An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media

AbstractOffline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.

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Coarse-Grid Multiscale Model Reduction Techniques for Flows in Heterogeneous Media and Applications

Lecture Notes in Computational Science and Engineering - Numerical Analysis of Multiscale Problems ◽

10.1007/978-3-642-22061-6_4 ◽

2011 ◽

pp. 97-125 ◽

Cited By ~ 10

Author(s):

Yalchin Efendiev ◽

Juan Galvis

Keyword(s):

Model Reduction ◽

Heterogeneous Media ◽

Multiscale Model ◽

Coarse Grid ◽

Reduction Techniques

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A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

Communications in Computational Physics ◽

10.4208/cicp.071211.070912a ◽

2013 ◽

Vol 14 (2) ◽

pp. 370-392 ◽

Cited By ~ 24

Author(s):

Eric T. Chung ◽

Wing Tat Leung

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Heterogeneous Media ◽

Numerical Solutions ◽

Solution Space ◽

Convection Diffusion ◽

Diffusion And Convection ◽

Convection Dominated ◽

Diffusion Problems

AbstractIn this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.

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Systematic development of upwind numerical fluxes for the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media with physical interfaces

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2020.113352 ◽

2020 ◽

Vol 372 ◽

pp. 113352 ◽

Cited By ~ 1

Author(s):

B. Tie ◽

A.-S. Mouronval

Keyword(s):

Wave Propagation ◽

Elastic Wave ◽

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Heterogeneous Media ◽

Elastic Wave Propagation ◽

Physical Interfaces ◽

Numerical Fluxes ◽

Systematic Development

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Numerical dispersion analysis of discontinuous Galerkin method with different basis functions for acoustic and elastic wave equations

Geophysics ◽

10.1190/geo2017-0485.1 ◽

2018 ◽

Vol 83 (3) ◽

pp. T87-T101 ◽

Cited By ~ 6

Author(s):

Weijuan Meng ◽

Li-Yun Fu

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Basis Function ◽

Wave Equations ◽

Basis Functions ◽

Numerical Dispersion ◽

Dispersion Property ◽

Lagrange Polynomial ◽

Equidistant Nodes

The discontinuous Galerkin method (DGM) has been applied to investigate seismic wave propagation recently. However, few studies have examined the dispersion property of DGM with different basis functions. Therefore, three common basis functions, Legendre polynomial, Lagrange polynomial with equidistant nodes, and Lagrange polynomial with Gauss-Lobatto-Legendre (GLL) nodes, are used for numerical approximation. The numerical dispersion and anisotropy numerical behavior of acoustic and elastic waves are compared, and the numerical errors of different order methods are analyzed. The result shows that the dispersion errors for all basis functions reduce generally with increasing interpolation orders, but with large differences in different directions. Specifically, the Legendre basis function and Lagrange basis function with GLL nodes have attractive advantages over the Lagrange polynomial with equidistant nodes for numerical computation. We verified the dispersion properties by theoretical and numerical analyses.

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