Computational multiscale methods for first-order wave equation using mixed CEM-GMsFEM

Chung, Eric T.; Pun, Sai-Mang

doi:10.1016/j.jcp.2020.109359

Cited by 14 publications

(7 citation statements)

References 30 publications

(46 reference statements)

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“…Lemma 4.7. Let (u glo , p glo ) be the solution in (10) and (u ms , p ms ) be the solution in (16). Suppose 0 < τ ≤ 2(µ min + µ max ) −1 .…”

Section: Discussionmentioning

confidence: 99%

“…Here, (u ms , p ms ) is the multiscale solution obtained by solving (16) pre A, respectively. We denote k ∈ N the number of iteration level.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In [14], a variation of GMsFEM based on a constraint energy minimization (CEM) strategy for mixed formulation has been developed. This approach is inspired by the work on localization [36,38,39] and makes use of the ideas of oversampling to compute multiscale basis functions in oversampled subregions with the satisfaction of an appropriate orthogonality condition, where similar ideas have been applied for various numerical discretization and model problems [17,10,9,16,33,11]. The method proposed in [14] provides a mass conservative velocity field and allows one to identify some nonlocal information depending on the inputs of the problem.…”

Section: Introductionmentioning

confidence: 99%

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Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

2018

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This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that can not be localized, in general. This is built on our previous work on the Generalized Multiscale Finite Element Method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast-dependence in the convergence. The method's convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.

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“…Lemma 4.7. Let (u glo , p glo ) be the solution in (10) and (u ms , p ms ) be the solution in (16). Suppose 0 < τ ≤ 2(µ min + µ max ) −1 .…”

Section: Discussionmentioning

confidence: 99%

“…Here, (u ms , p ms ) is the multiscale solution obtained by solving (16) pre A, respectively. We denote k ∈ N the number of iteration level.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

2018

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“…In the multiscale methods, a central research problem is constructing the coarse grid approximation for faster results, where multiscale basis functions are computed on a fine grid to capture the influence of fractures and other heterogeneity [9][10][11][12][13][14]. We consider seismic waves in fractured media and construct multiscale basis functions for coarse grid simulations in the two-dimensional formulation [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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.

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“…These approaches require a careful design of multiscale dominant modes. We remark that the applications of these methods to hyperbolic equations are challenging [4,9] due to distant temporal effects.…”

Section: Introductionmentioning

confidence: 99%

Temporal Splitting algorithms for non-stationary multiscale problems

Efendiev,

Pun,

Vabishchevich

2020

Preprint

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In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smallerdimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a "good" splitting of the solution space, one can achieve an efficient temporal splitting. We present a theoretical result, which was derived in our earlier paper and adopted in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.

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Computational multiscale methods for first-order wave equation using mixed CEM-GMsFEM

Cited by 14 publications

References 30 publications

Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

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

Temporal Splitting algorithms for non-stationary multiscale problems

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