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Multiscale Finite Element Methods for Flows on Rough Surfaces

Published online by Cambridge University Press:  03 June 2015

Yalchin Efendiev ,
Juan Galvis  and
M. Sebastian Pauletti
Yalchin Efendiev*
Affiliation:
Department of Mathematics, TAMU, College Station, TX 77843-3368, USA
Juan Galvis*
Affiliation:
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
M. Sebastian Pauletti*
Affiliation:
Department of Mathematics, TAMU, College Station, TX 77843-3368, USA
*
Email:efendiev@math.tamu.edu
Corresponding author.Email:jcgalvisa@unal.edu.co
Email:pauletti@math.tamu.edu
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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 nodal multiscale 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 for multiscale basis construction. The first deformation employs local information and the second deformation employs a global information. Our numerical results show that one can improve the accuracy of the simulations when a global information is used.

Keywords

65N3058E2076M5037E35Multiscale finite elements on surfacesLaplace Beltramiresonance errorharmonic maps
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 4 , October 2013 , pp. 979 - 1000
Copyright
Copyright © Global Science Press Limited 2013

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