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multiscale basis functions for Singular Perturbation on Adaptively Graded Meshes

Advances in Applied Mathematics and Mechanics 2014年第5期6卷 604-614页

作者： Mei-Ling Sun Shan Jiang College of Mathematics Science Yangzhou UniversityYangzhou 225002China Education and Technology Center Nantong Vocational CollegeNantong 226007China Department of Mathematics Texas A&M UniversityCollege StationTX 77843USA

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational *** multiscale space is built through standard finite element basis functions enriched with multiscale basis *** multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given *** numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.

关键词： multiscale basis functions singular perturbation boundary layer adaptively graded meshes.

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Coupling finite element and multiscale finite element methods for the non-stationary Stokes-Darcy model

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JOURNAL OF COMPUTATIONAL PHYSICS 2025年 530卷

作者： Hong, Yachen Zhang, Wenhan Zhao, Lina Zheng, Haibiao East China Normal Univ Sch Math Sci Shanghai Peoples R China City Univ Hong Kong Dept Math Kowloon Tong Hong Kong Peoples R China East China Normal Univ Sch Math Sci Minist Educ Key Lab Math & Engn Applicat Shanghai Key Lab PMMP Shanghai Peoples R China

In this paper, we propose and analyze the multiscale finite element method for the non-stationary Stokes-Darcy model, where the permeability coefficient in the Darcy region exhibits multiscale characteristics. Our algorithm involves two main steps: offline, where we solve the multiscale basis functions in the Darcy region via parallel computation;Online, where we decouple the Stokes-Darcy equations in an implicit-explicit manner. One significant feature of the algorithm is that it solves problems on relatively coarse grids while maintaining accuracy, thus significantly reducing computational costs compared to the standard finite element method. Moreover, under the same coarse grid size, it exhibits a significant improvement in accuracy compared to the standard finite element method. Under the assumption that the permeability coefficient is periodic and independent of time, we demonstrate the stability and convergence of the resulting sequential algorithm. Finally, the capability and effectiveness of the algorithm are verified through three numerical experiments, with computational results consistent with theoretical analysis.

关键词： Stokes-Darcy multiscale characteristics multiscale finite element method multiscale basis functions

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multiscale model reduction for stochastic elasticity problems using ensemble variable-separated method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2023年 421卷

作者： Guan, Xiaofei Jiang, Lijian Wang, Yajun Tongji Univ Sch Math Sci Shanghai Peoples R China

Elasticity is a fundamental model in mechanics and material sciences. In the article, we present an ensemble variable-separated multiscale method for elasticity problems in random media. A variable-separated method is utilized to provide a low-rank approximation for random fields. The variable-separated method is integrated with the Generalized multiscale Finite Element Method (GMsFEM) to obtain an efficient reduced model. To this end, a few local problems are solved for snapshots to construct the multiscale basis functions in GMsFEM. An ensemble method is used to construct stochastic basis functions shared by a series of elasticity equations, and a residual decomposition is adopted to calculate the deterministic physical components for all ensemble members. The ensemble variable-separation is used to solve the local problems of the multiscale basis functions and achieves efficient online-computation for the basis. Then, under the Galerkin projection of the generalized finite element method, the solutions of stochastic elasticity problems are projected onto the low-dimensional space spanned by the online multiscale basis functions. This renders an effective coarse model. The ensemble variable-separated multiscale method is applied to steady-state elastic deformation and elastic wave propagation in random media. A few numerical results are carried out to demonstrate the accuracy and efficiency of the presented method. (c) 2022 Elsevier B.V. All rights reserved.

关键词： multiscale basis functions Elasticity problems Random media Ensemble variable-separated method

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The multiscale perturbation method for two-phase reservoir flow problems

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APPLIED MATHEMATICS AND COMPUTATION 2022年 421卷 126908-126908页

作者： Rocha, Franciane F. Mankad, Het Sousa, Fabricio S. Pereira, Felipe Univ Sao Paulo Inst Ciencias Matemat & Comput Av Trabalhador Sao Carlense BR-13566590 Sao Carlos SP Brazil Univ Texas Dallas Dept Math Sci 800 W Campbell Rd Richardson TX 75080 USA

