An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media

Direct numerical simulation (DNS) of fluid flow in porous media with many scales is often not feasible, and an effective or homogenized description is more desirable. To construct the homogenized equations, effective properties must be computed. Computation of effective properties for nonperiodic...

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Main Authors: Brown, Donald L., Efendiev, Yalchin, Hoang, Viet Ha
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2014
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Online Access:https://hdl.handle.net/10356/100579
http://hdl.handle.net/10220/18461
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spelling sg-ntu-dr.10356-1005792023-02-28T19:39:42Z An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media Brown, Donald L. Efendiev, Yalchin Hoang, Viet Ha School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Applied mathematics Direct numerical simulation (DNS) of fluid flow in porous media with many scales is often not feasible, and an effective or homogenized description is more desirable. To construct the homogenized equations, effective properties must be computed. Computation of effective properties for nonperiodic microstructures can be prohibitively expensive, as many local cell problems must be solved for different macroscopic points. In addition, the local problems may also be computationally expensive. When the microstructure varies slowly, we develop an efficient numerical method for two scales that achieves essentially the same accuracy as that for the full resolution solve of every local cell problem. In this method, we build a dense hierarchy of macroscopic grid points and a corresponding nested sequence of approximation spaces. Essentially, solutions computed in high accuracy approximation spaces at select points in the the hierarchy are used as corrections for the error of the lower accuracy approximation spaces at nearby macroscopic points. We give a brief overview of slowly varying media and formal Stokes homogenization in such domains. We present a general outline of the algorithm and list reasonable and easily verifiable assumptions on the PDEs, geometry, and approximation spaces. With these assumptions, we achieve the same accuracy as the full solve. To demonstrate the elements of the proof of the error estimate, we use a hierarchy of macro–grid points in [0, 1]2 and finite element (FE) approximation spaces in [0, 1]2. We apply this algorithm to Stokes equations in a slowly porous medium where the microstructure is obtained from a reference periodic domain by a known smooth map. Using the arbitrary Lagrange–Eulerian (ALE) formulation of the Stokes equations (cf. [G. P. Galdi and R. Rannacher, Fundamental Trends in Fluid-Structure Interaction, Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications 1, World Scientific, Singapore, 2010]), we obtain modified Stokes equations with varying coefficients in the periodic domain. We show that the algorithm can be utilized in this setting. Finally, we implement the algorithm on the modified Stokes equations, using a simple stretch deformation mapping, and compute the effective permeability. We show that our efficient computation is of the same order as the full solve. Published version 2014-01-13T06:47:32Z 2019-12-06T20:24:44Z 2014-01-13T06:47:32Z 2019-12-06T20:24:44Z 2013 2013 Journal Article Brown, D. L., Efendiev, Y., & Hoang, V. H. (2013). An Efficient Hierarchical Multiscale Finite Element Method for Stokes Equations in Slowly Varying Media. Multiscale Modeling & Simulation, 11(1), 30-58. https://hdl.handle.net/10356/100579 http://hdl.handle.net/10220/18461 10.1137/110858525 en Multiscale modeling & simulation © 2013 Society for Industrial and Applied Mathematics. This paper was published in Multiscale Modeling & Simulation and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/110858525]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic DRNTU::Science::Mathematics::Applied mathematics
spellingShingle DRNTU::Science::Mathematics::Applied mathematics
Brown, Donald L.
Efendiev, Yalchin
Hoang, Viet Ha
An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
description Direct numerical simulation (DNS) of fluid flow in porous media with many scales is often not feasible, and an effective or homogenized description is more desirable. To construct the homogenized equations, effective properties must be computed. Computation of effective properties for nonperiodic microstructures can be prohibitively expensive, as many local cell problems must be solved for different macroscopic points. In addition, the local problems may also be computationally expensive. When the microstructure varies slowly, we develop an efficient numerical method for two scales that achieves essentially the same accuracy as that for the full resolution solve of every local cell problem. In this method, we build a dense hierarchy of macroscopic grid points and a corresponding nested sequence of approximation spaces. Essentially, solutions computed in high accuracy approximation spaces at select points in the the hierarchy are used as corrections for the error of the lower accuracy approximation spaces at nearby macroscopic points. We give a brief overview of slowly varying media and formal Stokes homogenization in such domains. We present a general outline of the algorithm and list reasonable and easily verifiable assumptions on the PDEs, geometry, and approximation spaces. With these assumptions, we achieve the same accuracy as the full solve. To demonstrate the elements of the proof of the error estimate, we use a hierarchy of macro–grid points in [0, 1]2 and finite element (FE) approximation spaces in [0, 1]2. We apply this algorithm to Stokes equations in a slowly porous medium where the microstructure is obtained from a reference periodic domain by a known smooth map. Using the arbitrary Lagrange–Eulerian (ALE) formulation of the Stokes equations (cf. [G. P. Galdi and R. Rannacher, Fundamental Trends in Fluid-Structure Interaction, Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications 1, World Scientific, Singapore, 2010]), we obtain modified Stokes equations with varying coefficients in the periodic domain. We show that the algorithm can be utilized in this setting. Finally, we implement the algorithm on the modified Stokes equations, using a simple stretch deformation mapping, and compute the effective permeability. We show that our efficient computation is of the same order as the full solve.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Brown, Donald L.
Efendiev, Yalchin
Hoang, Viet Ha
format Article
author Brown, Donald L.
Efendiev, Yalchin
Hoang, Viet Ha
author_sort Brown, Donald L.
title An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
title_short An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
title_full An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
title_fullStr An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
title_full_unstemmed An efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
title_sort efficient hierarchical multiscale finite element method for stokes equations in slowly varying media
publishDate 2014
url https://hdl.handle.net/10356/100579
http://hdl.handle.net/10220/18461
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