An Efficient Hierarchical Multiscale Finite Element Method for Stokes Equations in Slowly Varying Media
KAUST Grant NumberKUS-C1-016-04
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AbstractDirect 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. © 2013 Society for Industrial and Applied Mathematics.
CitationBrown DL, Efendiev Y, Hoang VH (2013) An Efficient Hierarchical Multiscale Finite Element Method for Stokes Equations in Slowly Varying Media. Multiscale Model Simul 11: 30–58. Available: http://dx.doi.org/10.1137/110858525.
SponsorsReceived by the editors December 12, 2011; accepted for publication (in revised form) September 4, 2012; published electronically January 10, 2013. This publication is based in part on work supported by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), a National Science Foundation East Asia Pacific Summer Institute (EAPSI) Award, and a National Science Foundation Integrated Graduate Education and Research Traineeship (IGERT) Award.Department of Mathematics, Texas A&M University, College Station, TX 77843 (firstname.lastname@example.org). This author's work was partially supported by the US Army (62151-MA), the DOE, and the NSF (DMS 0934837, DMS 0724704, and DMS 0811180).
JournalMultiscale Modeling & Simulation