In this work we formulate and test a new procedure, the multiscale Perturbation Method for Two-Phase Flows (MPM-2P), for the fast, accurate and naturally parallelizable numerical solution of two-phase, incompressible, immiscible displacement in porous media approximated by an operator splitting method. The proposed procedure is based on domain decomposition and combines the multiscale Perturbation Method (MPM) [Ali, et al., Appl. Math. and Comput., 387 (2020) pp. 125023] with the multiscale Robin Coupled Method (MRCM) [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21]. When an update of the velocity field is called for by the operator splitting algorithm, the MPM-2P may provide, depending on the magnitude of a dimensionless algorithmic parameter, an accurate and computationally inexpensive approximation for the velocity field by reusing previously computed multiscale basis functions. Thus, a full update of all multiscale basis functions required by the MRCM for the construction of a new velocity field is avoided. There are two main steps in the formulation of the MPM-2P. Initially, for each subdomain one local boundary value problem with trivial Robin boundary conditions is solved (instead of a full set of multiscale basis functions, that would be required by the MRCM). Then, the solution of an inexpensive interface problem provides the velocity field on the skeleton of the decomposition of the domain. The resulting approximation for the velocity field is obtained by downscaling. We consider challenging two-phase flow problems, with high-contrast permeability fields and water-oil finger growth in homogeneous media. Our numerical experiments show that the use of the MPM-2P gives exceptional speed-up -almost 90% of reduction in computational cost -of two-phase flow simulations. Hundreds of MRCM solutions can be replaced by inexpensive MPM-2P solutions, and water breakthrough can be simulated with very few updates of the MRCM set of multiscale basis functions

关键词： Two-phase flows Porous media multiscale perturbation method multiscale basis functions Robin boundary conditions

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Probabilistic Godunov-type hydrodynamic modelling under multiple uncertainties: robust wavelet-based formulations

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ADVANCES IN WATER RESOURCES 2020年 137卷 103526-103526页

作者： Shaw, James Kesserwani, Georges Pettersson, Per Univ Sheffield Dept Civil & Struct Engn Western Bank Sheffield S10 2TN S Yorkshire England NORCE Norwegian Res Ctr N-5838 Bergen Norway

Intrusive stochastic Galerkin methods propagate uncertainties in a single model run, eliminating repeated sampling required by conventional Monte Carlo methods. However, an intrusive formulation has yet to be developed for probabilistic hydrodynamic modelling incorporating robust wetting-and-drying and stable friction integration under joint uncertainties in topography, roughness, and inflow. Robustness measures are well-developed in deterministic models, but rely on local, nonlinear operations that can introduce additional stochastic errors that destabilise an intrusive model. This paper formulates an intrusive hydrodynamic model using a multidimensional tensor product of Haar wavelets to capture fine-scale variations in joint probability distributions and extend the validity of robustness measures from the underlying deterministic discretisation. Probabilistic numerical tests are designed to verify intrusive model robustness, and compare accuracy and efficiency against a conventional Monte Carlo approach and two other alternatives: a nonintrusive stochastic collocation formulation sharing the same tensor product wavelet basis, and an intrusive formulation that truncates the basis to gain efficiency under multiple uncertainties. Tests reveal that (i) A full tensor product basis is required to preserve intrusive model robustness, while the nonintrusive counterpart achieves identically accurate results at a reduced computational cost;and, (ii) the Haar wavelet basis requires at least three levels of refinements per uncertainty dimension to reliably capture complex probability distributions. Accompanying model software and simulation data are openly available online.

关键词： Probabilistic finite volume formulation Intrusive and nonintrusive stochastic approaches multiscale basis functions Multidimensional uncertainty space Robust wetting-and-drying and friction treatments Accuracy, efficiency and reliability analyses

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Constraint energy minimizing generalized multiscale discontinuous Galerkin method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年 380卷 112960-112960页

作者： Cheung, Siu Wun Chung, Eric T. Leung, Wing Tat Texas A&M Univ Dept Math College Stn TX 77843 USA Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing high-contrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media. (C) 2020 Elsevier B.V. All rights reserved.

关键词： multiscale methods Discontinuous Galerkin methods multiscale basis functions

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The multiscale perturbation method for second order elliptic equations

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APPLIED MATHEMATICS AND COMPUTATION 2020年 387卷 125023-125023页

作者： Ali, Alsadig Mankad, Het Pereira, Felipe Sousa, Fabricio S. Univ Texas Dallas Dept Math Sci 800 W Campbell Rd Richardson TX 75080 USA Univ Sao Paulo Inst Ciencias Matemat & Comp Av Trabalhador Sao Carlense 400 BR-13566590 Sao Carlos SP Brazil

In the numerical solution of elliptic equations, multiscale methods typically involve two steps: the solution of families of local solutions or multiscale basis functions (an embarrassingly parallel task) associated with subdomains of a domain decomposition of the original domain, followed by the solution of a global problem. In the solution of multiphase flow problems approximated by an operator splitting method one has to solve an elliptic equation every time step of a simulation, that would require that all multiscale basis functions be recomputed. In this work, we focus on the development of a novel method that replaces a full update of local solutions by reusing multiscale basis functions that are computed at an earlier time of a simulation. The procedure is based on classical perturbation theory. It can take advantage of both an offline stage (where multiscale basis functions are computed at the initial time of a simulation) as well as of a good initial guess for velocity and pressure. The formulation of the method is carefully explained and several numerical studies are presented and discussed. They provide an indication that the proposed procedure can be of value in speeding-up the solution of multiphase flow problems by multiscale methods. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Porous media Domain decomposition multiscale basis functions Robin boundary conditions Multiphase flows

