A multiscale finite element method for elliptic problems in composite materials and porous media

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DOI10.1006/jcph.1997.5682zbMath0880.73065OpenAlexW2029918260MaRDI QIDQ1360447

Xiao-Hui Wu, Thomas Yizhao Hou

Publication date: 11 February 1998

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://semanticscholar.org/paper/7739e08ec0dcf73355e2b2a87037176910106061

zbMATH Keywords

random coefficients oversampling technique continuous scales large scale behavior multiscale finite element base functions resonant sampling

Mathematics Subject Classification ID

Inhomogeneity in solid mechanics (74E05) Finite element methods applied to problems in solid mechanics (74S05) Composite and mixture properties (74E30) Flows in porous media; filtration; seepage (76S05)

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Degenerate multiphase flow equations in porous media, A multiscale method for model order reduction in PDE parameter estimation, Inverse modeling of tracer flow via a mass conservative generalized multiscale finite volume/element method and stochastic collocation, Error estimation and adaptive meshing in strongly nonlinear dynamic problems, New scheme of finite difference heterogeneous multiscale method to solve saturated flow in porous media, Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations, Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner, Finite element modeling of a multi-physics poro-elastic problem in multiscale media, A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems, Efficient methods for the estimation of homogenized coefficients, \textit{A posteriori} error estimation and adaptive strategy for the control of MsFEM computations, Constraint energy minimizing generalized multiscale finite element method, Local-global model reduction method for stochastic optimal control problems constrained by partial differential equations, Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates, Smart finite elements: a novel machine learning application, A spatially adaptive high-order meshless method for fluid-structure interactions, Estimation of a permeability field within the two-phase porous media flow using nonlinear multigrid method, Non-local multi-continua upscaling for flows in heterogeneous fractured media, High dimensional finite elements for time-space multiscale parabolic equations, Generalized multiscale finite element methods for space-time heterogeneous parabolic equations, Multiscale reconstruction in physics for compositional simulation, Multiscale computation of pore-scale fluid dynamics: single-phase flow, A residual-driven local iterative corrector scheme for the multiscale finite element method, A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients, High order FEM for multigrid homogenization, The hole-filling method and the multiscale computation for the wave equations in perforated domains, Error estimates for an immersed finite element method for second order hyperbolic equations in inhomogeneous media, A new tailored finite point method for strongly anisotropic diffusion equation on misaligned grids, Unstructured grid adaptation for multiscale finite volume method, A hybrid model reduction method for stochastic parabolic optimal control problems, A novel second-order reduced homogenization approach for nonlinear thermo-mechanical problems of axisymmetric structures with periodic micro-configurations, Nonlocal multicontinua with representative volume elements. Bridging separable and non-separable scales, Multiscale VEM for the Biot consolidation analysis of complex and highly heterogeneous domains, Semi-Lagrangian subgrid reconstruction for advection-dominant multiscale problems with rough data, An adaptive multiscale hybrid-mixed method for the Oseen equations, A subcell-enriched Galerkin method for advection problems, FV-MHMM method for reservoir modeling, An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain, A multi-stage deep learning based algorithm for multiscale model reduction, Adaptive generalized multiscale finite element methods for \(\mathrm{H(curl)}\)-elliptic problems with heterogeneous coefficients, High dimensional finite element method for multiscale nonlinear monotone parabolic equations, Additive Schwarz preconditioners for a localized orthogonal decomposition method, The discontinuous Petrov-Galerkin methodology for the mixed multiscale finite element method, Multiscale mortar mixed domain decomposition approximations of nonlinear parabolic equations, Generalized multiscale discontinuous Galerkin method for convection-diffusion equation in perforated media, A contribution to wavelet-bases subgrid modeling, Tailored finite point method for diffusion equations with interfaces on distorted meshes, Analysis of non-local multicontinuum upscaling for dual continuum model, The multiscale perturbation method for two-phase reservoir flow problems, NH-PINN: neural homogenization-based physics-informed neural network for multiscale problems, An online generalized multiscale finite element method for heat and mass transfer problem with artificial ground freezing, Multiscale model reduction for stochastic elasticity problems