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    Multiscale Finite Element Methods for Flows on Rough Surfaces

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    Type
    Article
    Authors
    Efendiev, Yalchin R. cc
    Galvis, Juan
    Pauletti, M. Sebastian
    KAUST Grant Number
    KUS-C1-016-04
    Date
    2015-06-03
    Online Publication Date
    2015-06-03
    Print Publication Date
    2013-10
    Permanent link to this record
    http://hdl.handle.net/10754/598915
    
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    Abstract
    In this paper, we present the Multiscale Finite Element Method (MsFEM) for problems on rough heterogeneous surfaces. We consider the diffusion equation on oscillatory surfaces. Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid. This problem arises in many applications where processes occur on surfaces or thin layers. We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface. The main ingredients of MsFEM are (1) the construction of multiscale basis functions and (2) a global coupling of these basis functions. For the construction of multiscale basis functions, our approach uses the transformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solved along the edges of coarse-grid blocks to calculate nodalmultiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration. We discuss the use of appropriate transformation operators that improve the accuracy of the method. The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition. In this paper, we consider such transformations based on harmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and Applied Math., LX(2007), pp. 675-723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when two types of deformations are used formultiscale basis construction. The first deformation employs local information and the second deformation employs a global information. Our numerical results showthat one can improve the accuracy of the simulations when a global information is used. © 2013 Global-Science Press.
    Citation
    Efendiev Y, Galvis J, Pauletti MS (2013) Multiscale Finite Element Methods for Flows on Rough Surfaces. CiCP. Available: http://dx.doi.org/10.4208/cicp.170512.310113a.
    Sponsors
    YE's work is partially supported by the US Army 62151-MA, DOE and NSF (DMS 0934837, DMS 0724704, and DMS 0811180). This publication is based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
    Publisher
    Global Science Press
    Journal
    Communications in Computational Physics
    DOI
    10.4208/cicp.170512.310113a
    ae974a485f413a2113503eed53cd6c53
    10.4208/cicp.170512.310113a
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