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multiscale simulation of inelastic creep deformation for geological rocks

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JOURNAL OF COMPUTATIONAL PHYSICS 2021年 440卷 110439-110439页

作者： Kumar, Kishan Ramesh Hajibeygi, Hadi Delft Univ Technol Fac Civil Engn & Geosci Dept Geosci Engn Stevinweg 1 NL-2628 CV Delft Netherlands

Subsurface geological formations provide giant capacities for large-scale (TWh) storage of renewable energy, once this energy (e.g. from solar and wind power plants) is converted to green gases, e.g. hydrogen. The critical aspects of developing this technology to full-scale will involve estimation of storage capacity, safety, and efficiency of a subsurface formation. Geological formations are often highly heterogeneous and, when utilized for cyclic energy storage, entail complex nonlinear rock deformation physics. In this work, we present a novel computational framework to study rock deformation under cyclic loading, in presence of nonlinear time-dependent creep physics. Both classical and relaxation creep methodologies are employed to analyze the variation of the total strain in the specimen over time. Implicit time-integration scheme is employed to preserve numerical stability, due to the nonlinear process. Once the computational framework is consistently defined using finite element method on the fine scale, a multiscale strategy is developed to represent the nonlinear deformation not only at fine but also coarser scales. This is achieved by developing locally computed finite element basis functions at coarse scale. The developed multiscale method also allows for iterative error reduction to any desired level, after being paired with a fine-scale smoother. Numerical test cases are studied to investigate various aspects of the developed computational workflow, from benchmarking with experiments to analysing the impact of nonlinear deformation for a field-scale relevant environment. Results indicate the applicability of the developed multiscale method in order to employ nonlinear physics in their laboratory-based scale of relevance (i.e., fine scale), yet perform field-relevant simulations. The developed simulator is made publicly available at https://***/ADMIRE_Public/mechanics. (C) 2021 The Author(s). Published by Elsevier Inc.

关键词： Subsurface energy storage Geomechanics Nonlinear material deformation multiscale basis functions Algebraic multiscale method Scalable physics-based nonlinear simulation

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Deep learning nonlinear multiscale dynamic problems using Koopman operator

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JOURNAL OF COMPUTATIONAL PHYSICS 2021年 446卷 110660-110660页

作者： Li, Mengnan Jiang, Lijian Hunan Univ Sch Math Changsha 410082 Peoples R China Tongji Univ Sch Math Sci Shanghai 200092 Peoples R China

In this paper, a deep learning method using Koopman operator is presented for modeling nonlinear multiscale dynamical problems. Koopman operator is able to transform a nonlinear dynamical system into a linear system in a Koopman invariant subspace. However, it is usually very challenging to choose a set of suitable observation functions spanning the Koopman invariant subspace when only data is available for the model. It is practically important for us to predict the evolution of the state of the dynamical system from the Koopman invariant subspace. To this end, we introduce a reconstruction operator that maps the observation function space to the model's state space. Incorporating measurement data, a set of neural networks are constructed to learn the Koopman invariant subspace and the reconstruction operator. The loss function not only considers the properties of Koopman invariant subspace, but also reflects the prediction of future state, which makes the proposed method can realize the prediction of future state for a relatively long time. It may be experimentally expensive to collect the fine-scale data. It will be challenging to use limited computational resources to generate sufficient fine-scale data for neural network training. To overcome this difficulty, we use the data in a coarse-scale and learn effective coarse models for the nonlinear multiscale dynamical problems. In order to make the learned coarse model effectively capture fine-scale information, the loss functions for the neural networks are constructed using a set of multiscale basis functions, which are assumed to be given as a prior. In this case, an accurate fine-scale model can be derived by downscaling the learned coarse model. The deep learning multiscale models using Koopman operator can achieve a relatively long-time prediction for the evolution of the state of the nonlinear multiscale dynamical problems. A few numerical examples are presented to show that the effectiveness of learning mul

关键词： Nonlinear multiscale dynamic systems Deep learning Koopman operator multiscale basis functions

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Generalized multiscale Finite Element Methods with energy minimizing oversampling

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2019年第3期117卷 316-343页

作者： Chung, Eric Efendiev, Yalchin Leung, Wing Tat Chinese Univ Hong Kong Shatin Dept Math Hong Kong Peoples R China Texas A&M Univ Dept Math College Stn TX 77843 USA Texas A&M Univ Inst Sci Computat College Stn TX USA

In this paper, we propose a general concept for constructing multiscale basis functions within Generalized multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due to using multiscale glueing functions.

关键词： discontinuous Galerkin multiscale basis functions multiscale problems oversampling

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