using ensemble variable-separated method, A multiscale rectangular element method for elliptic problems with entirely small periodic coefficients, Convergence analysis of the localized orthogonal decomposition method for the semiclassical Schrödinger equations with multiscale potentials, Recursive formulation and parallel implementation of multiscale mixed methods, Partially explicit time discretization for time fractional diffusion equation, Multicontinuum homogenization and its relation to nonlocal multicontinuum theories, Meshfree generalized multiscale finite element method, Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media, Determination of thermal, stiffness and electrical effective tensors in composite media, Fast and multiscale formation of isogeometric matrices of microstructured geometric models, Numerical methods for elliptic partial differential equations with rapidly oscillating coefficients, A residual-free bubble formulation for nonlinear elliptic problems with oscillatory coefficients, Nonoverlapping domain decomposition method with preconditioner from asymptotic analysis of steady flow in high contrast media, A numerical upscaling method for an elliptic equation with heterogeneous tensorial coefficients, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Multi-Modes Multiscale Approach of Heat Transfer Problems in Heterogeneous Solids with Uncertain Thermal Conductivity, A Global-Local Approach for the Construction of Enrichment Functions for the Generalized FEM and Its Application to Three-Dimensional Cracks, A BV functional and its relaxation for joint motion estimation and image sequence recovery, Manifold Learning and Nonlinear Homogenization, A brief introduction to homogenization and miscellaneous applications, Numerical homogenization: survey, new results, and perspectives, Bridging the multiscale hybrid-mixed and multiscale hybrid high-order methods, Nonconforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part I: Methodologies and Numerical Experiments, Analysis of two-scale finite volume element method for elliptic problem, FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica, On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model, Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media, Formulation and Analysis of a Parameter-Free Stabilized Finite Element Method, Implementation of the Multiscale Stochastic Finite Element Method on Elliptic PDE Problems, Multiscale scattering in nonlinear Kerr-type media, Finite Element Approximation of Electromagnetic Fields Using Nonfitting Meshes for Geophysics, A Space-Time Multiscale Method for Parabolic Problems, The Heterogeneous Multiscale Method for Dispersive Maxwell Systems, An MsFEM Approach Enriched Using Legendre Polynomials, Error estimates for discrete generalized FEMs with locally optimal spectral approximations, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media: fully discrete scheme, Multiscale Finite Elements for Transient Advection-Diffusion Equations through Advection-Induced Coordinates, Contrast Independent Localization of Multiscale Problems, Computation of Quasi-Local Effective Diffusion Tensors and Connections to the Mathematical Theory of Homogenization, An Asymptotic Preserving Method for Transport Equations with Oscillatory Scattering Coefficients, Complementary finite element methods applied to the numerical homogenization of 3D absolute permeability, Improving the Parallel Performance of Overlapping Schwarz Methods by Using a Smaller Energy Minimizing Coarse Space, Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems, Numerical method for elliptic multiscale problems, Randomized Sampling for Basis Function Construction in Generalized Finite Element Methods, Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point, Random Sampling and Efficient Algorithms for Multiscale PDEs, Additive and Hybrid Nonlinear Two-Level Schwarz Methods and Energy Minimizing Coarse Spaces for Unstructured Grids, Numerical Upscaling of Perturbed Diffusion Problems, On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows, A Multiscale Reduced Basis Method for the Schrödinger Equation With Multiscale and Random Potentials, The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media, A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities, Analysis of three-dimensional fracture mechanics problems: A two-scale approach using coarse-generalized FEM meshes, A variational subgrid scale model for transient incompressible flows, Staggered DG Method for Coupling of the Stokes and Darcy--Forchheimer Problems, A Localization Strategy for Data Assimilation; Application to State Estimation and Parameter Estimation, A Low-Rank Approximated Multiscale Method for Pdes With Random Coefficients, A review of numerical methods for nonlinear partial differential equations, Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo, An Introduction to Multi-point Flux (MPFA) and Stress (MPSA) Finite Volume Methods for Thermo-poroelasticity, Multiscale Finite Element Modeling of Nonlinear Magnetoquasistatic Problems using Magnetic Induction Conforming Formulations, Bayesian Inference Using Intermediate Distribution Based on Coarse Multiscale Model for