2 Homogenization of parabolic equations
2.1.3 Homogenization of convection-diffusion equation
2.1.4 Homogenization of convection-diffusion reaction equations
2.2 Numerical homogenization for linear problems with oscillatory …
2.2.1 A motivation
2.2.2 Local problems and macroscopic equations
2.2.3 Convergence results for numerical homogenization
2.2.4 The choice of boundary conditions in numerical homogenization. Oversampling
2.2.5 Increasing representative volume size
2.2.6 Improving numerical homogenization
2.2.7 Numerical homogenization for space-time heterogeneous problems
2.3 Homogenization in perforated regions
2.3.1 Homogenization of Stokes equations
2.4 Numerical homogenization in perforated domains
3 Local model reduction. Introduction to Multiscale Finite Element Methods
3.1 Multiscale finite element methods
3.1.1 Finite element with multiscale basis functions
3.1.2 Basic idea of MsFEM
3.1.3 Using smaller regions in computing multiscale basis functions
3.2 Reducing boundary effects
3.2.1 Motivation
3.2.2 Oversampling technique
3.3 Comparison to other multiscale methods
3.3.1 Comparison to numerical homogenization
3.3.2 Comparison to variational multiscale
3.3.3 Comparison to heterogeneous multiscale method
3.4 Performance and implementation issues
3.4.1 Cost and performance
3.5 Convergence of multiscale finite element methods
3.5.1 The analysis of conforming multiscale finite element method
3.6 Mixed MsFEM
3.7 MsFEM for parabolic equations
3.8 MsFEM using limited global information
4 Generalized multiscale finite element methods. Main concepts and overview
4.1 Introduction
4.1.1 Overview
4.2 Setup
4.3 Parameter-independent case
4.3.1 Examples of snapshot spaces. Oversampling and non-oversampling
4.3.2 Offline spaces
4.3.3 A numerical example
4.4 Online space for parameter-dependent case
4.5 An example of enrichment. The importance of local spectral problem
4.5.1 Reduced-dimensional coarse spaces
4.6 Iterative solvers - online correction of fine-grid solution
4.7 Some numerical studies
4.7.1 Case with no parameter
4.7.2 Elliptic equation with the parameter
4.8 Randomized snapshots
4.8.1 Overview
4.8.2 Randomized oversampling
4.8.3 Numerical results
5 Adaptive strategies
5.1 Introduction
5.2 Preliminaries
5.3 A-posteriori error estimates and adaptive enrichment
5.4 Numerical results for offline adaptivity
5.4.1 Comparison with uniform enrichment
5.4.2 Performance study
5.5 Residual-based online adaptivity
5.6 Numerical results for online adaptivity
5.6.1 Comparison of using different numbers of initial basis
5.6.2 Adaptive online enrichment
6 Selected global formulations for GMsFEM and energy stable oversampling
6.1 Introduction
6.2 Global formulations
6.2.1 Preliminaries
6.2.2 Mixed GMsFEM
6.2.3 GMsDGM
6.2.4 Nonconforming GMsFEM
6.2.5 GMsHDG
6.2.6 General concept of energy stable (minimizing) oversampling
6.3 Basis construction
6.3.1 Multiscale basis functions in mixed GMsFEM
6.3.2 Multiscale basis functions in GMsDGM
6.3.3 Multiscale basis functions in nonconforming GMsFEM
6.3.4 Multiscale basis functions in GMsHDG
6.4 Numerical results
6.4.1 Mixed GMsFEM
6.4.2 GMsDGM
7 Constraint energy minimizing concepts
7.1 Introduction
7.2 Preliminaries
7.3 Construction of multiscale basis functions
7.4 Numerical results
7.5 Relaxed CEM-GMsFEM
7.6 Construction of online basis functions
7.7 Numerical results using online basis functions
8 Non-local multicontinua upscaling
8.1 Introduction
8.2 Preliminaries
8.3 The non-local multicontinua upscaling
8.3.1 Multicontinua functions
8.3.2 Transmissibility computations
8.3.3 Approximation using local multiscale basis
8.4 Time-dependent problem
8.5 Numerical results
8.5.1 Steady state case
8.5.2 Time-dependent case
8.6 Coupled GMsFEM-NLMC at different resolutions
9 Space-time GMsFEM
9.1 Introduction
9.2 Space-time GMsFEM
9.2.1 Preliminaries and motivation
9.2.2 Construction of offline basis functions
9.2.3 Error estimates
9.3 Numerical results for offline GMsFEM
9.4 Residual-based online adaptive procedure
9.5 Numerical results for online GMsFEM
10 Multiscale methods for perforated domains
10.1 Introduction
10.2 Preliminaries
10.2.1 Problem setting
10.2.2 Coarse- and fine-grid notations
10.2.3 Outline of GMsFEM
10.3 The construction of offline and online basis functions
