Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

Spiridonov, Denis; Huang, Jian; Vasilyeva, Maria; Huang, Yunqing; Chung, Eric T.

doi:10.3390/math7121212

Cited by 8 publications

(5 citation statements)

References 92 publications

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“…In the following examples, we set µ = 1 and ρ = 1. The Darcy-Forchheimer coefficient are taken to be β = β 0 κ −1 [13,29,30], where the parameter β 0 control the influence of the nonlinear term and we will test cases with β 0 = 1, β 0 = 10, β 0 = 100, β 0 = 1000 and β 0 = 10000, respectively. Denote the fine-grid solution by (p f , u f ), suppose the multiscale solution is denoted by (p ms , u ms ), then the relative L 2 errors for pressure and velocity are denoted as follows Erp(p ms ) := p ms − p f / p f and Eru(u ms ) := u ms − u f / u f .…”

Section: Numerical Testsmentioning

confidence: 99%

“…Model reduction techniques are required to reduce computational complexity. Spiridonov et al [13] utilized the mixed generalized multiscale finite element method (mixed GMsFEM) to approximate the Darcy-Forchheimer model on the coarse grid. The mixed GMsFEM is originally developed in [14] for Darcy's flow in heterogeneous media, the multiscale basis functions for velocity are constructed following the GMsFEM framework [15,16,17,18,19,20] which generalizes the multiscale finite element method (MsFEM) [21] by enriching the coarse-grid space systematically with additional multiscale basis functions that can help to reduce the error efficiently and substantially.…”

Section: Introductionmentioning

confidence: 99%

“…In the online stage, firstly, we exploit the derived offline space to solve the nonlinear problem on the coarse grid and find out the offline solution. Different from [13], Newton iterative algorithm is used to handle the nonlinear term, which will result in much fewer iterations than the use of Picard iterative algorithm when the nonlinearity is strong. The offline solution has a good approximation of the fine-grid solution.…”

Section: Introductionmentioning

confidence: 99%

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Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model

Chung

Chen

et al. 2020

Preprint

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In this paper, we propose a multiscale method for the Darcy-Forchheimer model in highly heterogeneous porous media. The problem is solved in the framework of generalized multiscale finite element methods (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. We consider the MFMFE method that utilizes the lowest order Brezzi-Douglas-Marini (BDM 1 ) mixed finite element spaces for the velocity and pressure approximation. The symmetric trapezoidal quadrature rule is employed for the integration of bilinear forms relating to the velocity variables so that the local velocity elimination is allowed and leads to a cellcentered system for the pressure. We construct multiscale space for the pressure and solve the problem on the coarse grid following the GMsFEM framework. In the offline stage, we construct local snapshot spaces and perform spectral decompositions to get the offline space with a smaller dimension. In the online stage, we use the Newton iterative algorithm to solve the nonlinear problem and obtain the offline solution, which reduces the iteration times greatly comparing to the standard Picard iteration. Based on the offline space and offline solution, we calculate online basis functions which contain important global information to enrich the multiscale space iteratively. The online basis functions are efficient and accurate to reduce relative errors substantially. Numerical examples are provided to highlight the performance of the proposed multiscale method.

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Section: Numerical Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model

Chung

Chen

et al. 2020

Preprint

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“…The effective characteristics of the heterogeneous medium are calculated in local domains to describe the process on a coarse grid. The gas filtration model is a nonlinear problem, and to resolve the nonlinearity, we use the Picard iteration method [6,5,4]. In this work, we present the numerical experiments for several problems with the different physical properties.…”

Section: Introductionmentioning

confidence: 99%

Numerical homogenization of the gas filtration problem

Kachalin,

Spiridonov

2024

EJMCA

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The non-stationary gas flow equation in homogeneous and heterogeneous media is considered. Areas of this type are often encountered when considering gas production processes from a gas-bearing reservoir. The heterogeneous structure of the reservoir can strongly affect the extraction processes. For the numerical solution of the problem under consideration, we construct an approximation of the equations on a coarse grid using the numerical averaging method. The method allows us to solve the problem using less computational power and in a shorter time. A numerical comparison of the results of solving the model problem is carried out in a two-dimensional domain for the cases of linear and nonlinear variants of the equations, as well as in a homogeneous and heterogeneous medium. The finite element solution on a fine mesh was taken as a reference solution. The computational realization was performed using the FEniCS library. The construction of the geometric computational domain was performed in the program Gmsh. Paraview and the Matplotlib library in Python were used for data visualization.

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“…As for the parameter-dependent model and the thermoelastic model with phase flow, the strategy of multiscale model reduction was addressed in [8][9][10][11][12]. Besides, there were other techniques such as multiscale reduced-basis method [8,10,13,14], localized orthogonal decomposition [15], weak Galerkin GMsFEM [16], mixed GMsFEM [17], multiscale-spectral GFEM [18], meshfree GMsFEM [19], higher-order three-scale method [20] and local discontinuous Galerkin method [21], were applied in a large number of interested fields.…”

Section: Introductionmentioning

confidence: 99%

Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems

Jiang,

Cheng,

Cheng

et al. 2023

Mathematics

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We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users’ demand. Based upon the generalized multiscale simulation on the coarse scale, the balanced truncation is applied to solve its Lyapunov equations on the reduced scale for further savings while ensuring high accuracy. A θ-implicit scheme is utilized for the fully discretization process. Finally, numerical results validate the uniform stability and robustness of our proposed method.

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Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

Cited by 8 publications

References 92 publications

Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model

Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model

Numerical homogenization of the gas filtration problem

Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems

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