Time Fractional Diffusion Equations, Low-Rank Approximation to Heterogeneous Elliptic Problems, Corrector Estimates for a Thermodiffusion Model with Weak Thermal Coupling, A special finite element method based on component mode synthesis, Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem, Optimal Convergence Rates for Elliptic Homogenization Problems in Nondivergence-Form: Analysis and Numerical Illustrations, An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media, A Multiscale Algorithm for Heat Conduction-Radiation Problems in Porous Materials with Quasi-Periodic Structures, Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach, An efficient coarse-grained parallel algorithm for global-local multiscale computations on massively parallel systems, Generalized Multiscale Inversion for Heterogeneous Problems, A Three-Level Multi-Continua Upscaling Method for Flow Problems in Fractured Porous Media, Constraint Energy Minimizing Generalized MultiscaleFiniteElementMethodforHigh-Contrast Linear Elasticity Problem, Time-Lapse 3-D Seismic Wave Simulation via the Generalized Multiscale Finite Element Method, Sparse dynamics for partial differential equations, Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading, Uniformly Convergent Scheme for Strongly Anisotropic Diffusion Equations with Closed Field Lines, Unnamed Item, An adaptive variational multiscale method for convection-diffusion problems, Two-scale FEM for homogenization problems, On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems Without Oversampling Techniques, Numerical Homogenization of Elliptic PDEs with Similar Coefficients, A Model Reduction Method for Multiscale Elliptic Pdes with Random Coefficients Using an Optimization Approach, A Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Parabolic Equations, An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems, Numerical Homogenization of Heterogeneous Fractional Laplacians, Multiscale Methods for Subsurface Flow, Wavelets andWavelet Based Numerical Homogenization, Analytical Effective Coefficient and First-Order Approximation to Linear Darcy’s Law through Block Inclusions, Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids, Numerical upscaling for heterogeneous materials in fractured domains, A LOCAL CONSERVATIVE MULTISCALE METHOD FOR ELLIPTIC PROBLEMS WITH OSCILLATING COEFFICIENTS, Online Adaptive Basis Enrichment for Mixed CEM-GMsFEM, A MIXED ADAPTIVE VARIATIONAL MULTISCALE METHOD WITH APPLICATIONS IN OIL RESERVOIR SIMULATION, Hierarchical model reduction at multiple scales, A High-Order Approach to Elliptic Multiscale Problems with General Unstructured Coefficients, Scaling up through domain decomposition, A Low-Rank Schwarz Method for Radiative Transfer Equation With Heterogeneous Scattering Coefficient, A Multiscale Finite Element Method for the Schrödinger Equation with Multiscale Potentials, Adaptive GDSW Coarse Spaces for Overlapping Schwarz Methods in Three Dimensions, Multiscale modeling, homogenization and nonlocal effects: Mathematical and computational issues, High-Order Multiscale Finite Element Method for Elliptic Problems, Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation, Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions, Wavelet-based Edge Multiscale Finite Element Method for Helmholtz problems in perforated domains, Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, A parabolic local problem with exponential decay of the resonance error for numerical homogenization, A combined finite element method for parabolic equations posted in domains with rough boundaries, Generalized Multiscale Finite Element Method for Highly Heterogeneous Compressible Flow, Numerical homogenization beyond scale separation, Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition, A Multiscale Finite Element Method for Oscillating Neumann Problem on Rough Domain, A generalized multiscale independent cover method for nonlocal damage simulation, Optimal Control for Multiscale Elliptic Equations with Rough Coefficients, Solving multiscale elliptic problems by sparse radial basis function neural networks, ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications, <scp>A</scp> second‐order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations, Resolving vibro‐acoustics in poroelastic media via a multiscale virtual element method, A multiscale method for inhomogeneous elastic problems with high contrast coefficients, An online generalized multiscale approximation of the multipoint flux mixed finite element method, Non-local multi-continuum method (NLMC) for Darcy-Forchheimer flow in fractured media, Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction, Multiscale modeling and analysis for some special additive noises driven stochastic partial differential equations, P1$$ {P}_1 $$‐nonconforming quadrilateral finite element space with periodic boundary conditions: Part II. Application to the nonconforming heterogeneous multiscale method, P1$$ {P}_1 $$–Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis, Analysis and Simulation of Optimal Control for a Two-Time-Scale Fractional Advection-Diffusion-Reaction Equation with Space-Time-Dependent Order and Coefficients, A second-order strain