10.3.1 Elasticity problem
10.3.2 Stokes problem
10.4 Numerical results
10.4.1 Elasticity equations in perforated domain
10.4.2 Stokes equations in perforated domain
10.5 Convergence results
11 Multiscale stabilization
11.1 Introduction
11.2 Preliminaries
11.3 Generalized multiscale finite element method for Petrov-Galerkin approximations
11.3.1 Construction of the multiscale trial space
11.3.2 Construction of the multiscale test space
11.3.3 Global coupling
11.3.4 Summary of the procedures for the offline method
11.3.5 Discussion
11.3.6 Online test basis construction (residual-driven correction)
11.4 Numerical results
12 GMsFEM for selected applications
12.1 Multiscale methods for elasticity equations
12.1.1 Preliminaries
12.1.2 Construction of multiscale basis functions
12.1.3 Numerical result
12.2 Multiscale methods for multi-phase flow and transport
12.3 Multiscale methods for acoustic wave propagation: Mixed formulation
12.3.1 Problem description
12.3.2 Multiscale basis functions
12.3.3 The mixed GMsFEM
12.3.4 Numerical results
12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport
12.4.1 Model problem
12.4.2 Fine-scale discretization
12.4.3 Coarse-grid discretization using GMsFEM: Offline spaces
12.4.4 Randomized oversampling GMsFEM
12.4.5 Residual-based adaptive online GMsFEM
12.5 Non-local multicontinua upscaling for poroelasticity in fractured media
12.5.1 Embedded fracture model for poroelastic medium
12.5.2 Fine-grid approximation of the coupled system
12.5.3 Coarse-grid upscaled model for coupled problem
12.5.4 Numerical results
12.6 Multiscale methods for elastic wave propagation in fractured media
12.6.1 Problem formulation
12.6.2 Fine-scale discretization
12.6.3 Coarse-scale discretization
12.6.4 Numerical results
12.7 GMsFEM for stochastic problems using clustering
12.7.1 Preliminaries
12.7.2 Outline of the method
12.7.3 The construction of offline space
12.7.4 Numerical results
12.8 GMsFEM for uncertainty quantification in inverse problems
12.8.1 Preliminaries
12.8.2 GMsFEM for parameter-dependent problem
12.8.3 Multilevel Monte Carlo methods
12.8.4 Multilevel Markov chain Monte Carlo
12.8.5 Numerical results
12.9 Other applications
13 Homogenization and numerical homogenization of nonlinear equations
13.1 Monotone and pseudomonotone operators
13.2 Homogenization
13.3 Numerical homogenization (computation of effective parameters)
13.3.1 Pre-computing the effective coefficients
13.3.2 Parabolic equation
13.4 MsFEM for nonlinear problems
13.4.1 Multiscale finite volume element method (MsFVEM)
13.4.2 Examples of VH
13.4.3 MsFEM for parabolic equations
13.5 Remark on the analysis of MsFEM for nonlinear problems
14 GMsFEM for nonlinear problems
14.1 Introduction
14.2 Preliminaries and motivation
14.2.1 Preliminaries and notations
14.2.2 Motivation
14.3 The GMsFEM
14.3.1 Partition of unity functions
14.3.2 Multiscale basis
14.4 Convergence of the method
14.5 Numerical implementation and results
14.5.1 Numerical results
15 Nonlinear non-local multicontinua upscaling
15.1 Introduction
15.2 Preliminaries
15.2.1 Preliminaries. A brief overview of NLMC for linear problems
15.3 Nonlinear non-local multicontinua model (NLNLMC)
15.3.1 General concept
15.3.2 Nonlinear non-local multicontinuum approach
15.4 Linear approach
15.4.1 Linear transport
15.4.2 Single-phase flow
15.4.3 Two-phase flow
15.5 Nonlinear approach
15.6 RVE-based non-local multicontinua approaches
15.6.1 NLNLMC on RVE-scale
15.6.2 RVE-based NLNLMC
15.6.3 Examples
16 Global-local multiscale model reduction using GMsFEM
16.1 Introduction
16.2 Preliminaries
16.2.1 Model problem
16.2.2 Discrete empirical interpolation method (DEIM)
16.2.3 Generalized multiscale finite element method (GMsFEM)
16.3 Global-local nonlinear model reduction
16.3.1 Local multiscale model reduction
16.3.2 Global-local nonlinear model reduction approach
16.4 Numerical results
16.4.1 Single offline parameter
16.4.2 Multiple offline parameters
17 Multiscale methods in temporal splitting. Efficient implicit-explicit methods for multiscale problems
17.1 Introduction
17.2 Partially explicit temporal splitting scheme
17.3 Spaces construction
17.3.1 Construction of VH,1
17.3.2 Construction of VH,2
17.4 Numerical results
Appendix References
Index