A Multiscale Finite Element Method for Computing Wave Propagation and Scattering in Heterogeneous Media.

1999 ◽  
Author(s):  
Thomas Y. Hou
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Heterogeneous Media ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

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.

scholarly journals Generalized Multiscale Finite Element Method for Elastic Wave Propagation in the Frequency Domain

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 63 ◽  
Author(s):  
Uygulana Gavrilieva ◽  
Maria Vasilyeva ◽  
Eric T. Chung
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Elastic Wave ◽  
Coarse Grid ◽  
Basis Functions ◽  
Interface Conditions ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this work, we consider elastic wave propagation in fractured media. The mathematical model is described by the Helmholtz problem related to wave propagation with specific interface conditions (Linear Slip Model, LSM) on the fracture in the frequency domain. For the numerical solution, we construct a fine grid that resolves all fracture interfaces on the grid level and construct approximation using a finite element method. We use a discontinuous Galerkin method for the approximation by space that helps to weakly impose interface conditions on fractures. Such approximation leads to a large system of equations and is computationally expensive. In this work, we construct a coarse grid approximation for an effective solution using the Generalized Multiscale Finite Element Method (GMsFEM). We construct and compare two types of the multiscale methods—Continuous Galerkin Generalized Multiscale Finite Element Method (CG-GMsFEM) and Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM). Multiscale basis functions are constructed by solving local spectral problems in each local domains to extract dominant modes of the local solution. In CG-GMsFEM, we construct continuous multiscale basis functions that are defined in the local domains associated with the coarse grid node and contain four coarse grid cells for the structured quadratic coarse grid. The multiscale basis functions in DG-GMsFEM are discontinuous and defined in each coarse grid cell. The results of the numerical solution for the two-dimensional Helmholtz equation are presented for CG-GMsFEM and DG-GMsFEM for different numbers of multiscale basis functions.

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.

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