gradient fracture model for the brittle materials with micro-cracks by a multiscale asymptotic homogenization, Adaptive Nonlinear Domain Decomposition Methods with an Application to the $p$-Laplacian, Localized Model Reduction for Nonlinear Elliptic Partial Differential Equations: Localized Training, Partition of Unity, and Adaptive Enrichment, Online Adaptive Algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method, Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Inhomogeneous Boundary Value Problems with High Contrast Coefficients, An Elliptic Local Problem with Exponential Decay of the Resonance Error for Numerical Homogenization, An Online Efficient Two-Scale Reduced Basis Approach for the Localized Orthogonal Decomposition, Empirical interscale finite element method (EIFEM) for modeling heterogeneous structures via localized hyperreduction, EMsFEM based concurrent topology optimization method for hierarchical structure with multiple substructures, An adaptive global-local generalized FEM for multiscale advection-diffusion problems, Wavelet-based edge multiscale parareal algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain, A novel vertex-centered finite volume method for solving Richards' equation and its adaptation to local mesh refinement, Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media, Partially explicit splitting method for a multi-physics problem, Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited, An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media, Non-conforming multiscale finite element method for Stokes flows in heterogeneous media. II: Error estimates for periodic microstructure, Efficient decoupling schemes for multiscale multicontinuum problems in fractured porous media, Multiscale asymptotic analysis and algorithm for the quadratic eigenvalue problem in composite materials, A transient global-local generalized FEM for parabolic and hyperbolic PDEs with multi-space/time scales, Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs, Novel Design and Analysis of Generalized Finite Element Methods Based on Locally Optimal Spectral Approximations, Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Convection Diffusion Equation, Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations, Trefftz Approximation Space for Poisson Equation in Perforated Domains, A multiscale continuous Galerkin method for stochastic simulation and robust design of photonic crystals, Adaptive multilevel space-time-stepping scheme for transport in heterogeneous porous media (ADM-LTS), Adaptive dynamic multilevel simulation of fractured geothermal reservoirs, A relaxed localized trust-region reduced basis approach for optimization of multiscale problems, Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model, Numerical upscaling via the wave equation with perfectly matched layers, Order reduction of fracture mechanics in porous microstructures: a multiscale computing framework, Algorithmic monotone multiscale finite volume methods for porous media flow, A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks, Second-order three-scale asymptotic analysis and algorithms for Steklov eigenvalue problems in composite domain with hierarchical cavities, Algebraic multiscale grid coarsening using unsupervised machine learning for subsurface flow simulation, Coarsened \(C^1\)-continuous shape functions using B-spline patches for simulating nonlinear heterogeneous structures, On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators, MULTISCALE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS, Numerical Methods for Constrained Elliptic Optimal Control Problems with Rapidly Oscillating Coefficients, A Sparse Decomposition of Low Rank Symmetric Positive Semidefinite Matrices, Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations, Exploring the Locally Low Dimensional Structure in Solving Random Elliptic PDEs, Reduction in the resonance error in numerical homogenization. II: Correctors and extrapolation, Reduced multiscale finite element basis methods for elliptic PDEs with parameterized inputs, An Adaptive Multiscale Finite Element Method, A Global Jacobian Method for Mortar Discretizations of a Fully Implicit Two-Phase Flow Model, A Combined Finite Element and Multiscale Finite Element Method for the Multiscale Elliptic Problems, High Dimensional Finite Elements for Multiscale Wave Equations, Mass Conservative Domain Decomposition Preconditioners for Multiscale Finite Volume Method, On the Compressive Spectral Method, Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods, Enriched Nonconforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media Based on High-order Weighting Functions, Optimal Local Approximation Spaces for Parabolic Problems, Numerical homogenization of fractal interface problems, Reduced Basis Multiscale Finite Element Methods for Elliptic Problems, Mixed Generalized Multiscale Finite Element Methods and Applications, On a Multiscale Hybrid-Mixed Method for Advective-Reactive Dominated Problems with Heterogeneous Coefficients, Bayesian Numerical Homogenization, Pressure and flow statistics of Darcy flow from simulated annealing

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