scholarly journals Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1212
Author(s):  
Denis Spiridonov ◽  
Jian Huang ◽  
Maria Vasilyeva ◽  
Yunqing Huang ◽  
Eric T. Chung
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heterogeneous Media ◽  
Mixed Finite Element ◽  
Picard Iteration ◽  
Fine Grid ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Forchheimer Model ◽  
Element Method

In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method.

scholarly journals Mixed Generalized Multiscale Finite Element Method for a Simplified Magnetohydrodynamics Problem in Perforated Domains

Computation ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 58
Author(s):  
Valentin Alekseev ◽  
Qili Tang ◽  
Maria Vasilyeva ◽  
Eric T. Chung ◽  
Yalchin Efendiev
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Model Problem ◽  
Mixed Finite Element ◽  
Basis Functions ◽  
Fine Grid ◽  
Perforated Domains ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions.

scholarly journals Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 298
Author(s):  
Aleksei Tyrylgin ◽  
Maria Vasilyeva ◽  
Dmitry Ammosov ◽  
Eric T. Chung ◽  
Yalchin Efendiev
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heterogeneous Media ◽  
Coupled System ◽  
Coarse Grid ◽  
Basis Functions ◽  
Fine Grid ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution.

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